hep-th/0311107 KUL-TF-03/16

Polyvector Super-Poincaré Algebras

Dmitri V. Alekseevsky

Dept. of Mathematics, University of Hull, Cottingham Road, Hull, HU6 7RX, UK D.V.Alekseevsky@maths.hull.ac.uk

Vicente Cortés

Institut Élie Cartan de Mathématiques, Université Henri Poincaré - Nancy 1, B.P. 239,

F-54506 Vandoeuvre-lès-Nancy Cedex, France cortes@iecn.u-nancy.fr

Chandrashekar Devchand

Mathematisches Institut der Universität Bonn, Beringstraße 1, D-53115 Bonn, Germany devchand@math.uni-bonn.de

Antoine Van Proeyen

Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D,

B-30001 Leuven, Belgium Antoine.VanProeyen@fys.kuleuven.ac.be

Abstract

A class of ℤ_{2}-graded Lie algebra and Lie superalgebra extensions of the pseudoorthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form 𝔤 = 𝔤_{0} + 𝔤_{1}, with 𝔤_{0} = 𝔰𝔬(V) + W_{0} and 𝔤_{1} = W_{1}, where the algebra of generalized translations W = W_{0} + W_{1} is the maximal solvable ideal of 𝔤, W_{0} is generated by W_{1} and commutes with W. Choosing W_{1} to be a spinorial 𝔰𝔬(V)-module (a sum of an arbitrary number of spinors and semispinors), we prove that W_{0} consists of polyvectors, i.e. all the irreducible 𝔰𝔬(V)-submodules of W_{0} are submodules of ∧V. We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of 𝔰𝔬(V)-invariant ∧^{k}V -valued bilinear forms on the spinor module S.

Contents

1 Introduction 1 2 ϵ-extensions of 𝔰𝔬(V ) 5 3 Extensions of translational type and ϵ-transalgebras 6 4 Extended polyvector Poincaré algebras 9 5 Decomposition of S ⊗ S: complex case 12 6 Decomposition of S ⊗ S: real case 16

6.1 Odd dimensional case: dim V = 2m + 1 . . . . . . . . . . . . . . . . . . . . . . . 18 6.2 Even dimensional case: dim V = 2m . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.3 Decomposition of tensor square of semi-spinors . . . . . . . . . . . . . . . . . . . 20

7 N-extended polyvector Poincaré algebras 22 A Admissible ∧^{k}V -valued bilinear forms on S 24 B Reformulation for physicists 32

B.1 Complex and real Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 32 B.2 Summary of the results for the algebras . . . . . . . . . . . . . . . . . . . . . . . 35

1 Introduction

A superextension of a Lie algebra 𝔥 is a Lie superalgebra 𝔤 = 𝔤_{0} + 𝔤_{1} , such that 𝔥 ⊂ 𝔤_{0}. If the Lie algebra 𝔤_{0} ⊃ 𝔥 and a 𝔤_{0}-module 𝔤_{1} are given, then a superextension is determined by the Lie superbracket in the odd part, which is a 𝔤_{0}-equivariant linear map

∨^{2}𝔤_{1} → 𝔤_{0} ,  (1.1)

satisfying the Jacobi identity for X, Y, Z ∈ 𝔤_{1}, where ∨ denotes the symmetric tensor product. Similarly, a ℤ_{2}-graded extension (or simply Lie extension) of 𝔥 is a ℤ_{2}-graded Lie algebra 𝔤, i.e. a Lie algebra with a ℤ_{2}-grading 𝔤 = 𝔤_{0} + 𝔤_{1} compatible with the Lie bracket: [𝔤_{α}, 𝔤_{β}] ⊂ 𝔤_{α+β} , α, β ∈ ℤ/2ℤ, such that 𝔤_{0} ⊃ 𝔥. As above, a ℤ_{2}-graded extension is determined by the Lie bracket in 𝔤_{1}, which defines a 𝔤_{0}-equivariant linear map,

∧^{2}𝔤_{1} → 𝔤_{0} ,  (1.2)

satisfying the Jacobi identity. For instance, consider a super vector space V_{0} +V_{1} endowed with a scalar superproduct g = g_{0} + g_{1}, i.e. g_{0} is a (possibly indefinite) scalar product on V_{0} and g_{1} is a nondegenerate skewsymmetric bilinear form on V_{1}. The Lie algebra 𝔥 = 𝔤_{0} = 𝔰𝔬(V_{0}) ⊕ 𝔰𝔭(V_{1}) of infinitesimal even automorphisms of (V_{0} + V_{1}, g) has a natural extension with 𝔤_{1} = V_{0} ⊗ V_{1}, where the Lie superbracket is given by:

[v_{0} ⊗ v_{1}, v′_{0} ⊗ v′_{1}] := g_{1}(v_{1}, v′_{1})v_{0} ∧ v′_{0} + g_{0}(v_{0}, v′_{0})v_{1} ∨ v′_{1} .

This is the orthosymplectic Lie superalgebra 𝔬𝔰𝔭(V_{0}|V_{1}). One can also define an analogous Lie superalgebra 𝔰𝔭𝔬(V_{0}|V_{1}), starting from a symplectic super vector space (V_{0} + V_{1}, ω = ω_{0} + ω_{1}), such that 𝔰𝔭𝔬(V_{0}|V_{1}) = 𝔬𝔰𝔭(V_{1}|V_{0}).

Similarly, for a ℤ_{2}-graded vector space V_{0}+V_{1} endowed with a scalar product g = g_{0}+g_{1} (respectively, a symplectic form ω = ω_{0} + ω_{1}) we have a natural ℤ_{2}-graded extension 𝔤 = 𝔤_{0} + 𝔤_{1} = 𝔰𝔬(V_{0} + V_{1}) (respectively, 𝔤 = 𝔰𝔭(V_{0} + V_{1})) of the Lie algebra 𝔥 = 𝔤_{0} = 𝔰𝔬(V_{0}) ⊕ 𝔰𝔬(V_{1}) (respectively, of 𝔥 = 𝔰𝔭(V_{0}) ⊕ 𝔰𝔭(V_{1})).

For a pseudo-Euclidean space-time V = ℝ^{p,q} (with p positive and q negative directions), Nahm [N] classified superextensions 𝔤 of the pseudo-orthogonal Lie algebra 𝔰𝔬(V) under the assumptions that q ≤ 2, 𝔤 is simple, 𝔤_{0} is a direct sum of ideals, 𝔤_{0} = 𝔰𝔬(V)⊕𝔨, where 𝔨 is reductive and 𝔤_{1} is a spinorial module (i.e. its irreducible summands are spinors or semi-spinors). These algebras for q = 2 are usually considered as superconformal algebras for Minkowski spacetimes, in virtue of the identification 𝔠𝔬𝔫𝔣(p − 1, 1) = 𝔰𝔬(p, 2).

In this paper, we shall consider both super and Lie extensions (which we call ϵextensions) of the pseudo-orthogonal Lie algebra 𝔰𝔬(V), with ϵ= + 1 corresponding to superextensions and ϵ= − 1 to Lie extensions. Here V = ℝ^{p,q} or V = ℂ^{n} is a vector space endowed with a scalar product. In the case 𝔤_{0} = 𝔰𝔬(V) + V (Poincaré Lie algebra), ϵ-extensions 𝔤 = 𝔤_{0} + 𝔤_{1} such that 𝔤_{1} is a spinorial module and [𝔤_{1}, 𝔤_{1}] ⊂ V were classified in [AC]. In the case ϵ = −1 such extensions clearly do not respect the conventional field theoretical spin–statistics relationship. However, in order to classify super-Poincaré algebras (ϵ = +1) with an arbitrary number of irreducible spinorial submodules in 𝔤_{1} we need to classify Lie extensions as well as superextensions with irreducible 𝔤_{1}.

We study ℤ_{2}-graded Lie algebras and Lie superalgebras, 𝔤 = 𝔤_{0} + 𝔤_{1} where 𝔤_{0} = 𝔰𝔬(V) + W_{0} , 𝔤_{1} = W_{1}, such that 𝔰𝔬(V) is a maximal semisimple Lie subalgebra of 𝔤 and W = W_{0} + W_{1} is its maximal solvable ideal. If W_{0} contains [W_{1}, W_{1}] and commutes with W, we call 𝔤 an ϵ-extension of translational type. If moreover, W_{0} = [W_{1}, W_{1}], we call 𝔤 an ϵ-transalgebra. Our main result is the classification of ϵ-extended polyvector Poincaré algebras, i.e. ϵ-extensions of translational type in the case when W_{1} = S, the spinor 𝔰𝔬(V)-module, or, more generally, an arbitrary spinorial module. Here V is an arbitrary pseudo-Euclidean vector space ℝ^{p,q}. We prove that, under these assumptions, any irreducible 𝔰𝔬(V)-submodule of W_{0} is of the form ∧^{k}V or ∧_{±}^{m}V, where m = (p + q)/2 and ∧_{±}^{m}V are the eigenspaces of the Hodge star operator on ∧^{m}V.

If 𝔤 = 𝔰𝔬(V) + W_{0} + S is an ϵ-transalgebra, then the (super) Lie bracket defines an 𝔰𝔬(V)-equivariant surjective map Γ_{W_{0}} : S ⊗ S → W_{0}. If K is the kernel of this map, then there exists a complementary submodule W̃_{0} such that S ⊗ S = W̃_{0} + K and we can identify W̃_{0} with W_{0}. We note that we can choose W̃_{0} ⊂ S ∧ S in the Lie algebra case and W̃_{0} ⊂ S ∨ S in the Lie superalgebra case. Conversely, if we have a decomposition S ⊗ S = W_{0} + K into a sum of two 𝔰𝔬(V)-submodules and moreover W_{0} ⊂ S ∧ S or W_{0} ⊂ S ∨ S, then the projection Γ_{W_{0}} onto W_{0} with the kernel K defines an 𝔰𝔬(V)- equivariant bracket

\begin{aligned}[ , ] : & S ⊗ S → W_{0} \\ & [s, t] = Γ_{W_{0}}(s ⊗ t) \end{aligned}  (1.3)

which is skewsymmetric or symmetric, respectively. More generally, if A is an endomorphism of W_{0} that commutes with 𝔰𝔬(V), then the twisted projection A ∘ Γ_{W_{0}} is another 𝔰𝔬(V)-equivariant bracket and any bracket can be obtained in this way. Together with the action of 𝔰𝔬(V) on W_{0} and S, this defines the structure of an ϵ-transalgebra 𝔤 = 𝔰𝔬(V)+W_{0}+S, since the Jacobi identity for X, Y, Z ∈ 𝔤_{1} follows from [𝔤_{1}, [𝔤_{1}, 𝔤_{1}]] = 0. The classification problem then reduces essentially to the decomposition of S∧S, S∨S into irreducible 𝔰𝔬(V)-submodules and the description of the projection Γ_{W_{0}}. In this paper, we resolve both these matters. In all cases the irreducible 𝔰𝔬(V)-submodules occurring in the tensor product S ⊗ S are k-forms ∧^{k}V, with the exception of the case of even dimensions n = p+q = 2m with signature s = p−q divisible by 4. In the latter case the m-form module splits into irreducible selfdual and anti-selfdual submodules ∧_{±}^{m}V. The multiplicities of any irreducible 𝔰𝔬(V)-submodules of S ⊗ S take values 1,2,4 or 8. For example if V = ℂ^{n} , n = 2m + 1 or if V = ℝ^{m,m+1}, we have (c.f. [OV])

S ⊗ S = ∑_{k=0}^{m} ∧^{k}V ,

S ∨ S = ∑_{k=0}^{[m/4]} ∧^{m−4k}V + ∑_{k=0}^{[(m−3)/4]} ∧^{m−3−4k}V ,

S ∧ S = ∑_{k=0}^{[(m−2)/4]} ∧^{m−2−4k}V + ∑_{k=0}^{[(m−1)/4]} ∧^{m−1−4k}V .

The vector space of ϵ=−1-extensions of translational type of the form 𝔤 = 𝔰𝔬(V) + ∧^{k}V + S is identified with the vector space Bil_{−}^{k}(S)^{𝔰𝔬(V)} := Hom_{𝔰𝔬(V)}(S ∧ S , ∧^{k}V) of ∧^{k}V -valued invariant skewsymmetric bilinear forms on S. Similarly, the vector space of ϵ=+1-extensions of translational type of the form 𝔤 = 𝔰𝔬(V) + ∧^{k}V + S is identified with the vector space Bil_{+}^{k}(S)^{𝔰𝔬(V)} :=Hom_{𝔰𝔬(V)}(S ∨ S , ∧^{k}V).

The main problem is the description of these spaces of invariant ∧^{k}V -valued bilinear forms. For k = 0, 1 this problem was solved in [AC], where three invariants, σ, τ and ι, were defined for bilinear forms on the spinor module.

Following [AC], a nondegenerate 𝔰𝔬(V)-invariant (scalar) bilinear form β on the spinor module S is called admissible if it has the following properties:

1) β is either symmetric or skewsymmetric, β(s, t) = σ(β)β(t, s) , s, t ∈ S , σ(β) = ±1. We define σ(β) to be the symmetry of β.

2) Clifford multiplication by v ∈ V,

γ(v) : S → S , s ↦ γ(v)s = v · s ,

is either β-symmetric or β-skewsymmetric, i.e.

β(vs, t) = τ(β)β(s, vt) , s, t ∈ S ,

with τ(β) = +1 or −1, respectively. We define τ(β) to be the type of β.

3) If the spinor module is reducible, S = S^{+} + S^{−}, then the semispinor modules S^{+} and S^{−} are either mutually orthogonal or isotropic. We define the isotropy of β to be ι(β) = +1 if β(S_{+}, S_{−}) = 0 or ι(β) = −1 if β(S_{±}, S_{±}) = 0.

In [AC], a basis β_{i} of the space Bil(S)^{𝔰𝔬(V)} := Bil^{0}(S)^{𝔰𝔬(V)} of scalar-valued invariant forms was constructed explicitly, which consists of admissible forms. These are tabulated in the appendix (Table A3). The dimension N(s) = dim Bil(S)^{𝔰𝔬(V)} depends only on the signature s = p − q of V (see Table A1 in the Appendix). We associate with a bilinear form β on S the ∧^{k}V -valued bilinear form Γ_{β}^{k} : S ⊗ S → ∧^{k}V, defined by the following fundamental formula

⟨Γ_{β}^{k}(s ⊗ t) , v_{1} ∧ ⋯ ∧ v_{k}⟩ = ∑_{π∈𝔖_{k}} sgn(π)β (γ(v_{π(1)}) ⋯ γ(v_{π(k)})s , t) s, t ∈ S , v_{i} ∈ V ,

which extends the formula given in [AC] from k = 1 to arbitrary k. For k = 0 we have that Γ_{β}^{0} = β.

We shall prove that the map β ↦ Γ_{β}^{k} is 𝔰𝔬(V)-equivariant and induces an isomorphism

Γ^{k} : Bil(S)^{𝔰𝔬(V)} →^{∼} Bil^{k}(S)^{𝔰𝔬(V)}

onto the vector space of ∧^{k}V -valued invariant bilinear forms on S. This was proven for k = 1 in [AC].

The definitions of the invariants σ, τ, ι make sense for ∧^{k}V -valued bilinear forms as well. If σ(Γ_{β}^{k}) = −1, the form Γ_{β}^{k} is skewsymmetric and hence defines a Lie algebra structure on 𝔤 = 𝔰𝔬(V) + ∧^{k}V + S. If σ(Γ_{β}^{k}) = +1, it defines a Lie superalgebra structure on 𝔤 = 𝔰𝔬(V) + ∧^{k}V + S. We shall prove, for admissible β, that

σ(Γ_{β}^{k}) = σ(β)τ(β)^{k}(−1)^{k(k−1)/2} .  (1.4)

In the cases when semi-spinors exist, we shall prove that

ι(Γ_{β}^{k}) = ι(β)(−1)^{k} .  (1.5)

For k > 0 the bilinear forms Γ_{β}^{k} associated with an admissible bilinear form β have neither value of the type τ. Clearly, the formulae for the invariants show that σ(Γ_{β}^{k}) and ι(Γ_{β}^{k}) depend only on k modulo 4. We tabulate these invariants for Γ_{β_{i}}^{k} for k = 0, 1, 2, 3 in the Appendix.

Let the number N_{k}^{ϵ}(s, n) denote the dimension of the vector space Bil_{ϵ}^{k}(S)^{𝔰𝔬(V)} of ϵextended k-polyvector Poincaré algebra structures on 𝔤 = 𝔰𝔬(V) + ∧^{k}V + S . We shall see that the sum

N_{k}(s, n) = N_{k}^{+}(s, n) + N_{k}^{−}(s, n) = N(s) = dim Bil(S)^{𝔰𝔬(V)}

depends only on the signature s. We shall also verify the following remarkable shift formula

N_{k}^{±}(s, n + 2k) = N_{0}^{±}(s, n) ,  (1.6)

which reduces the calculation of these numbers to the case of zero forms. The function N^{±}(s, n) := N_{0}^{±}(s, n) has the following symmetries:

a) Periodicity modulo 8 in s and n:

N^{±}(s + 8a, n + 8b) = N^{±}(s, n) , a, b ∈ ℤ .

Using this, we can extend the functions N^{±}(s, n) to all integer values of s and n.

b) Symmetry with respect to reflection in signature 3:

N^{±}(−s + 6, n) = N^{±}(s, n) .

c) The mirror symmetries:

N^{±}(s, n + 4) = N^{∓}(s, n) ,  (1.7)
N^{±}(s, −n + 4) = N^{∓}(s, n) .  (1.8)

Due to the shift formula (1.6), all these identities yield corresponding identities for N_{k}^{±}(s, n) for any k. For example the mirror identity (1.8) gives the mirror symmetry for k=1 (reflection with respect to zero dimension),

N_{1}^{±}(s, −n) = N_{1}^{∓}(s, n) ,

which was discovered in [AC].

In Appendix B, we summarise our results in language more familiar to the physics community.

Recently, there have been many discussions (e.g. [AI, CAIP, DFLV, DN, FV, Sc, Sh, V, VV]) of generalizations of spacetime supersymmetry algebras which go beyond Nahm’s classification. Of particular interest, has been the M-theory algebra, which extends the d=11 super Poincaré algebra by two-form and five-form brane charges. In the important paper [DFLV], the authors study superconformal Lie algebras and polyvector super-Poincaré algebras 𝔤 = 𝔰𝔬(V) + ∧^{k}V + W_{1}, where W_{1} = S or W_{1} = S_{±}. They propose an approach for the classification of such Lie superalgebras 𝔤 which consists essentially of the following two steps: first describe the space Hom_{𝔰𝔬(V^{ℂ})}(𝕊 ∨ 𝕊 , ∧^{k}V^{ℂ}), if the complex spinor module 𝕊 is irreducible, and the spaces Hom_{𝔰𝔬(V^{ℂ})}(𝕊_{±} ∨ 𝕊_{±} , ∧^{k}V^{ℂ}) and Hom_{𝔰𝔬(V^{ℂ})}(𝕊_{+} ⊗ 𝕊_{−} , ∧^{k}V^{ℂ}) if the complex spinor module 𝕊 = 𝕊_{+} + 𝕊_{−} is reducible, then describe 𝔰𝔬(V)-invariant reality conditions. They determine the dimension of the above vector spaces, which is always zero or one and discuss the second problem. In the present paper we start from the real spinor module S and, in particular, describe explicitly the real vector space H = Hom_{𝔰𝔬(V)}(W_{1} ∨ W_{1} , ∧^{k}V) for an arbitrary spinorial module W_{1}. We shall see that even if W_{1} is an irreducible spinor module S, the dimension of H can be 0, 1, 2 or 3. Polyvector super-Poincaré algebras were also considered in [CAIP] for Lorentzian signature (1, q) in the dual language of left-invariant one-forms on the supergroup of supertranslations.

2 ϵ-extensions of 𝔰𝔬(V )

Let V be a real or complex vector space endowed with a scalar product and W_{1} an 𝔰𝔬(V)-module.

Definition 1 A superextension (ϵ = +1-extension) of 𝔰𝔬(V ) of type W_{1} is a Lie superalgebra 𝔤 satisfying the conditions

i) 𝔰𝔬(V) ⊂ 𝔤_{0} as a subalgebra ii) 𝔤_{1} = W_{1}, a 𝔤_{0}-module. A Lie extension (ϵ = −1-extension) of 𝔰𝔬(V ) of type W_{1} is a ℤ_{2}-graded Lie algebra 𝔤 = 𝔤_{0} + 𝔤_{1}, also satisfying i) and ii). Further, an ϵ-extension is called minimal if it does not contain a proper subalgebra which is also an ϵ-extension of type W_{1}; more precisely, if 𝔤′ = 𝔤′_{0} + 𝔤_{1} ⊂ 𝔤, 𝔰𝔬(V) ⊂ 𝔤′_{0}, then 𝔤′ = 𝔤.

The Lie superalgebras classified by Nahm are examples of superextensions of 𝔰𝔬(ℝ^{p,q}) of spinor type W_{1} = S.

Let 𝔤 = 𝔤_{0} + 𝔤_{1}, be an ϵ-extension of 𝔰𝔬(V), 𝔤_{0} = 𝔰𝔬(V) + W_{0}, with W_{0} an 𝔰𝔬(V)- submodule that is complementary to 𝔰𝔬(V) in 𝔤_{0} and 𝔤_{1} = W_{1}. There are two extremal classes of such algebras:

E1: 𝔤 is semi-simple, i.e. does not contain any proper solvable ideal,

E2: 𝔤 is of semi-direct type, i.e. 𝔤 is maximally non semi-simple, in the sense that 𝔰𝔬(V) is its largest semi-simple super Lie subalgebra, 𝔤 = 𝔰𝔬(V)+W_{0}+W_{1} and W = W_{0}+W_{1} is a solvable ideal.

3 Extensions of translational type and ϵ-transalgebras

Definition 2 Let 𝔤 = 𝔰𝔬(V) + W_{0} + W_{1} be an ϵ-extension of 𝔰𝔬(V). If [W_{0}, W] = 0 and [W_{1}, W_{1}] ⊂ W_{0} then the extension 𝔤 = 𝔰𝔬(V) + W_{0} + W_{1} is called an ϵ-extension of 𝔰𝔬(V ) of translational type and the (nilpotent) ideal W = W_{0} + W_{1} is called the algebra of generalized translations. If it is minimal, in the sense of Definition 1, then it is called an ϵ-transalgebra.

We note that such an extension is automatically of semi-direct type, provided that dim V ≥ 3, which we assume in this section. We also assume for definiteness that V is a real vector space. The minimality condition is equivalent to [W_{1}, W_{1}] = W_{0} and means that even translations are generated by odd translations. The construction of ϵextensions of 𝔰𝔬(V) of translational type with given 𝔰𝔬(V)-modules W_{0} and W_{1} reduces to the construction of 𝔰𝔬(V)-equivariant linear maps ∨^{2}W_{1} → W_{0} and ∧^{2}W_{1} → W_{0}. The Jacobi identity for the Lie bracket associated to such a map follows from the 𝔰𝔬(V)- equivariance. Now we show that the description of ϵ-extensions of 𝔰𝔬(V) of translational type reduces to that of minimal ones (i.e. transalgebras). Let 𝔤 = 𝔰𝔬(V) + W_{0} + W_{1} be an ϵ-extension of 𝔰𝔬(V) of translational type. Then 𝔤′ := 𝔰𝔬(V) + [W_{1}, W_{1}] + W_{1} is an ϵ-transalgebra and 𝔤 = 𝔤′ + 𝔞 is the semi-direct sum of the subalgebra 𝔤′ and an (even) Abelian ideal 𝔞, where 𝔞 ⊂ W_{0} is an 𝔰𝔬(V)-submodule complementary to [W_{1}, W_{1}] ⊂ W_{0}. Conversely, if 𝔤′ = 𝔰𝔬(V) + W′_{0} + W_{1} is an ϵ-transalgebra and 𝔞 is an 𝔰𝔬(V)-module then the semi-direct sum 𝔤 := 𝔤′ + 𝔞 is an ϵ-extension of 𝔰𝔬(V) of translational type, where W_{0} := W′_{0} + 𝔞.

Proposition 1 Let W_{1} be an 𝔰𝔬(V)-module. Then there exists a unique (up to isomorphism) ϵ-transalgebra of maximal dimension with 𝔤_{1} = W_{1}:

𝔤^{ϵ} = 𝔤^{ϵ}(W_{1}) = 𝔤_{0}^{ϵ} + 𝔤_{1}^{ϵ} = (𝔰𝔬(V) + W_{0}^{ϵ}) + W_{1} ,

where W_{0}^{+} = ∨^{2}W_{1} and W_{0}^{−} = ∧^{2}W_{1}. The Lie (super)bracket [·, ·] : W_{1} ⊗ W_{1} → W_{0}^{ϵ} is the projection onto the corresponding summand of W_{1} ⊗ W_{1} = ∨^{2}W_{1} ⊕ ∧^{2}W_{1}. Moreover, any ϵ-transalgebra with 𝔤_{1} = W_{1} is isomorphic to a contraction of 𝔤^{ϵ}(W_{1}).

Proof: It is clear that 𝔤^{ϵ} is a maximal ϵ-transalgebra. Let 𝔤 = 𝔤_{0} + 𝔤_{1} be a maximal ϵ-transalgebra with 𝔤_{1} = W_{1} and 𝔤_{0} = 𝔰𝔬(V) + W_{0}. The Lie (super)bracket [·, ·] : W_{1} ⊗ W_{1} → W_{0} defines an 𝔰𝔬(V)-equivariant isomorphism from ∨^{2}W_{1} or ∧^{2}W_{1} onto W_{0}. This isomorphism extends to an isomorphism 𝔤^{ϵ} → 𝔤, which is the identity on 𝔰𝔬(V) + W_{1}. Similarly for any ϵ-transalgebra with 𝔤_{1} = W_{1} the (super) Lie bracket [·, ·] : W_{1}⊗W_{1} → W_{0} defines an 𝔰𝔬(V)-equivariant epimorphism φ from ∨^{2}W_{1} or ∧^{2}W_{1} onto W_{0}. The kernel K is an 𝔰𝔬(V)-submodule of ∨^{2}W_{1} or ∧^{2}W_{1}, respectively. Since 𝔰𝔬(V) is semi-simple, there exists a complementary submodule W̃_{0} isomorphic to W_{0}. We can identify the 𝔰𝔬(V)- module W̃_{0} with W_{0} by means of the isomorphism φ|W̃_{0}. Then the Lie bracket corresponds

to the projection π+ ; W̃_{0} : ∨^{2}W_{1} = K + W̃_{0} → W̃_{0} or π− ; W̃_{0} : ∧^{2}W_{1} = K + W̃_{0} → W̃_{0}. This

defines an ϵ-transalgebra 𝔤^{ϵ}(W_{1}, W̃_{0}) = 𝔰𝔬(V) + W̃_{0} + W_{1}, whose bracket is the above projection πϵ ; W̃_{0}. The isomorphism φ|W̃_{0} : W̃_{0} → W_{0} of 𝔰𝔬(V)-modules extends trivially

to an isomorphism 𝔤^{ϵ}(W_{1}, W̃_{0}) → 𝔤. This shows that any ϵ-transalgebra is isomorphic to an ϵ-transalgebra of the form 𝔤^{ϵ}(W_{1}, W̃_{0}), where W̃_{0} ⊂ W_{1} ⊗ W_{1} is an 𝔰𝔬(V)-submodule contained in ∨^{2}W_{1} or ∧^{2}W_{1} , respectively. Consider now the one-parameter family of Lie brackets [·, ·]_{t} := t(Id−πϵ ; W̃_{0})+πϵ ; W̃_{0}. This defines a family of ϵ-transalgebras (𝔤^{ϵ}(W_{1}), [·, ·]_{t}). For t ≠ 0 they are isomorphic to the original (t = 1) ϵ-transalgebra. In the limit t → 0 we obtain the ϵ-transalgebra 𝔤^{ϵ}(W_{1}, W̃_{0}) as a contraction of 𝔤^{ϵ}(W_{1}).

The following proposition describes the structure of extensions of semi-direct type under the additional assumption that the 𝔰𝔬(V)-module W_{1} is irreducible. We denote by ρ : 𝔤_{0} → 𝔤𝔩(W_{1}) the adjoint representation of 𝔤_{0} = 𝔰𝔬(V)+W_{0} on W_{1} and by K the kernel of ρ|_{W_{0}}, which is clearly an ideal of 𝔤. Thus, ρ is the action of ad_{W_{0}} on W_{1} and K are all the generators of W_{0} that commute with W_{1}.

Proposition 2 Let 𝔤 = 𝔰𝔬(V) + W_{0} + W_{1} be an ϵ-extension of semi-direct type. Assume that the 𝔰𝔬(V)-module W_{1} is irreducible of dimension at least 3 if it does not admit an 𝔰𝔬(V)-invariant complex structure and dim_{ℂ} W_{1} ≥ 3 if it does and that dim V ≥ 3. Consider the decomposition of 𝔤 into a direct sum of 𝔰𝔬(V)-submodules, 𝔤 = 𝔰𝔬(V) + A + K + W_{1} , where A is an 𝔰𝔬(V)-invariant complement to K in W_{0} = A+K. Then the dimension dim A = 0, 1, 2 and the irreducible linear Lie algebra ρ(𝔤_{0}) = ρ(𝔰𝔬(V)) + Z, where the centre Z ≅ W_{0}/K is either 0, ℝ·Id or ℂ·Id. Moreover,

[A, A] ⊂ K , [𝔰𝔬(V), A] = 0 , [W_{1}, W_{1}] ⊂ K .

Proof: By assumption, the linear Lie algebra ρ(𝔤_{0}) ⊂ 𝔤𝔩(W_{1}) is irreducible and hence reductive. Since any solvable ideal of a reductive Lie algebra belongs to the centre, we conclude that the solvable ideal ρ(W_{0}) ⊂ ρ(𝔤_{0}) is in fact Abelian and consists of operators commuting with 𝔰𝔬(V). Now Schur’s Lemma implies that ρ(W_{0}) = 0, ℝ·Id, or ℂ·Id. The inclusion [A, A] ⊂ K follows from the fact that ρ(A) is in the centre of 𝔤_{0}. Since the restriction of ρ to 𝔰𝔬(V) + A is faithful and [ρ(𝔰𝔬(V)), ρ(A)] = 0, we conclude that [𝔰𝔬(V), A] = 0. From the assumptions it follows that there exist three vectors x, y, z ∈ W_{1}, which are linearly independent over the reals if W_{1} has no invariant complex structure and over the complex numbers if W_{1} has an invariant complex structure J. For any three linearly independent vectors (over ℝ or ℂ) x, y, z ∈ W_{1}, the Jacobi identity gives

\begin{aligned}0 & = & [[x, y], z] + [[y, z], x] + [[z, x], y] \\ & = & ρ([x, y])z + ρ([y, z])x + ρ([z, x])y . \end{aligned}

Since [W_{1}, W_{1}] ⊂ W_{0} and ρ(W_{0}) = ρ(A) consists of scalar operators (over ℝ or ℂ), we have that ρ([x, y]) = 0, i.e. [W_{1}, W_{1}] ⊂ K = ker ρ.

Note that 𝔤 is a transalgebra if and only if A=0.

The following corollary gives sufficient conditions for extensions of semi-direct type to be transalgebras.

Corollary 1 Under the assumptions of the previous proposition, assume moreover that 𝔤 = 𝔰𝔬(V)+W_{0}+W_{1} is a minimal extension of type W_{1}. Then 𝔤 is a transalgebra.

Proof: Minimality implies W_{0} = [W_{1}, W_{1}] and, by the above Proposition, [W_{1}, W_{1}] commutes with W_{1}. Now the Jacobi identity for x, y ∈ W_{1} and z ∈ W_{0} yields [W_{0}, W_{0}] = 0.

Instead of minimality we may assume the irreducibility of the 𝔰𝔬(V)-module W_{0}.

Proposition 3 Let 𝔤 = 𝔰𝔬(V) + W_{0} + W_{1} be an ϵ-extension of semi-direct type, with dim V ≥ 3. Assume that W_{0} and W_{1} are irreducible 𝔰𝔬(V)-modules. Then either 𝔤 is of translational type, i.e. [W_{0}, W] = 0, or W_{0} ≅ ℝ (considered as a real Lie algebra) is the centre of 𝔤_{0} = 𝔰𝔬(V) + W_{0} and ad _{W_{0}} acts on W_{1} by scalars.

Proof: Let W_{0}, W_{1} be irreducible 𝔰𝔬(V) modules. Since the algebra W = W_{0} + W_{1} is solvable, [W_{0}, W_{0}] is a proper 𝔰𝔬(V) submodule of W_{0}, hence [W_{0}, W_{0}] = 0. The kernel K of the adjoint representation ρ : W_{0} → 𝔤𝔩(W_{1}) is an 𝔰𝔬(V)-submodule of W_{0}. Hence K = W_{0} or 0. In the first case, 𝔤 is of translational type. In the second case, the representation ρ is faithful and ρ(W_{0}) commutes with ρ(𝔰𝔬(V)), hence [𝔰𝔬(V), W_{0}] = 0. On the other hand the 𝔰𝔬(V)-module W_{0} is irreducible, so W_{0} ≅ ℝ.

4 Extended polyvector Poincaré algebras and ∧^{k}V - valued invariant bilinear forms on the spinor module S

In this and the next two sections, we devote ourselves to the classification of ϵ-transalgebras 𝔤 = 𝔤_{0} + 𝔤_{1} with 𝔤_{1} = W_{1} = S, the spinor 𝔰𝔬(V)-module. We take V to be the pseudo-Euclidean space ℝ^{p,q} of dimension n = p+q and signature s = p−q. In other words, we consider (super) Lie algebras 𝔤 = (𝔰𝔬(V) + W_{0}) + S with

[W_{0} , W_{0} + S] = 0 , W_{0} = [S , S] .

The (super) Lie bracket defines an 𝔰𝔬(V)-equivariant surjective map Γ_{W_{0}} : S ⊗ S → W_{0}. If K is the kernel of this map, then S ⊗ S = W̃_{0} + K, where W̃_{0} is an 𝔰𝔬(V)-submodule equivalent to W_{0} such that W_{0} ⊂ S ∧ S in the Lie algebra case and W_{0} ⊂ S ∨ S in the superalgebra case. Conversely, if we have a decomposition S ⊗ S = W_{0} + K into a sum of two 𝔰𝔬(V)-submodules and moreover W_{0} ⊂ S ∧ S or W_{0} ⊂ S ∨ S, then the projection Γ_{W_{0}} onto W_{0} with the kernel K defines an 𝔰𝔬(V)-equivariant bracket

\begin{aligned}[ , ] : & S ⊗ S → W_{0} \\ & [s, t] = Γ_{W_{0}}(s ⊗ t) \end{aligned}  (4.1)

which is skewsymmetric or symmetric, respectively. More generally, if A is an endomorphism of W_{0} which commutes with 𝔰𝔬(V), then the twisted projection A∘Γ_{W_{0}} is another 𝔰𝔬(V)-equivariant bracket and any bracket can be obtained in this way. Together with the action of 𝔰𝔬(V) on W_{0} and S, this defines the structure of an ϵ-transalgebra 𝔤 = 𝔰𝔬(V) + W_{0} + S. We therefore have a 1–1 correspondence between ϵ-transalgebras of the form 𝔤 = 𝔰𝔬(V) + W_{0} + S, where W_{0} is a submodule of S ∨ S (for ϵ = 1) or S ∧ S (for ϵ = −1), and equivariant surjective maps Γ_{W_{0}} : S ⊗ S → W_{0}, whose kernel contains S ∨ S if ϵ = −1 and S ∧ S if ϵ = 1. The problem thus reduces to the description of the decomposition of S ∧ S and S ∨ S into irreducible 𝔰𝔬(V)-submodules and the determination of the twisted projections A∘Γ_{W_{0}}. We consider these projections as equivariant W_{0}-valued symmetric or skewsymmetric bilinear forms on S. In the next section we show that the irreducible submodules of S ⊗ S are of the form ∧^{k}V or ∧_{±}^{m}V ((anti)selfdual m-forms) if n = 2m and s is divisible by 4. We denote by

Bil^{k}(S) = Hom(S ⊗ S , ∧^{k}V) ,

the vector space of ∧^{k}V -valued bilinear forms on S. It can be decomposed, Bil^{k}(S) = Bil_{+}^{k}(S)⊕Bil_{−}^{k}(S), into the sum of the vector spaces of symmetric (+) and skewsymmetric (−) bilinear forms.

For W_{0} = ∧^{k}V, the space of ϵ-transalgebras (ϵ = ±) is identified with the space Bil_{ϵ}^{k}(S)^{𝔰𝔬(V)} of 𝔰𝔬(V)-invariant symmetric (ϵ = +) or skewsymmetric (ϵ = −) bilinear forms. Hence: The classification of ϵ-transalgebras 𝔤 = 𝔤_{0} + 𝔤_{1} with 𝔤_{1} = S reduces to the description of the spaces Bil_{ϵ}^{k}(S)^{𝔰𝔬(V)} of ∧^{k}V -valued invariant bilinear forms on the spinor module S.

The following formula associates a ∧^{k}V -valued bilinear form Γ_{β}^{k} ∈ Bil^{k}(S) to every (scalar) bilinear form β ∈ Bil(S).

⟨Γ_{β}^{k}(s ⊗ t) , v_{1} ∧ ⋯ ∧ v_{k}⟩ = ∑_{π∈𝔖_{k}} sgn(π)β (γ(v_{π(1)}) ⋯ γ(v_{π(k)})s , t) s, t ∈ S , v_{i} ∈ V ,

where the sum is over permutations π of {1, …, k}. Our classification is based on the following theorem.

Theorem 1 For any pseudo-Euclidean vector space V ≅ ℝ^{p,q}, the map

Γ^{k} : Bil(S) → Bil^{k}(S)
β ↦ Γ_{β}^{k}

is a Spin(V)-equivariant monomorphism and it induces an isomorphism

Γ^{k} : Bil(S)^{𝔰𝔬(V)} →^{∼} Bil^{k}(S)^{𝔰𝔬(V)}

of vector spaces.

Proof: It is known that Clifford multiplication γ : V → End S is Spin(V)–equivariant, i.e.

γ(gv) = gγ(v)g^{−1}, g ∈ Spin(V), v ∈ V .

Using this property we now check that the map Γ^{k} is also Spin(V)–equivariant:

Γ_{g·β}^{k} = g · Γ_{β}^{k} ,

where (g · β)(s, t) = β(g^{−1}s, g^{−1}t) and (g · Γ_{β}^{k})(s, t) = gΓ_{β}^{k}(g^{−1}s, g^{−1}t). We calculate

⟨Γ_{g·β}^{k}(s, t) , v_{1} ∧ ⋯ ∧ v_{k}⟩ = ∑_{π∈𝔖_{k}} sgn(π)β (g^{−1}γ(v_{π(1)}) ⋯ γ(v_{π(k)})s , g^{−1}t)
= ∑_{π∈𝔖_{k}} sgn(π)β (γ(g^{−1}v_{π(1)}) ⋯ γ(g^{−1}v_{π(k)})g^{−1}s , g^{−1}t)
= ⟨Γ_{β}^{k}(g^{−1}s, g^{−1}t) , g^{−1}v_{1} ∧ ⋯ ∧ g^{−1}v_{k}⟩
= ⟨gΓ_{β}^{k}(g^{−1}s, g^{−1}t) , v_{1} ∧ ⋯ ∧ v_{k}⟩
= ⟨(g · Γ_{β}^{k})(s, t) , v_{1} ∧ ⋯ ∧ v_{k}⟩ .  (4.2)

Next, we prove that Γ is injective. For β ∈ Bil(S) the bilinear form Γ_{β}^{k} is zero if and only if

β(∑_{π} sgn(π)γ(v_{π(1)}) ⋯ γ(v_{π(k)})S, S) = 0

or

∑_{π} sgn(π)γ(v_{π(1)}) ⋯ γ(v_{π(k)})S ⊂ ker(β)

for any vectors v_{1}, …, v_{k}. If the vectors v_{1}, …, v_{k} are orthogonal, then the endomorphisms γ(v_{1}), …, γ(v_{k}) anticommute and the endomorphism ∑_{π} sgn(π)γ(v_{π(1)}) ⋯ γ(v_{π(k)}) = k! γ(v_{1}) ⋯ γ(v_{k}) is invertible. This implies that ker(β) = S and so β = 0.

To complete the proof of the theorem, we need to check that

dim Bil^{k}(S)^{𝔰𝔬(V)} = dim Bil(S)^{𝔰𝔬(V)} =: N(p − q) .

In fact, dim Bil^{k}(S)^{𝔰𝔬(V)} = μ(k) dim 𝒞(∧^{k}V), where μ(k) is the multiplicity of ∧^{k}V in S ⊗ S and 𝒞(M) = End_{𝔰𝔬(V)}(M) denotes the Schur algebra of an 𝔰𝔬(V)-module M. If the signature s = p−q is divisible by 4 and k = m = n/2, then ∧^{m}V = ∧_{+}^{m}V ⊕∧_{−}^{m}V is the sum of two inequivalent irreducible 𝔰𝔬(V)-modules of real type and hence 𝒞(∧^{m}V) ≅ ℝ ⊕ ℝ. If the signature s is even but not divisible by 4 and k = m = n/2, then ∧^{m}V is an irreducible 𝔰𝔬(V)-module of complex type, with the complex structure defined by the Hodge star operator and hence 𝒞(∧^{m}V) ≅ ℂ. In both cases

dim Bil^{m}(S)^{𝔰𝔬(V)} = μ(m) dim 𝒞(∧^{m}V) = 2μ(m) = N(s) ,

where the last equation follows from Table A1 in the appendix. In all other cases, ∧^{k}V is an irreducible module of real type and 𝒞(∧^{k}V) = ℝ. Therefore, using Table A1, we obtain

dim Bil^{k}(S)^{𝔰𝔬(V)} = μ(k) dim 𝒞(∧^{k}V) = μ(k) = N(s) .

In the Introduction we defined the three ℤ/2ℤ-valued invariants for ∧^{k}V -valued bilinear forms on the spinor module: symmetry, type and isotropy. We say that a non-zero ∧^{k}V -valued bilinear form Γ ∈ Bil^{k}(S), k > 0, is admissible if it is either symmetric or skewsymmetric and, in the cases when semispinor modules exist, if the two semispinor modules are either isotropic or mutually orthogonal with respect to Γ. Recall that in the case of scalar-valued bilinear forms (k = 0), admissibility requires, in addition, that the bilinear form has a specific type τ. The invariants of admissible ∧^{k}V -valued bilinear forms in terms of the invariants of the scalar-valued admissible bilinear forms are given by:

Proposition 4 Let β ∈ Bil(S) be a an admissible scalar bilinear form and Γ_{β}^{k} the associated ∧^{k}V -valued bilinear form. Then Γ_{β}^{k} is admissible and its invariants, the symmetry σ(Γ_{β}^{k}) and the isotropy ι(Γ_{β}^{k}), can be calculated as follows

σ(Γ_{β}^{k}) = σ(β)τ(β)^{k}(−1)^{k(k−1)/2} ,  (4.3)
ι(Γ_{β}^{k}) = ι(β)(−1)^{k} .  (4.4)

For k > 0 the bilinear forms Γ_{β}^{k} ≠ 0 have neither type.

Proof: Let s, t ∈ S and e_{1}, …, e_{k} ∈ V be orthogonal vectors. We put γ_{i} := γ_{e_{i}} and compute

⟨Γ_{β}^{k}(s ⊗ t), e_{1} ∧ ⋯ ∧ e_{k}⟩ = k!β(γ_{1} ⋯ γ_{k}s, t)
= k!τ(β)^{k}β(s, γ_{k} ⋯ γ_{1}t)
= k!τ(β)^{k}(−1)^{k(k−1)/2}β(s, γ_{1} ⋯ γ_{k}t)
= k!τ(β)^{k}(−1)^{k(k−1)/2}σ(β)β(γ_{1} ⋯ γ_{k}t, s)
= k!τ(β)^{k}(−1)^{k(k−1)/2}σ(β)⟨Γ_{β}^{k}(t ⊗ s), e_{1} ∧ ⋯ ∧ e_{k}⟩ .

This proves equation (4.3). Equation (4.4) follows from the fact that Clifford multiplication γ_{v} maps S_{+} to S_{−} and vice versa.

5 Decomposition of the tensor square of the spinor module of Spin(V ) into irreducible components: complex case

In this section we consider the spinor module S of a complex Euclidean vector space V = ℂ^{n} and we derive the decompositions of S⊗S, S∨S and S∧S into inequivalent irreducible Spin(V)–submodules. These decompositions also yield the corresponding decompositions for the cases when S is a spinor module of a real vector space V = ℝ^{m,m} if n = 2m and V = ℝ^{m,m+1} if n = 2m+1. We shall use the well known facts summarised in the following lemma, see e.g. [OV].

Lemma 1 Let V be an n-dimensional complex Euclidean vector space or a real pseudo-Euclidean vector space of signature (p, q) , p + q=n , p − q=s. If n = 2m+1, then the decomposition of ∧V into irreducible pairwise inequivalent 𝔰𝔬(V)- submodules is given by

∧V = ∑_{k=0}^{n} ∧^{k}V = ∑_{k=0}^{m} ∧^{k}V + ∑_{k=0}^{m} ∗ ∧^{k} V = 2 ∑_{k=0}^{m} ∧^{k}V .  (5.1)

If n = 2m then we have the following decomposition into irreducible pairwise inequivalent 𝔰𝔬(V)-submodules

∧V = \begin{cases}2∑_{k=0}^{m−1} ∧^{k} V + ∧^{m}V & if s/2 is odd \\ 2∑_{k=0}^{m−1} ∧^{k} V + ∧_{+}^{m}V + ∧_{−}^{m}V & if s/2 is even or if V is complex. \end{cases}  (5.2)

Here ∧_{±}^{m}V are selfdual and anti-selfdual m-forms, the ±1-eigenspaces of the Hodge ∗- operator, which acts isometrically on ∧^{m}V, with ∗^{2} = (−1)^{m+q} = (−1)^{s/2} = +1 if s/2 is even .

In particular, the 𝔰𝔬(V)-module ∧^{k}V is irreducible, unless n = 2m, s/2 is even and k = m, in which case ∧^{m}V = ∧_{+}^{m}V + ∧_{−}^{m}V, where ∧_{+}^{m}V and ∧_{−}^{m}V are irreducible inequivalent modules.

Theorem 2

(i) The Spin(V)-module S ⊗ S contains all modules ∧^{k}V which are irreducible.

(ii) If V is a complex vector space of dimension n = 2m + 1 or if V is real of signature

(m, m + 1) then
S ⊗ S = ∑_{k=0}^{m} ∧^{k}V ,

S ∨ S = ∑_{i=0}^{[m/4]} ∧^{m−4i}V + ∑_{i=0}^{[(m−3)/4]} ∧^{m−3−4i}V ,

S ∧ S = ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V + ∑_{i=0}^{[(m−1)/4]} ∧^{m−1−4i}V .

Proof: (i) Theorem 1 associates a Spin(V)-equivariant linear map

(Γ_{β}^{k})^{∗} : ∧^{k}V ≅ ∧^{k}V^{∗} → S^{∗} ⊗ S^{∗} ≅ S ⊗ S

with any invariant bilinear form β on S. In particular, if ∧^{k}V is irreducible and β ≠ 0 then (Γ_{β}^{k})^{∗} embeds ∧^{k}V into S ⊗ S as a submodule. It was proven in [AC] that a non-zero invariant bilinear form β on S always exists. This shows that S ⊗ S ⊃ ∑_{k=0}^{m} ∧^{k}V. (ii) If n = 2m + 1 then the right hand side has dimension \frac{1}{2}2^{n} = 4^{m} and under the assumptions on V we have that dim S = 2^{m}. Hence dim S ⊗ S = 4^{m}, so the inclusion is an equality. The decompositions of S ∨ S and S ∧ S can either be read off the tables in [OV] or they follow from Proposition 4 using the invariants of the admissible scalar-valued form, which in this case is unique up to scale [AC] (see the tables in the appendix).

Now, we consider the case when V is complex of dimension n = 2m or real of signature (m, m). In this case, ∧^{m}V = ∧_{+}^{m}V ⊕ ∧_{−}^{m}V and the spinor module splits as a sum S = S_{+} + S_{−} of inequivalent irreducible semi-spinor modules S_{±} of dimension 2^{m−1}.

Theorem 3 Let V be complex of dimension n = 2m or real of signature (m, m). Then the decompositions of the Spin(V)-modules S_{+} ⊗ S_{−} and S_{±} ⊗ S_{±} into inequivalent irreducible submodules are given by:

S_{+} ⊗ S_{−} = ∑_{i=0}^{[(m−1)/2]} ∧^{m−1−2i}V ,  (5.3)

S_{±} ⊗ S_{±} = ∧_{±}^{m}V + ∑_{i=0}^{[(m−2)/2]} ∧^{m−2−2i}V ,  (5.4)
S ⊗ S = S_{+} ⊗ S_{+} + 2S_{+} ⊗ S_{−} + S_{−} ⊗ S_{−} = ∧V .  (5.5)

Further, for any admissible bilinear form β on S, the equivariant maps Γ_{β}^{k}|_{S_{±}⊗S_{±}} and Γ_{β}^{k}|_{S_{+}⊗S_{−}} have the following images:

Γ_{β}^{m}(S_{±} ⊗ S_{±}) = ∧_{±}^{m}V ,  (5.6)
Γ_{β}^{m±(2i+2)}(S_{±} ⊗ S_{±}) = ∧^{m±(2i+2)}V , 0 ≤ i ≤ [\frac{m−2}{2}] ,  (5.7)
Γ_{β}^{m±(2i+1)}(S_{+} ⊗ S_{−}) = ∧^{m±(2i+1)}V , 0 ≤ i ≤ [\frac{m−1}{2}] ,  (5.8)
Γ_{β}^{m±(2i+1)}(S_{±} ⊗ S_{±}) = 0 , 0 ≤ i ≤ [\frac{m−1}{2}] ,  (5.9)
Γ_{β}^{m±2i}(S_{+} ⊗ S_{−}) = 0 , 0 ≤ i ≤ [\frac{m}{2}] .  (5.10)

Proof: To prove the theorem we use the following model for the spinor module of an even dimensional complex Euclidean space V or of a pseudo-Euclidean space V with split signature (m, m): V = U ⊕ U^{∗}, where U is an m-dimensional vector space and the scalar product is defined by the natural pairing between U and the dual space U^{∗}. Then the spinor module is given by S = ∧U = ∧^{ev}U + ∧^{odd}U = S_{+} + S_{−}, where the semi-spinor modules S_{±} consist of even and odd forms. The Clifford multiplication is given by exterior and interior multiplication:

u · s := u ∧ s for u ∈ U, s ∈ S ,
u^{∗} · s := ι_{u}^{∗}s for u^{∗} ∈ U^{∗}, s ∈ S .

There exist exactly two independent admissible bilinear forms f and f_{E} = f(E· , ·) on the spinor module, where E|_{S_{±}} = ±Id, and the form f is given by

f(∧^{i}U , ∧^{j}U) = 0 , if i + j ≠ m ,
f(s, t) vol _{U} = (−1)^{i(i+1)/2} s ∧ t , s ∈ ∧^{i}U , t ∈ ∧^{m−i}U ,  (5.11)

where vol _{U} ∈ ∧^{m}U is a fixed volume form of U^{∗}. We note that the symmetry, type and isotropy of the admissible basis (f, f_{E}) of Bil(S)^{𝔰𝔬(V)} are given by

σ(f) = (−1)^{m(m+1)/2} , σ(f_{E}) = (−1)^{m(m−1)/2} ,
τ(f) = −1 , τ(f_{E}) = +1 , ι(f) = ι(f_{E}) = (−1)^{m} .

From this and Proposition 4 it follows that

σ(Γ_{f}^{k}) = (−1)^{(m(m+1)+k(k+1))/2} , σ(Γ_{f_{E}}^{k}) = (−1)^{(m(m−1)+k(k−1))/2} ,  (5.12)
ι(Γ_{f}^{k}) = ι(Γ_{f_{E}}^{k}) = (−1)^{m+k} .  (5.13)

The formulae (5.7)-(5.10) and the fact that Γ_{β}^{m}(S_{±} ⊗ S_{±}) ≠ 0 follow from the formulae for the isotropy of Γ_{f}^{k} and Γ_{f_{E}}^{k}.

To prove (5.6) we first show that for any admissible form β, the image Γ_{β}^{m}(S_{+} ⊗ S_{+}) contains ∧_{+}^{m}V and the image Γ_{β}^{m}(S_{−} ⊗ S_{−}) does not contain ∧_{+}^{m}V. For this we need to show that for any a ∈ ∧_{+}^{m}V there exist spinors s, t ∈ S_{+} = ∧^{ev}U such that the scalar product ⟨Γ_{β}^{m}(s ⊗ t), a⟩ ≠ 0 , and that there exists an element a ∈ ∧_{+}^{m}V such that ⟨Γ_{β}^{m}(s ⊗ t), a⟩ = 0 for any s, t ∈ S_{−} = ∧^{odd}U. Since ∧_{+}V is an irreducible 𝔰𝔬(V)-module, it follows that if a single element a of ∧_{+}^{m}V is contained in the 𝔰𝔬(V)-module Γ_{β}(S_{+} ⊗S_{−}), then all of ∧_{+}^{m}V is contained in it. Therefore, it will suffice to prove the first statement for just one choice of a.

We shall use the following lemma.

Lemma 2 Let V = U ⊕ U^{∗} as above. Then ∧^{m}U ⊂ ∧_{+}^{m}V.

Proof: Let (u_{1}, …, u_{m}) be a basis of U and (u_{1}^{∗}, …, u_{m}^{∗}) the dual basis of U^{∗}. Then, up to a sign factor, the volume form is given by vol = u_{1} ∧ ⋯ ∧ u_{m} ∧ u_{1}^{∗} ∧ ⋯ ∧ u_{m}^{∗}. Now, using the definition of the Hodge star operator, ⟨∗α, β⟩vol = α ∧ β, we may immediately check that ∗ (u_{1} ∧ ⋯ ∧ u_{m}) = u_{1} ∧ ⋯ ∧ u_{m}.

Let us consider a = vol_{U}. By the lemma, a ∈ ∧_{+}^{m}V. Then for s = t = 1 ∈ S_{+} we have

⟨Γ_{β}^{m}(s ⊗ t), a⟩ = β (a ∧ s , t) = β (a , 1) = ±1 ≠ 0 .

Similarly, for any s, t ∈ S_{−}

⟨Γ_{β}^{m}(s ⊗ t), a⟩ = β (a ∧ s , t) = 0 ,

since deg(a∧s) > m = dim U and hence a∧s = 0. This proves both the above statements and hence Γ_{β}^{m}(S_{+} ⊗ S_{+}) = ∧_{+}^{m}V. Since the image Γ_{β}^{m}(S_{−} ⊗ S_{−}) is nonzero and does not contain ∧_{+}^{m}V, we also have Γ_{β}^{m}(S_{−} ⊗ S_{−}) = ∧_{−}^{m}V. This proves (5.6).

We now prove (5.3). By (5.8), we have the inclusion ∑_{i=0}^{[(m−1)/2]} ∧^{m−1−2i}V ⊂ S_{+} ⊗ S_{−}. To prove equality we compare dimensions. Using the identity \binom{2m}{m−1−2i} = \binom{2m−1}{m−1−2i} + \binom{2m−1}{m−2−2i} , we calculate:

dim ( ∑_{i=0}^{[(m−1)/2]} ∧^{m−1−2i}V ) = ∑_{i=0}^{[(m−1)/2]} \binom{2m}{m − 1 − 2i} = ∑_{i=0}^{m−1} \binom{2m − 1}{i} (5.14)

= \frac{1}{2} ∑_{i=0}^{2m−1} \binom{2m − 1}{i} = 2^{2m−2} = dim(S_{+} ⊗ S_{−}) (5.15)

since dim S_{±} = 2^{m−1}. This proves (5.3).

Similarly, by (5.6) and (5.7), we have S_{±} ⊗ S_{±} ⊃ ∧_{±}^{m}V + ∑_{i=0}^{[(m−2)/2]} ∧^{m−2i−2}V. To prove (5.4) we compare dimensions:

dim (∧_{±}^{m}V + ∑_{i=0}^{[(m−2)/2]} ∧^{m−2i−2}V ) = ∑_{i=0}^{[m/2]} \binom{2m}{m − 2i} − \frac{1}{2}\binom{2m}{m}

= ∑_{i=0}^{[m/2]} (\binom{2m − 1}{m − 2i} + \binom{2m − 1}{m − 2i − 1}) − \frac{1}{2}\binom{2m}{m}

= ∑_{i=0}^{m} \binom{2m − 1}{m − i} − \frac{1}{2}\binom{2m}{m}

= \frac{1}{2} ∑_{i=0}^{2m−1} \binom{2m − 1}{i} = 2^{2m−2} = 2^{m−1} · 2^{m−1}
= dim(S_{±} ⊗ S_{±}) .

This proves (5.4) and (5.5).

Corollary 2

(i) Let V be either complex of even dimension or real of signature (m, m) and β an admissible bilinear form on the spinor module S = S_{+} + S_{−}. Then for all k the image of Γ_{β}^{k} restricted to S_{+} ⊗ S_{+}, S_{−} ⊗ S_{−} and S_{+} ⊗ S_{−} is an irreducible Spin(V)- module and the Spin(V)-module S ⊗ S is isomorphic to ∧V.

(ii) Let V be either complex of odd dimension or real of signature (m, m + 1) and β an admissible bilinear form on the spinor module S. Then for all k the image Γ_{β}^{k}(S ⊗S) is irreducible and the Spin(V)-module 2S ⊗ S is isomorphic to ∧V.

Corollary 3 Let V be complex of dimension n = 2m or real of signature (m, m). Then we have

S_{±} ∨ S_{±} = ∧_{±}^{m}V + ∑_{i=0}^{[(m−4)/4]} ∧^{m−4−4i}V ,  (5.16)

S_{±} ∧ S_{±} = ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V .  (5.17)

Proof: These decompositions follow from (5.4) and (5.12).

6 Decomposition of the tensor square of the spinor module of Spin(V ) into irreducible components: real case

In this section we describe the decompositions of S ⊗ S, S ∨ S and S ∧ S into inequivalent irreducible Spin(V)–submodules, where S is the spinor module of a pseudo-Euclidean vector space V = ℝ^{p,q} of arbitrary signature s = p − q and dimension n = p + q. We obtain these decompositions in two steps: First, we describe the complexification S^{ℂ} of the spinor module S. Second, using the decomposition of the tensor square 𝕊 ⊗ 𝕊 of the complex spinor module 𝕊, we decompose S^{ℂ} ⊗ S^{ℂ} into complex irreducible Spin(V^{ℂ})– submodules and then we take real forms. We recall that the complex spinor module 𝕊 associated to the complex Euclidean space 𝕍 = V^{ℂ} = V ⊗ ℂ = ℂ^{n} is the restriction to Spin(𝕍) of an irreducible representation of the complex Clifford algebra 𝒞l(𝕍).

Depending on the signature s ≡ p − q (mod 8), the complexification S^{ℂ} of the spinor module S is given by either

i) S^{ℂ} = 𝕊, where we denote by 𝕊 the spinor module of the complex Euclidean space 𝕍 = V^{ℂ} = V ⊗ ℂ = ℂ^{n}, or

ii) S^{ℂ} = 𝕊 + \overline{𝕊}, where \overline{𝕊} is the complex conjugated module of 𝕊.

In the latter case S admits a Spin(V)-invariant complex structure J and 𝕊 is identified with the complex space (S, J) and \overline{𝕊} with (S, −J). In the next lemma we specify the signatures for which the cases i) or ii) occur. For this we use Table 1, in which we have collected important information about the real and complex Clifford algebras and spinor modules.

Now, we define the notion of Type_{𝒞l^{0}(V)}(𝕊, 𝕊_{±}) used in Table 1. If s is odd, then the complex spinor module 𝕊 is irreducible (as a complex module of the real even Clifford algebra 𝒞l^{0}(V)). In this case we define Type_{𝒞l^{0}(V)}(𝕊) := 𝕂 ∈ {ℝ, ℍ} if the 𝒞l^{0}(V)-module 𝕊 is of real or quaternionic type, i.e. it admits a real or quaternionic structure commuting with 𝒞l^{0}(V). For even s the complex spinor module 𝕊 = 𝕊_{+} + 𝕊_{−} and 𝕊_{±} are irreducible complex 𝒞l^{0}(V)-modules. We put Type_{𝒞l^{0}(V)}(𝕊, 𝕊_{±}) = (l𝕂, 𝕂′), where 𝕂 and 𝕂′ are the types of 𝕊 and 𝕊_{±}, respectively, further l = 1 if 𝕊 is irreducible and l = 2 if 𝕊_{+} and 𝕊_{−} are not equivalent as complex 𝒞l^{0}(V)-modules. Note that if the semispinor modules are of complex type (s = 2, 6), then they are complex-conjugates of each other: \overline{𝕊}_{±} ≅ 𝕊_{∓}. If 𝕊_{±} are of real (s = 0) or quaternionic (s = 4) type, then they are selfconjugate: \overline{𝕊}_{±} ≅ 𝕊_{±}.

We now explain how Table 1 has been obtained. The first two columns have been extracted from [LM] and imply the third column. Passing to the complexification of the Clifford algebras we have: 𝒞l(V)⊗ℂ = 𝒞l(V ⊗ℂ) and 𝒞l^{0}(V)⊗ℂ = 𝒞l^{0}(V ⊗ℂ). From this we can describe the complex spinor module 𝕊 and semispinors modules 𝕊_{±} and determine the relation between S, S_{±} and 𝕊, 𝕊_{±}. This gives the fourth, fifth and sixth columns of the table. Using this table we prove the following lemma, which describes the complex Spin(𝕍)–module S^{ℂ}:

Lemma 3

S^{ℂ} = \{ \begin{matrix}𝕊 + \overline{𝕊} & if s = p − q ≡ 1, 2, 3, 4, 5 (mod 8) \\ 𝕊 & if s ≡ 6, 7, 8 (mod 8) . \end{matrix}

Proof: According to Table 1, if s ≡ 6, 7, 8 (mod 8) we have 𝕊 = S ⊗ ℂ. In all other cases there exists a Spin(V)-invariant complex structure J and the complex space (S, J) is identified with 𝕊. Then S^{ℂ} = 𝕊 + \overline{𝕊}.

Remark 1: We note that a Spin(V)–invariant real or quaternionic structure φ on 𝕊 (i.e. an antilinear map with φ^{2} = +1 or −1, respectively) defines an isomorphism φ : 𝕊 → \overline{𝕊}. From Table 1 it follows that if s ≡ 1, 2 (mod 8), then there exists a Spin(V)–invariant real structure and if s ≡ 3, 4, 5 (mod 8), then there exists a Spin(V)–invariant quaternionic structure on 𝕊.

Now, using the results of the previous section for the complex case, we decompose S^{ℂ}⊗S^{ℂ} into complex irreducible Spin(𝕍)–submodules. If S^{ℂ}⊗S^{ℂ} = ∑ 𝕎_{i} and all submodules 𝕎_{i} are of real type (i.e. complexifications of irreducible real Spin(V)–submodules W_{i}), then S ⊗ S = ∑ W_{i} is the desired decomposition. In odd dimensions all modules 𝕎_{i} = ∧^{i}𝕍 are of real type. This is also the case in even dimensions n = 2m, with one exception: the modules ∧_{±}^{m}𝕍 ⊂ S^{ℂ} ⊗ S^{ℂ} are not of real type if ∗^{2} = −1, i.e. if s/2 is odd. Then, ∧^{m}𝕍 = ∧_{+}^{m}𝕍 + ∧_{−}^{m}𝕍 is the complexification of the irreducible Spin(V)–module ∧^{m}V, which has the Spin(V)–invariant complex structure ∗.

The decompositions of S ∨S and S ∧S can be obtained using the same method. Using this approach, we describe in detail all these decompositions for any pseudo-Euclidean vector space V = ℝ^{p,q} in the next three subsections.

s  𝒞l_{p,q}  𝒞l_{p,q}^{0}  𝒞  𝕊  𝕊_{±}  Type_{𝒞l^{0}(V)}(𝕊, 𝕊_{±}) Name
0 ℝ(N) 2ℝ(N/2) 2ℝ  S ⊗ ℂ  S_{±} ⊗ ℂ  (2ℝ, ℝ)  M-W
1 ℂ(N)  ℝ(N) ℝ(2) S = S_{±} ⊗ ℂ  ℝ  M
2 ℍ(N/2) ℂ(N/2) ℂ(2) S = S_{±} ⊗ ℂ S_{±}  (2ℂ, ℂ)  M, W
3 2ℍ(N/2) ℍ(N/2) ℍ  S  ℍ  Symp
4 ℍ(N/2) 2ℍ(N/4) 2ℍ  S  S_{±}  (2ℍ, ℍ)  Symp-W
5 ℂ(N) ℍ(N/2) ℍ  S  ℍ  Symp
6 ℝ(N) ℂ(N/2) ℂ  S ⊗ ℂ  S  (ℝ, ℂ)  M, W
7 2ℝ(N)  ℝ(N)  ℝ  S ⊗ ℂ  ℝ  M

Table 1: Clifford Modules 𝒞l_{p,q}, their even parts 𝒞l_{p,q}^{0}, the Schur algebra 𝒞 = End _{𝒞l^{0}(V)}(S), the complex Spinor Module 𝕊, the complex Semispinor Modules 𝕊_{±}, the Type of these 𝒞l^{0}(V)-modules and physics terminology: M stands for Majorana, W for Weyl and Symp for symplectic (i.e. quaternionic) spinors; s = p − q mod 8 , n = p + q , N = 2^{[n/2]}. Note that p is the number of negative eigenvalues of the product of two gamma matrices, and q the number of positive eigenvalues, see appendix B.1.

6.1 Odd dimensional case: dim V = 2m + 1

We now describe the decomposition of S ⊗S = S ∨S +S ∧S for all signatures s = 1, 3, 5, 7 (mod 8) in the odd dimensional case.

Theorem 4 Let V = ℝ^{p,q} be a pseudo-Euclidean vector space of dimension n = p + q = 2m+1. Then the decompositions of the Spin(V)-modules S ⊗ S, S ∨ S and S ∧ S into inequivalent irreducible submodules is given by the following:

If the signature s = p−q ≡ 1, 3, 5 (mod 8) , we have

S ⊗ S = 2(∧V) = 4 ∑_{i=0}^{m} ∧^{i}V ,  (6.1)

S ∨ S = 3 ∑_{i=0}^{[m/4]} ∧^{m−4i}V + 3 ∑_{i=0}^{[(m−3)/4]} ∧^{m−3−4i}V

+ ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V + ∑_{i=0}^{[(m−1)/4]} ∧^{m−1−4i}V ,  (6.2)

S ∧ S = 3 ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V + 3 ∑_{i=0}^{[(m−1)/4]} ∧^{m−1−4i}V

+ ∑_{i=0}^{[m/4]} ∧^{m−4i}V + ∑_{i=0}^{[(m−3)/4]} ∧^{m−3−4i}V .  (6.3)

If the signature s ≡ 7 (mod 8) , we have

S ⊗ S = ∑_{i=0}^{m} ∧^{i}V ,  (6.4)

S ∨ S = ∑_{i=0}^{[m/4]} ∧^{m−4i}V + ∑_{i=0}^{[(m−3)/4]} ∧^{m−3−4i}V ,  (6.5)

S ∧ S = ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V + ∑_{i=0}^{[(m−1)/4]} ∧^{m−1−4i}V .  (6.6)

Moreover, S is an irreducible Spin(V)-module for s = 3, 5, 7 and for s = 1, S = S_{+} + S_{−} is the sum of two equivalent semi-spinor modules.

Proof: The signature s ≡ 7 (mod 8) corresponds to ℝ^{m,m+1}, which was already discussed in Theorem 2.

For s ≡ 1, 3, 5 (mod 8), the spinor module S has an invariant complex structure J and (S, J) is identified with the complex spinor module 𝕊. We denote by \overline{𝕊} = (S, −J) the module conjugate to 𝕊. According to Lemma 3 and Remark 1, 𝕊 and \overline{𝕊} are equivalent as complex modules of the real spin group Spin(V) and 𝕊^{ℂ} = 𝕊 + \overline{𝕊} = 2𝕊. Hence,

(S ⊗_{ℝ} S)^{ℂ} = S^{ℂ} ⊗_{ℂ} S^{ℂ} = 4𝕊 ⊗_{ℂ} 𝕊 ,
(∨^{2}S)^{ℂ} = ∨^{2}(𝕊 + \overline{𝕊}) = ∨^{2}(2𝕊) = 3 ∨^{2} 𝕊 + ∧^{2}𝕊 ,
(∧^{2}S)^{ℂ} = 3 ∧^{2} 𝕊 + ∨^{2}𝕊 .

This implies the theorem. For example,

(S ⊗_{ℝ} S)^{ℂ} = 4𝕊 ⊗_{ℂ} 𝕊 = 4 ∑_{i=0}^{m} ∧^{i}𝕍 = 4 ∑_{i=0}^{m}(∧^{i}V)^{ℂ} ,

in virtue of Theorem 2 and the real part gives (6.1).

6.2 Even dimensional case: dim V = 2m

In this subsection, we describe the decomposition of S⊗S = S∨S+S∧S for all signatures s = 0, 2, 4, 6 (mod 8) in the even dimensional case.

Theorem 5 Let V = ℝ^{p,q} be a pseudo-Euclidean vector space of dimension n = p + q = 2m. Then the decompositions of the Spin(V)-module S ⊗ S into inequivalent irreducible submodules is given by the following:

If the signature s = p−q ≡ 2, 4 (mod 8), we have

S ⊗ S = 4(∧V) = \begin{cases}8∑_{i=0}^{m−1} ∧^{i} V + 4 ∧^{m} V & if s = 2 (mod 8) \\ 8∑_{i=0}^{m−1} ∧^{i} V + 4 ∧_{+}^{m} V + 4 ∧_{−}^{m} V & if s = 4 (mod 8). \end{cases}  (6.7)

If the signature s ≡ 0, 6 (mod 8) , we have

S ⊗ S = ∧V = \begin{cases}2∑_{i=0}^{m−1} ∧^{i} V + ∧_{+}^{m}V + ∧_{−}^{m}V & if s = 0 (mod 8) \\ 2∑_{i=0}^{m−1} ∧^{i} V + ∧^{m}V & if s = 6 (mod 8). \end{cases}  (6.8)

Proof: Similarly to the odd dimensional case, we have

(S ⊗_{ℝ} S)^{ℂ} = 𝕊 ⊗_{ℂ} 𝕊 = ∧𝕍 = 2 ∑_{i=0}^{m} ∧^{i}𝕍 + ∧_{+}^{m}𝕍 + ∧_{−}^{m}𝕍 .  (6.9)

Now, we note that (∧^{m}V)^{ℂ} = ∧^{m}𝕍 and ∗^{2}|_{∧^{m}V} = (−1)^{s/2}. If s/2 is even, then ∧^{m}V = ∧_{+}^{m}V + ∧_{−}^{m}V, where ∧_{±}^{m}V are irreducible submodules, which are ±1–eigenspaces of ∗. Then, (∧_{±}^{m}V )^{ℂ} = ∧_{±}^{m}𝕍. In this case, the real part of (6.9) gives the first part of (6.8). If s/2 is odd, then ∗ is a complex structure on ∧^{m}V, which is irreducible since it has complex structure and its complexification ∧^{m}𝕍 has only two irreducible components ∧_{±}^{m}𝕍. In this case the real part of (6.9) gives the second part of (6.8).

The decompositions of S ∨ S and S ∧ S for the cases when semispinor modules exist, in particular for s = 0, 2, 4 (mod 8), will be given in the next subsection. Therefore it is now sufficient to determine these decompositions for s = 6 (mod 8).

Corollary 4 Let S be the spinor module of a pseudo-Euclidean vector space V of signature s ≡ 6 (mod 8) and dimension n = 2m. Then we have

S ∨ S = ∧^{m}V + 2 ∑_{i=0}^{[(m−4)/4]} ∧^{m−4−4i}V + ∑_{i=0}^{[(m−1)/2]} ∧^{m−1−2i}V ,  (6.10)

S ∧ S = 2 ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V + ∑_{i=0}^{[(m−1)/2]} ∧^{m−1−2i}V .  (6.11)

Proof: This follows by complexification of equation (6.8), using Lemma 3, equation (5.3) and Corollary 3.

6.3 Decomposition of tensor square of semi-spinors

According to Table 1, semi-spinor modules S_{±} exist if the signature s ≡ 0, 1, 2, 4 (mod 8). More precisely, we list below whether S_{±} are equivalent Spin(V)–modules and we give S_{±}^{ℂ}.

s  S_{±}  S_{±}^{ℂ}
0 inequivalent S_{±} ⊗ ℂ = 𝕊_{±}
1 equivalent  S_{±} ⊗ ℂ = 𝕊
2 equivalent  S_{±} ⊗ ℂ = 𝕊
4 inequivalent 𝕊_{±} + \overline{𝕊}_{±} = 2𝕊_{±}

For s ≡ 1, 2, 4 (mod 8) we have 𝕊 = S, whereas for s ≡ 0 (mod 8) we have 𝕊 = S^{ℂ}. We also note that for s ≡ 2 (mod 8), although the Spin(V)–modules S_{±} are equivalent, the Spin(𝕍)–modules 𝕊_{+} and 𝕊_{−} = \overline{𝕊}_{+} are not equivalent. For s = 4, S_{±} = 𝕊_{±} admits a Spin(V)-invariant quaternionic structure.

Using the above description for S_{±}^{ℂ} and the decompositions of the tensor squares of complex spinor and semi-spinor modules, we obtain the following:

Theorem 6 Let S = S_{+} + S_{−} be the spinor module of a pseudo-Euclidean vector space V of signature s ≡ 0, 1, 2, 4 (mod 8) and dimension n = 2m or n = 2m+1. Then we have the following decomposition of Spin(V)–modules S_{±} ⊗ S_{±} and S_{+} ⊗ S_{−} into inequivalent irreducible submodules:

For s ≡ 0 (mod 8):

S_{+} ⊗ S_{−} = ∑_{i=0}^{[(m−1)/2]} ∧^{m−1−2i}V ,

S_{±} ⊗ S_{±} = ∧_{±}^{m}V + ∑_{i=0}^{[(m−2)/2]} ∧^{m−2−2i}V ,

S_{±} ∨ S_{±} = ∧_{±}^{m}V + ∑_{i=0}^{[(m−4)/4]} ∧^{m−4−4i}V ,

S_{±} ∧ S_{±} = ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V .

For s ≡ 1 (mod 8):

S_{±} ⊗ S_{±} = S_{+} ⊗ S_{−} = ∑_{i=0}^{m} ∧^{i}V ,

S_{±} ∨ S_{±} = ∑_{i=0}^{[m/4]} ∧^{m−4i}V + ∑_{i=0}^{[(m−3)/4]} ∧^{m−3−4i}V ,

S_{±} ∧ S_{±} = ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V + ∑_{i=0}^{[(m−1)/4]} ∧^{m−1−4i}V .

For s ≡ 2 (mod 8):

S_{±} ⊗ S_{±} = S_{+} ⊗ S_{−} = ∧^{m}V + 2 ∑_{i=0}^{m−1} ∧^{i}V ,

S_{±} ∨ S_{±} = ∧^{m}V + 2 ∑_{i=0}^{[(m−4)/4]} ∧^{m−4−4i}V + ∑_{i=0}^{[(m−1)/2]} ∧^{m−1−2i}V ,

S_{±} ∧ S_{±} = 2 ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V + ∑_{i=0}^{[(m−1)/2]} ∧^{m−1−2i}V .

For s ≡ 4 (mod 8):

S_{+} ⊗ S_{−} = 4 ∑_{i=0}^{[(m−1)/2]} ∧^{m−1−2i}V ,

S_{±} ⊗ S_{±} = 4 ∧_{±}^{m} V + 4 ∑_{i=0}^{[(m−2)/2]} ∧^{m−2−2i}V ,

S_{±} ∨ S_{±} = 3 ∧_{±}^{m} V + 3 ∑_{i=0}^{[(m−4)/4]} ∧^{m−4−4i}V + ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V ,

S_{±} ∧ S_{±} = 3 ∑_{i=0}^{[(m−2)/4]} ∧^{m−2−4i}V + ∧_{±}^{m}V + ∑_{i=0}^{[(m−4)/4]} ∧^{m−4−4i}V .

Proof: The case s = 0 (mod 8) follows from the complex case (see Theorem 3 and Corollary 3). For s = 1, 2 (mod 8) the modules S_{+} and S_{−} are isomorphic. Hence S = S_{+} + S_{−} = 2S_{+} and S ⊗ S = 4S_{+} ⊗ S_{+}. Since S ⊗ S = 4 ∑_{i=0}^{m} ∧^{i}V we have

S_{+} ⊗ S_{+} = S_{−} ⊗ S_{−} = S_{+} ⊗ S_{−} = ∑_{i=0}^{m} ∧^{i}V .

The splitting into symmetric and skew parts of these tensor products follows that in the complex cases, see Theorem 2 and Corollary 3. For s = 4 (mod 8) the semi-spinor modules S_{±} are not equivalent but S_{±}^{ℂ} = 𝕊 = 𝕊_{+} + 𝕊_{−}. The result follows from the decomposition in the complex case (Theorem 3 and Corollary 3) on taking the real parts.

7 N-extended polyvector Poincaré algebras

In the previous sections we have classified ϵ-transalgebras of the form 𝔤 = 𝔤_{0} + 𝔤_{1}, 𝔤_{0} = 𝔰𝔬(V) + W_{0}, with 𝔤_{1} = W_{1} = S the spinor module. As in [AC] we can easily extend this classification to the case where W_{1} is a general spinorial module, i.e. W_{1} = NS = S ⊗ℝ^{N}, if S is irreducible, or W_{1} = N_{+}S_{+} ⊕ N_{−}S_{−} = S_{+} ⊗ ℝ_{+}^{N} ⊕ S_{−} ⊗ ℝ_{−}^{N} if semi-spinors exist.

An ϵ-extensions of translational type of the above form is called an N-extended polyvector Poincaré algebra if W_{1} = NS and an (N_{+}, N_{−})-extended polyvector Poincaré algebra if W_{1} = N_{+}S_{+} ⊕ N_{−}S_{−}. Consider first the case W_{1} = NS. As before, the classification reduces to the decomposition of ∨^{2}W_{1} and ∧^{2}W_{1} into irreducible submodules. These decompositions follow from the decompositions of ∨^{2}S and ∧^{2}S obtained in the previous sections, together with the decompositions

∨^{2}W_{1} = ∨^{2}S ⊗ ∨^{2}ℝ^{N} ⊕ ∧^{2}S ⊗ ∧^{2}ℝ^{N} ,
∧^{2}W_{1} = ∧^{2}S ⊗ ∨^{2}ℝ^{N} ⊕ ∨^{2}S ⊗ ∧^{2}ℝ^{N} .

In particular, this implies that the multiplicities μ_{+}(k, N) and μ_{−}(k, N) of the module ∧^{k}V in ∨^{2}W_{1} and ∧^{2}W_{1}, respectively, are given by

μ_{+}(k, N) = μ_{+}(k)\frac{N(N + 1)}{2} + μ_{−}(k)\frac{N(N − 1)}{2} ,
μ_{−}(k, N) = μ_{−}(k)\frac{N(N + 1)}{2} + μ_{+}(k)\frac{N(N − 1)}{2} ,

where μ_{+}(k) and μ_{−}(k) are the multiplicities of ∧^{k}V in ∨^{2}S and ∧^{2}S, respectively. The vector space of N-extended polyvector Poincaré ϵ-algebra structures with W_{0} = ∧^{k}V is identified with the space Bil_{ϵ}^{k}(W_{1})^{𝔰𝔬(V)} of invariant ∧^{k}V -valued bilinear forms on W_{1}. Its dimension is given by

dim Bil_{ϵ}^{k}(W_{1})^{𝔰𝔬(V)} = μ_{ϵ}(k, N) dim 𝒞(∧^{k}V) ,

where the Schur algebra 𝒞(∧^{k}V) = ℝ, ℝ ⊕ ℝ or ℂ, see the proof of Theorem 1. Any element Γ_{ϵ} ∈ Bil_{ϵ}^{k}(W_{1})^{𝔰𝔬(V)} can be represented as

Γ_{+} = ∑_{i} Γ_{β_{+}^{i}}^{k} ⊗ b_{+}^{i} + ∑_{j} Γ_{β_{−}^{j}}^{k} ⊗ b_{−}^{j} ,
Γ_{−} = ∑_{i} Γ_{β_{−}^{i}}^{k} ⊗ b_{+}^{i} + ∑_{j} Γ_{β_{+}^{j}}^{k} ⊗ b_{−}^{j} ,

where β_{±}^{i} ∈ Bil_{±}^{k}(S)^{𝔰𝔬(V)} and b_{+}^{i} and b_{−}^{i} are, respectively, symmetric and skewsymmetric bilinear forms on ℝ^{N}. We note also, that there exists a unique minimal (i.e. W_{0} = [W_{1}, W_{1}]) N-extended polyvector Poincaré ϵ-algebra with W_{0} = μ_{ϵ}(k, N) ∧^{k} V. The Lie (super)bracket is given, up to a twist by an invertible element of the Schur algebra of W_{0}, by the projection onto the corresponding maximal isotypical submodule of ∨^{2}W_{1} or ∧^{2}W_{1}, respectively.

Similarly, in the case when the spinor module S = S_{+} ⊕S_{−} is reducible, we can reduce the description of all (N_{+}, N_{−})-extended polyvector Poincaré ϵ-algebras 𝔤 = 𝔰𝔬(V) + ∧^{k}V + W_{1} such that W_{1} = N_{+}S_{+} + N_{−}S_{−} to the chiral cases (N_{+}, N_{−}) = (1, 0) or (0, 1) and to the isotropic case: (N_{+}, N_{−}) = (1, 1) and [S_{+}, S_{+}] = [S_{−}, S_{−}] = 0.

Let β be an admissible bilinear form on S = S_{+} ⊕ S_{−} and Γ_{β}^{k} ∈ Bil_{ϵ}^{k}(S)^{𝔰𝔬(V)} the corresponding admissible ∧^{k}V -valued bilinear form. Its restriction to S_{+} (or S_{−}) defines a (1, 0)-extended (respectively, (0, 1)-extended) k-polyvector Poincaré ϵ-algebra if and only if ι(Γ_{β}^{k}) = +1. If ι(Γ_{β}^{k}) = −1 then we obtain an isotropic (1, 1)-extended k-polyvector Poincaré ϵ-algebra, i.e. [S_{+}, S_{+}] = [S_{−}, S_{−}] = 0. The values of the invariants σ(Γ_{β}^{k}) = ϵ and ι(Γ_{β}^{k}) can be read off Tables A3–A6 in Appendix A.

Acknowledgments

This work was generously supported by the ‘Schwerpunktprogramm Stringtheorie’ of the Deutsche Forschungsgemeinschaft. Partial support by the European Commission’s 5th Framework Human Potential Programmes under contracts HPRN-CT-2000-00131 (Quantum Spacetime) and HPRN-CT-2002-00325 (EUCLID) is also acknowledged. A.V.P. is supported in part by the Belgian Federal Office for Scientific, Technical and Cultural Affairs through the Inter-University Attraction Pole P5/27. V.C, C.D. and A.V.P. are grateful to the University of Hull for hospitality during a visit in which the basis of this work was discussed.

A Admissible ∧^{k}V -valued bilinear forms on S

In Table A1 we give the numbers (N^{+}(s, n), N^{−}(s, n)) of independent symmetric (+) and skewsymmetric (−) invariant bilinear forms on S, i.e. N^{±}(s, n) = dim Bil_{±}(S)^{𝔰𝔬(V)}. They were computed in [AC] and are periodic with period 8 in the signature s = p − q and in the dimension n = p + q of V = ℝ^{p,q}. The entry N(s) = N^{+}(s, n) + N^{−}(s, n) is the total number of invariant bilinear forms and μ(k) is the multiplicity of the irreducible submodule ∧^{k}V in S ⊗ S. For k ≠ n/2, it does not depend on k, i.e. μ(k) = μ(0). In the case n = 2m the multiplicities μ(m) with a ∗ indicate that the module ∧^{m}V is reducible as ∧_{+}^{m}V + ∧_{−}^{m}V.

s\n 0 1 2 3 4 5 6 7 N(s) μ(k) = μ(0) μ(m)
0 2,0  1,1  0,2  1,1  2  2  1*
1  3,1  1,3  1,3  3,1 4  4
2 6,2  4,4  2,6  4,4  8  8  4
3  3,1  1,3  1,3  3,1 4  4
4 6,2  4,4  2,6  4,4  8  8  4*
5  3,1  1,3  1,3  3,1 4  4
6 2,0  1,1  0,2  1,1  2  2  1
7  1,0  0,1  0,1  1,0 1  1

Table A1: The numbers (N^{+}(s, n), N^{−}(s, n)) of independent symmetric and skewsymmetric bilinear forms on S From Table A1, we note the following symmetries:

a) Modulo 8-symmetry

N^{±}(s + 8a, n + 8b) = N^{±}(s, n) , a, b ∈ ℤ

b) Reflection with respect to the horizontal line s = 3

N^{±}(−s + 6, n) = N^{±}(s, n)

and reflection with respect to the vertical line n = 0

N^{±}(s, n) = N^{±}(s, −n) .

c) The reflection with respect to the vertical line n = 2 is a mirror symmetry, i.e. it interchanges N^{+} and N^{−}:

N^{±}(s, n) = N^{∓}(s, −n + 4) .

It is the same as the mirror symmetry with respect to n = 6. We note that the composition of reflections in n = 0 and n = 2 gives the translation n ↦ n + 4:

N^{±}(s, n) = N^{∓}(s, n + 4) .

More generally, we consider the dimension

Ñ_{k}^{±}(p, q) := N_{k}^{±}(s, n) := dim Bil_{±}^{k}(S)^{𝔰𝔬(V)}

of the vector space of invariant ∧^{k}V -valued bilinear forms on S. By Theorem 1, the sum N_{k}(s, n) = N_{k}^{+}(s, n) + N_{k}^{−}(s, n) = N(s, n) does not depend on k. From Table A1, we see that it depends only on the signature s. As a corollary of Proposition 4, the numbers Ñ_{k}^{±}(p, q) are periodic modulo 4 in k:

Ñ_{k}^{±}(p, q) = Ñ_{k+4}^{±}(p, q) .

Moreover, we have the following periodicities in (p, q):

Ñ_{k}^{±}(p, q) = Ñ_{k}^{±}(p + 8, q) = Ñ_{k}^{±}(p, q + 8) = Ñ_{k}^{±}(p + 4, q + 4) .

In fact, it was proven in [AC] that, for any given symmetry σ_{0}, type τ_{0} and isotropy ι_{0} (if defined), the number of bilinear forms β with σ(β) = σ_{0}, τ(β) = τ_{0} and ι(β) = ι_{0} in a basis (β_{i}) of Bil(S)^{𝔰𝔬(V)} consisting of admissible forms is (8, 0)-, (0, 8)- and (4, 4)-periodic in (p, q). By Proposition 4, this implies that for any given symmetry σ′_{0} and isotropy ι′_{0} (if defined), the number of ∧^{k}V -valued bilinear forms Γ_{β_{i}}^{k} with σ(Γ_{β_{i}}^{k}) = σ′_{0}, and ι(Γ_{β_{i}}^{k}) = ι′_{0} is (8, 0)-, (0, 8)- and (4, 4)-periodic in (p, q).

Finally, we have the following shift formula

N_{k}^{±}(s, n + 2k) = N_{0}^{±}(s, n) := N^{±}(s, n) ,

which we can write also as

Ñ_{k}^{±}(p + k, q + k) = Ñ_{0}^{±}(p, q) .

This shift formula follows from the tables below.

In Table A3 we describe a basis of Bil(S)^{𝔰𝔬(V)}, which consists of admissible forms and indicate the values of the three invariants σ, τ and ι. In the three following tables we give the invariants σ and ι for the corresponding bases of Bil^{k}(S)^{𝔰𝔬(V)}, k = 1, 2, 3 modulo 4, denoted for simplicity by the same symbols. Due to the above periodicity properties, we can calculate, from these tables, the values of the invariants for the corresponding bases of Bil^{k}(S)^{𝔰𝔬(V)} for all k ∈ ℕ and V = ℝ^{p,q}.

For any V = ℝ^{p,q} an explicit basis of Bil^{k}(S)^{𝔰𝔬(V)} consisting of admissible forms was constructed for k = 0 and k = 1, in terms appropriate models of the spinor module in [AC]. It was proven there that any admissible V -valued bilinear form on S is of the form Γ_{β}^{1}, where β is a linear combination of admissible scalar-valued forms. By Theorem 1 and Proposition 4, this result extends to k > 1, namely any admissible ∧^{k}V -valued bilinear form is of the form Γ_{β}^{k}, where β is a linear combination of admissible scalar-valued forms. This provides a basis of the vector space Bil^{k}(S)^{𝔰𝔬(V)} ≅ Bil(S)^{𝔰𝔬(V)} consisting of admissible forms. The dimension of this space is equal to the dimension of the Schur algebra 𝒞(S), which depends only on the signature s = p − q modulo 8, see [AC].

In the tables below we use the notation of [AC]. We use the fact that any pseudo-Euclidean vector space can be written as V = V_{1} ⊕ V_{2}, where V_{1} = ℝ^{r,r} and V_{2} = ℝ^{0,k} or V_{2} = ℝ^{k,0}. Then [LM]

𝒞l(V) ≅ 𝒞l(V_{1}) ˆ⊗ 𝒞l(V_{2}) ,

where ˆ⊗ denotes the ℤ/2ℤ-graded tensor product. Let S_{1} be the spinor module of Spin(V_{1}) and S_{2} the spinor module of Spin(V_{2}). Then we always have that S_{1} = S_{1}^{+} +S_{1}^{−} is a ℤ/2ℤgraded module of the ℤ/2ℤ-graded algebra 𝒞l(V_{1}). The spinor module S of Spin(V) can be described in terms of S_{1} and S_{2} as follows, see Proposition 2.3. of [AC]. Consider first the case when S_{2} = S_{2}^{+} +S_{2}^{−} is reducible. In this case S_{2} is a ℤ/2ℤ-graded 𝒞l(V_{2})-module and the spinor module S = S_{1} ˆ⊗ S_{2} of Spin(V) is obtained as the ℤ/2ℤ-graded tensor product of the modules S_{1} and S_{2}. It is again ℤ/2ℤ-graded with even part S_{+} = S_{1}^{+}⊗S_{2}^{+}+S_{1}^{−}⊗S_{2}^{−} and odd part S_{−} = S_{1}^{+} ⊗ S_{2}^{−} + S_{1}^{−} ⊗ S_{2}^{+}. If S_{2} is an irreducible Spin(V)-module, then the spinor module of Spin(V) is given by

S = S_{1} ˆ⊗ S_{2} = S_{1}^{+} ⊗ S_{2} + S_{1}^{−} ⊗ S_{2}

with the action

(a ˆ⊗ b) · (s_{1}^{±} ⊗ s_{2}) = (−1)^{deg(s_{1}^{±})deg(b)}as_{1}^{±} ⊗ bs_{2} ,

where a ∈ 𝒞l(V_{1}), b ∈ 𝒞l(V_{1}), s_{1}^{±} ∈ S_{1}^{±}, s_{2} ∈ S_{2}, deg(s_{1}^{+}) = 0 and deg(s_{1}^{−}) = 1. In this case S is an irreducible Spin(V)-module.

As discussed in Section 5, for the case of split signature, V = ℝ^{r,r}, there exist two independent admissible bilinear forms f and f_{E} = f(E· , ·) on the spinor module, where E|_{S_{±}} = ±Id. Their invariants are given in Table A3. If V is positive or negative definite, then there exists a unique (up to scale) Pin(V)-invariant scalar product g on the spinor module S and any admissible, hence invariant, bilinear form on S is of the form g_{A} = g(A·, ·), where A is an admissible element of the Schur algebra 𝒞(S). The admissibility of A means that A is either symmetric or skewsymmetric with respect to g, that it either commutes or anticommutes with the Clifford multiplication γ_{v} and that it preserves or interchanges S_{+} and S_{−} if they exist [AC].

s 𝒞(S)  basis of 𝒞(S)
0 2ℝ  Id, E
1 ℝ(2)  Id, E, I, EI
2 ℂ(2) Id, I, J, K, E, EI, EJ, EK
3 ℍ  Id, I, J, K
4 2ℍ Id, I, J, K, E, EI, EJ, EK
5 ℍ  Id, I, J, K
6 ℂ  Id, I
7 ℝ  Id

Table A2: Standard bases for the Schur algebras 𝒞(S)

Returning to the general case V = ℝ^{p,q} = V_{1} ⊕ V_{2}, S = S_{1} ˆ⊗ S_{2}, as above, the admissible bilinear forms on S can be described as follows. Let (g_{A_{i}}) be a basis of Bil(S_{2})^{𝔰𝔬(V_{2})} consisting of admissible elements. Inspection of Table A3 shows that for any g_{A_{i}} there exists a unique element ϕ_{i} ∈ {f, f_{E}} which satisfies the condition τ(ϕ_{i}) = ι(g_{A_{i}})τ(g_{A_{i}}). In virtue of Proposition 3.4 of [AC], the tensor products ϕ_{i}⊗g_{A_{i}} provide a basis of Bil(S)^{𝔰𝔬(V)} consisting of admissible elements. The corresponding basis Γ_{ϕ_{i}⊗g_{A_{i}}}^{k} of Bil^{k}(S)^{𝔰𝔬(V)} and its invariants are tabulated below. For simplicity the symbols Γ^{k} and ⊗ are omitted.

We use the following bases for the Schur algebra 𝒞(S) (see Table A2). If the signature s = 0, 1, 2 or 4, then S = S_{+} ⊕ S_{−}, and we put E = diag(Id_{S_{+}}, Id_{S_{−}}). In the cases s = 1 and s = 6 we denote by I the standard complex structure in 𝒞(S) = ℝ(2) and 𝒞(S) = ℂ, respectively. In fact, in the case s = 1, S = ℂ^{N} = ℝ^{N} ⊕ℝ^{N} has a 𝒞l(V)-invariant complex structure. In the cases s = 2, 3, 4 and 5, we denote by I, J, K = IJ ∈ 𝒞(S) the canonical 𝒞l^{0}(V)-invariant hypercomplex structure of S (3 anticommuting complex structures). In fact, in the case s = 4, S = ℍ^{N/2} = ℍ^{N/4} ⊕ ℍ^{N/4} has a 𝒞l(V)-invariant hypercomplex structure.

Table A3: Admissible bilinear forms β on S and their invariants (σ, τ, ι)

p\q  0  1  2  3
0 f  + − + g  ++  fg  −−  f_{E}g  −+
f_{E} + + +  f_{E}g_{I}  ++  f_{E}g_{I} ++
f_{E}g_{J} −−
f_{E}g_{K} −−
1 g  + − + f  − − − g  −−  f_{E}g  −+
g_{E}  + + + f_{E}  + + −  fg_{I}  −−
g_{I}  − − −
g_{IE} + + −
2 g  + − + f_{E}g  + + − f  − − + g  −+
g_{I}  − − + f_{E}g_{I} − + + f_{E}  − + +
g_{J}  − − − fg_{E}  − − −
g_{K} − − − fg_{IE} − − +
g_{E}  + + +
g_{IE} − + +
g_{JE} + + −
g_{KE} + + −
3 g  +−  f_{E}g  + + − fg  − − + f  + − −
g_{I}  −−  f_{E}g_{I} − + − fg_{I}  + − − f_{E}  − + −
g_{J}  −−  f_{E}g_{J} − + + f_{E}g_{E}  − + +
g_{K} −−  f_{E}g_{K} − + + f_{E}g_{IE} − + −
fg_{E}  − − −
fg_{IE} + − −
fg_{JE} − − +
fg_{KE} − − +
4 g  + − + f_{E}g  ++  fg  − − + f_{E}g  − + −
g_{I}  − − + f_{E}g_{I} −+  fg_{I}  + − + f_{E}g_{I} + + +
g_{J}  − − + f_{E}g_{J} −+  fg_{J}  + − − fg_{E}  + − −
g_{K} − − + f_{E}g_{K} −+  fg_{K}  + − − fg_{IE} + − +
g_{E}  + + +  f_{E}g_{E}  − + +
g_{IE} − + +  f_{E}g_{IE} + + +
g_{JE} − + +  f_{E}g_{JE} − + −
g_{KE} − + +  f_{E}g_{KE} − + −
5 g  +−  f_{E}g  + + − fg  −−  f_{E}g  − + −
g_{I}  −−  f_{E}g_{I} − + − fg_{I}  +−  f_{E}g_{I} + + −
g_{J}  −+  f_{E}g_{J} − + − fg_{J}  +−  f_{E}g_{J} + + +
g_{K} −+  f_{E}g_{K} − + − fg_{K}  +−  f_{E}g_{K} + + +
fg_{E}  − − −  fg_{E}  + − −
fg_{IE} + − −  fg_{IE} − − −
fg_{JE} + − −  fg_{JE} + − +
fg_{KE} + − −  fg_{KE} + − +
6 g  +−  f_{E}g  ++  fg  − − + f_{E}g  −+
g_{I}  −+  f_{E}g_{I} −+  fg_{I}  + − + f_{E}g_{I} ++
fg  +−  fg_{J}  + − + f_{E}g_{J} ++
fg_{I}  +−  fg_{K}  + − + f_{E}g_{K} ++
f_{E}g_{E}  − + +
f_{E}g_{IE} + + +
f_{E}g_{JE} + + +
f_{E}g_{KE} + + +
7 g  +−  f_{E}g  ++  fg  −−  f_{E}g  − + −
fg_{I}  +−  fg_{I}  +−  f_{E}g_{I} + + −
f_{E}g  ++  f_{E}g_{J} + + −
f_{E}g_{I}  ++  f_{E}g_{K} + + −
fg_{E}  + − −
fg_{IE} − − −
fg_{JE} − − −
fg_{KE} − − −

Table A4: ∧^{k}V -valued admissible bilinear forms on S and their invariants (σ, ι) for k≡1(4)

p\q  0  1  2  3
0 f  −− g  + fg  + f_{E}g  −
f_{E} +−  f_{E}g_{I}  + f_{E}g_{I} +
f_{E}g_{J} +
f_{E}g_{K} +
1 g  −− f  ++ g  + f_{E}g  −
g_{E}  +− f_{E}  ++  fg_{I}  +
g_{I}  ++
g_{IE} ++
2 g  −− f_{E}g  ++ f  +− g  −
g_{I}  +− f_{E}g_{I} −− f_{E}  −−
g_{J}  ++ fg_{E}  ++
g_{K} ++ fg_{IE} +−
g_{E}  +−
g_{IE} −−
g_{JE} ++
g_{KE} ++
3 g  − f_{E}g  ++ fg  +− f  −+
g_{I}  + f_{E}g_{I} −+ fg_{I}  −+ f_{E}  −+
g_{J}  + f_{E}g_{J} −− f_{E}g_{E}  −−
g_{K} + f_{E}g_{K} −− f_{E}g_{IE} −+
fg_{E}  ++
fg_{IE} −+
fg_{JE} +−
fg_{KE} +−
4 g  −− f_{E}g  + fg  +− f_{E}g  −+
g_{I}  +− f_{E}g_{I} − fg_{I}  −− f_{E}g_{I} +−
g_{J}  +− f_{E}g_{J} − fg_{J}  −+ fg_{E}  −+
g_{K} +− f_{E}g_{K} − fg_{K}  −+ fg_{IE} −−
g_{E} +− f_{E}g_{E} −−
g_{IE} −−  f_{E}g_{IE} +−
g_{JE} −−  f_{E}g_{JE} −+
g_{KE} −−  f_{E}g_{KE} −+
5 g  − f_{E}g  ++ fg  + f_{E}g  −+
g_{I}  + f_{E}g_{I} −+ fg_{I}  − f_{E}g_{I} ++
g_{J}  − f_{E}g_{J} −+ fg_{J}  − f_{E}g_{J} +−
g_{K} − f_{E}g_{K} −+ fg_{K}  − f_{E}g_{K} +−
fg_{E} ++ fg_{E} −+
fg_{IE} −+  fg_{IE} ++
fg_{JE} −+  fg_{JE} −−
fg_{KE} −+  fg_{KE} −−
6 g  − f_{E}g  + fg  +− f_{E}g  −
g_{I}  − f_{E}g_{I} − fg_{I}  −− f_{E}g_{I} +
fg  − fg_{J}  −− f_{E}g_{J} +
fg_{I}  − fg_{K}  −− f_{E}g_{K} +
f_{E}g_{E}  −−
f_{E}g_{IE} +−
f_{E}g_{JE} +−
f_{E}g_{KE} +−
7 g  − f_{E}g  + fg  + f_{E}g  −+
fg_{I}  − fg_{I}  − f_{E}g_{I} ++
f_{E}g  + f_{E}g_{J} ++
f_{E}g_{I}  + f_{E}g_{K} ++
fg_{E}  −+
fg_{IE} ++
fg_{JE} ++
fg_{KE} ++

Table A5: ∧^{k}V -valued admissible bilinear forms on S and their invariants (σ, ι) for k≡2(4)

p\q  0  1  2  3
0 f  −+ g  − fg  + f_{E}g  +
f_{E} −+  f_{E}g_{I}  − f_{E}g_{I} −
f_{E}g_{J} +
f_{E}g_{K} +
1 g  −+ f  +− g  + f_{E}g  +
g_{E}  −+ f_{E}  −−  fg_{I}  +
g_{I}  +−
g_{IE} −−
2 g  −+ f_{E}g  −− f  ++ g  +
g_{I}  ++ f_{E}g_{I} ++ f_{E}  ++
g_{J}  +− fg_{E}  +−
g_{K} +− fg_{IE} ++
g_{E}  −+
g_{IE} ++
g_{JE} −−
g_{KE} −−
3 g  − f_{E}g  −− fg  ++ f  −−
g_{I}  + f_{E}g_{I} +− fg_{I}  −− f_{E}  +−
g_{J}  + f_{E}g_{J} ++ f_{E}g_{E}  ++
g_{K} + f_{E}g_{K} ++ f_{E}g_{IE} +−
fg_{E}  +−
fg_{IE} −−
fg_{JE} ++
fg_{KE} ++
4 g  −+ f_{E}g  − fg  ++ f_{E}g  +−
g_{I}  ++ f_{E}g_{I} + fg_{I}  −+ f_{E}g_{I} −+
g_{J}  ++ f_{E}g_{J} + fg_{J}  −− fg_{E}  −−
g_{K} ++ f_{E}g_{K} + fg_{K}  −− fg_{IE} −+
g_{E} −+ f_{E}g_{E} ++
g_{IE} ++  f_{E}g_{IE} −+
g_{JE} ++  f_{E}g_{JE} +−
g_{KE} ++  f_{E}g_{KE} +−
5 g  − f_{E}g  −− fg  + f_{E}g  +−
g_{I}  + f_{E}g_{I} +− fg_{I}  − f_{E}g_{I} −−
g_{J}  + f_{E}g_{J} +− fg_{J}  − f_{E}g_{J} −+
g_{K} + f_{E}g_{K} +− fg_{K}  − f_{E}g_{K} −+
fg_{E} +− fg_{E} −−
fg_{IE} −−  fg_{IE} +−
fg_{JE} −−  fg_{JE} −+
fg_{KE} −−  fg_{KE} −+
6 g  − f_{E}g  − fg  ++ f_{E}g  +
g_{I}  + f_{E}g_{I} + fg_{I}  −+ f_{E}g_{I} −
fg  − fg_{J}  −+ f_{E}g_{J} −
fg_{I}  − fg_{K}  −+ f_{E}g_{K} −
f_{E}g_{E}  ++
f_{E}g_{IE} −+
f_{E}g_{JE} −+
f_{E}g_{KE} −+
7 g  − f_{E}g  − fg  + f_{E}g  +−
fg_{I}  − fg_{I}  − f_{E}g_{I} −−
f_{E}g  − f_{E}g_{J} −−
f_{E}g_{I}  − f_{E}g_{K} −−
fg_{E}  −−
fg_{IE} +−
fg_{JE} +−
fg_{KE} +−

Table A6: ∧^{k}V -valued admissible bilinear forms on S and their invariants (σ, ι) for k≡3(4)

p\q  0  1  2  3
0 f  +− g  − fg  − f_{E}g  +
f_{E} −−  f_{E}g_{I}  − f_{E}g_{I} −
f_{E}g_{J} −
f_{E}g_{K} −
1 g  +− f  −+ g  − f_{E}g  +
g_{E}  −− f_{E}  −+  fg_{I}  −
g_{I}  −+
g_{IE} −+
2 g  +− f_{E}g  −+ f  −− g  +
g_{I}  −− f_{E}g_{I} +− f_{E}  +−
g_{J}  −+ fg_{E}  −+
g_{K} −+ fg_{IE} −−
g_{E}  −−
g_{IE} +−
g_{JE} −+
g_{KE} −+
3 g  + f_{E}g  −+ fg  −− f  ++
g_{I}  − f_{E}g_{I} ++ fg_{I}  ++ f_{E}  ++
g_{J}  − f_{E}g_{J} +− f_{E}g_{E}  +−
g_{K} − f_{E}g_{K} +− f_{E}g_{IE} ++
fg_{E}  −+
fg_{IE} ++
fg_{JE} −−
fg_{KE} −−
4 g  +− f_{E}g  − fg  −− f_{E}g  ++
g_{I}  −− f_{E}g_{I} + fg_{I}  +− f_{E}g_{I} −−
g_{J}  −− f_{E}g_{J} + fg_{J}  ++ fg_{E}  ++
g_{K} −− f_{E}g_{K} + fg_{K}  ++ fg_{IE} +−
g_{E} −− f_{E}g_{E} +−
g_{IE} +−  f_{E}g_{IE} −−
g_{JE} +−  f_{E}g_{JE} ++
g_{KE} +−  f_{E}g_{KE} ++
5 g  + f_{E}g  −+ fg  − f_{E}g  ++
g_{I}  − f_{E}g_{I} ++ fg_{I}  + f_{E}g_{I} −+
g_{J}  + f_{E}g_{J} ++ fg_{J}  + f_{E}g_{J} −−
g_{K} + f_{E}g_{K} ++ fg_{K}  + f_{E}g_{K} −−
fg_{E} −+ fg_{E} ++
fg_{IE} ++  fg_{IE} −+
fg_{JE} ++  fg_{JE} +−
fg_{KE} ++  fg_{KE} +−
6 g  + f_{E}g  − fg  −− f_{E}g  +
g_{I}  + f_{E}g_{I} + fg_{I}  +− f_{E}g_{I} −
fg  + fg_{J}  +− f_{E}g_{J} −
fg_{I}  + fg_{K}  +− f_{E}g_{K} −
f_{E}g_{E}  +−
f_{E}g_{IE} −−
f_{E}g_{JE} −−
f_{E}g_{KE} −−
7 g  + f_{E}g  − fg  − f_{E}g  ++
fg_{I}  + fg_{I}  + f_{E}g_{I} −+
f_{E}g  − f_{E}g_{J} −+
f_{E}g_{I}  − f_{E}g_{K} −+
fg_{E}  ++
fg_{IE} −+
fg_{JE} −+
fg_{KE} −+

B Reformulation for physicists

In this appendix, we reformulate our results in a language that may be more familiar to physicists. It is useful first to review some properties of Clifford algebras, in particular those that concern the real Clifford algebras.

B.1 Complex and real Clifford algebras

We use here the terminology of Clifford algebras, spinors and gamma matrices as used in physics. Results for the real case are dependent on the signature. We remark that in the main text Clifford algebras have been taken with a minus sign:

γ^{a}γ^{b} + γ^{b}γ^{a} = −2η^{ab} .  (B.1)

The signature s has been introduced as p − q modulo 8, where p and q are the number of +1, respectively −1 eigenvalues of the metric η^{ab}. Thus,

p = number of negative eigenvalues of (γ^{a}γ^{b} + γ^{b}γ^{a})
q = number of positive eigenvalues of (γ^{a}γ^{b} + γ^{b}γ^{a})
s = p − q mod 8 ,  n = p + q .  (B.2)

This is important in order to interpret Table 1.

The bilinear form β corresponds to the charge conjugation matrix 𝒞, or for spinors s and t, we have β(s, t) = s^{T} 𝒞t. If v = e^{a}, a basis vector, then the operation γ(v) is the gamma matrix γ(v = e^{a}) = γ^{a}. The invariant σ(𝒞) indicates the symmetry of 𝒞, while τ(𝒞) indicates the symmetry of 𝒞γ^{a}:

𝒞^{T} = σ(𝒞) 𝒞 ,  (𝒞γ^{a})^{T} = τ(𝒞) 𝒞γ^{a} .  (B.3)

When we can define chiral spinors, called semi-spinors here, the invariant ι indicates whether the charge conjugation matrix maps between spinors of equal chirality ι(β) = 1 or different chirality ι(β) = −1.

For complex gamma matrices, there are many references, and one can compare e.g. with [VP]. The invariants σ and τ and ι are related to the two numbers ϵ and η of [VP] as

σ(𝒞) = −ϵ ,  τ(𝒞) = −η .  (B.4)

The main results depend on the dimension n. For n odd, there is one charge conjugation matrix (i.e. 1 bilinear form 𝒞) and

n = 1 mod 8 :  σ(𝒞) = 1 ,  τ(𝒞) = 1 ,
n = 3 mod 8 :  σ(𝒞) = −1 , τ(𝒞) = −1 ,
n = 5 mod 8 :  σ(𝒞) = −1 , τ(𝒞) = 1 ,
n = 7 mod 8 :  σ(𝒞) = 1 ,  τ(𝒞) = −1 .  (B.5)

For even n we can define a charge conjugation matrix for either sign of τ. We define

γ_{∗} ≡ (−i)^{n/2+p}γ_{1} … γ_{n} ,  γ_{∗}γ_{∗} = 1 .  (B.6)

Now, if 𝒞 is a good charge conjugation matrix, then 𝒞′ = 𝒞γ_{∗} is a charge conjugation matrix as well, with τ(𝒞′) = −τ(𝒞). The value of σ is

n = 0 mod 8 : σ(𝒞) = 1 , n = 2 mod 8 : σ(𝒞) = τ(𝒞) ,
n = 4 mod 8 : σ(𝒞) = −1 , n = 6 mod 8 : σ(𝒞) = −τ(𝒞) .  (B.7)

Using (1 ± γ_{∗})/2, we can define chiral spinors in this case, and we find

ι(𝒞) =n^{←}≡ (−1)^{n(n−1)/2} .  (B.8)

Here, n^{←} is the sign change on reversing n indices of an antisymmetric tensor.

In this paper we more often make use of real Clifford algebras. Explicit results on real Clifford algebras can be found in [O]. Here we give some key results. Only in the cases s = 0, 6, 7, can the matrices of the complex Clifford algebra (of dimension 2^{[n/2]}, where the Gauss bracket [x] denotes the integer part of x) be chosen to be real. This is called the normal type.

In the other cases, we can get real matrices of dimension twice that of the complex Clifford algebra. Many representations contain only pure real or pure imaginary gamma matrices. A simple way to obtain real matrices of double dimension is to use the matrices

Γ^{a} = γ^{a} ⊗ _{2} if γ^{a} is real ,  Γ^{a} = γ^{a} ⊗ σ_{2} if γ^{a} is imaginary .  (B.9)

In the cases s = 1, 5, called the almost complex type, there is one matrix that commutes with all gamma matrices. It is

J ≡ Γ^{1} … Γ^{n} ,  J^{2} = − .  (B.10)

Note that in the complex case the product of all the γ^{a}’s is proportional to the identity for odd dimensions, but this is not so for these larger and real Γ^{a}’s. (For s = 3, 7 the product of all real gamma matrices is also ± ). In this case there is always a charge conjugation matrix C with^{1}

σ(C) = σ(𝒞) ,  τ(C) = −1 ,  (B.11)

where 𝒞 indicates the charge conjugation matrix for the complex case. There is also a matrix D that satisfies the properties

DΓ^{a} + Γ^{a}D = 0 ,  D^{T} = CDC^{−1} ,  \begin{matrix}D^{2} = & if s = 1 , \\ D^{2} = − & if s = 5 . \end{matrix}  (B.12)

In the remaining cases, s = 2, 3, 4, called the quaternionic type, there are 3 matrices, which commute with all the Γ^{a}’s, denoted E_{i} for i = 1, 2, 3. They satisfy

[E_{i}, Γ^{a}] = 0 ,  E_{i}E_{j} = −δ_{ij} + ε_{ijk}E_{k} ,  E_{i}^{T} = −CE_{i}C^{−1} ,  (B.13)

where a charge conjugation matrix C is used that satisfies

σ(C) = −σ(𝒞) ,  τ(C) = τ(𝒞) .  (B.14)

1In this explanation of properties of real Clifford algebras, we will always denote by C a specific choice of charge conjugation matrix for the real Clifford algebra, and by 𝒞 the one for the complex gamma matrices. In general we use 𝒞 for any choice of charge conjugation matrix.

With these properties, we can obtain the following consequences for bilinear forms.

\underline{s = 0, 6}. We have the normal type. The two charge conjugation matrices of the complex Clifford algebra can be used (possibly multiplied by i to make them real, but an overall factor is not important), having opposite values of τ. For σ one can use (B.7). For s = 0 there is no imaginary factor in (B.6), and thus γ_{∗} is a real matrix that can be used to define real chiral spinors (Majorana-Weyl spinors). The value of ι is then as in the complex case, see (B.8). For s = 6 there is no projection possible in this real case. The fact that the Clifford algebra is real reflects that the irreducible spinors are Majorana spinors.

\underline{s = 7}. The real Clifford representation is also of the normal type. With the odd dimension there is only one charge conjugation matrix, and no chiral projection. The values of σ and τ are as in (B.5). Again, the reality reflects the property of Majorana spinors.

\underline{s = 1, 5}. The real Clifford representation is of the ‘almost complex type’. We have 4 choices for the charge conjugation matrix: C, CJ, CD and CDJ. We can derive from the given properties that

σ(𝒞) = σ(C) = − n^{←} σ(CJ) = σ(CD) =n^{←} σ(CDJ)

τ(C) = τ(CJ) = −1 ,  τ(CD) = τ(CDJ) = 1 .  (B.15)

If s = 1 then (B.12) says that \frac{1}{2}( ± D) are good projection operators, and can be used to define semispinors. These (real) semispinors have the same dimension as the original complex ones and are the Majorana spinors. It is straightforward to check that

ι(C) = ι(D) = 1 ,  ι(CJ) = ι(CDJ) = −1 .  (B.16)

If s = 5 no such projection is possible. The size of the spinor representation is doubled by the procedure (B.9), and this reflects the fact that we have symplectic-Majorana spinors.

\underline{s = 2, 4}. The real Clifford representation is of the quaternionic type. With dimension even, we start from the two charge conjugation matrices of the complex case. For each of them, we can construct 3 extra ones by multiplying with the imaginary units E_{i}, bringing the total to 8 invariant bilinear forms. From (B.13) and (B.14) it follows that

σ(𝒞) = −σ(C) = σ(CE_{i})
τ(𝒞) = τ(C) = τ(CE_{i}) .  (B.17)

The definition of chiral spinors as in the complex case is only possible if γ_{∗} in (B.6) is real. Thus if \frac{1}{2}n + p is even, i.e. s = 4, the product of all the Γ^{a}’s is a good chiral projection operator. The projected spinors are the components of symplectic Majorana-Weyl spinors. If γ_{∗} is imaginary, i.e. s = 2, the product of all the Γ^{a}’s squares to − . Using one of the complex structures, say E_{1}, then gives chiral projections of the form \frac{1}{2}( ±iΓ_{∗}E_{1}). In this case

ι(C) = ι(CE_{1}) = −ι(𝒞) ,  ι(CE_{2}) = ι(CE_{3}) = ι(𝒞) .  (B.18)

The projected spinors are the components of Majorana spinors.

\underline{s = 3}. Here also, the real Clifford algebra is of the quaternionic type, but since the dimension is odd, there is only one charge conjugation matrix in the complex Clifford algebra. For this, (B.17) applies. There is no chiral projection, and the components correspond to symplectic Majorana spinors.

The results can be seen in Table A3 (though the names for the different bilinear forms are unrelated to what has been explained here). Table A1 gives the number of solutions for charge conjugation matrices that have β = 1, β = −1.

The map Γ_{β}^{k} in the main text corresponds to the mapping from two spinors s and t to the form with components s^{T} 𝒞Γ_{a_{1}...a_{k}}t (where 𝒞 denotes now any choice as explained in footnote 1), and the number σ(Γ_{𝒞}^{k}) gives the symmetry of this bispinor (for commuting spinors) under interchange of s and t, while ι(Γ_{𝒞}^{k}) tells whether s and t have the same chirality. They are related to σ(𝒞), τ(𝒞) and ι(𝒞) by (1.4) and (1.5):

σ(Γ_{𝒞}^{k}) = σ(𝒞)τ^{k}(𝒞) k^{←} ,  ι(Γ_{𝒞}^{k}) = (−)^{k}ι(𝒞) .  (B.19)

For real Clifford algebras they are given explicitly in Tables A4– A6.

B.2 Summary of the results for the algebras

This paper treats algebras that consist of an even sector 𝔤_{0} = 𝔰𝔬(p, q) + W_{0}, and an odd sector 𝔤_{1} = W_{1} consisting of a representation of 𝔰𝔬(p, q). The group 𝔰𝔬(p, q) is denoted as 𝔰𝔬(V), and V denotes its vector representation. We consider either the usual case where the odd generators are fermionic (ϵ = 1, and we have a superalgebra), or they can be bosonic (ϵ = −1, and we have a ‘ℤ_{2}-graded Lie algebra’). We will use the word ‘commutator’ in all cases, though this is obviously an anticommutator for [W_{1}, W_{1}] in the superalgebra case. These algebras are called ϵ-extensions of 𝔰𝔬(V). We use the following terminologies for special cases

Poincaré superalgebras or Lie algebras: W_{0} are the translations in n dimensions (n = p + q), which are denoted by V [and thus 𝔤_{0} = 𝔰𝔬(V) + V] and W_{1} is a spinorial representation.

Algebra of translational type: all generators in [W_{1}, W_{1}] belong to W_{0}:

[W_{1}, W_{1}] ⊂ W_{0} ,  [W_{1}, W_{0}] = 0 ,  [W_{0}, W_{0}] = 0 .  (B.20)

This part W_{0} + W_{1} is called the ‘algebra of generalized translations’.

Transalgebra: algebra of translational type where all the generators of W_{0} appear in

[𝔤_{1}, 𝔤_{1}], i.e.

[W_{1}, W_{1}] = W_{0} .  (B.21)

ϵ-extended polyvector Poincaré algebras: Algebra of translational type where W_{1} is a (possibly reducible) spinorial representation (includes chiral and extended supersymmetry).

There are 2 extreme cases: one in which the full 𝔤 is semisimple, which is the case of the Nahm superalgebras, and the algebras of semi-direct type, where 𝔰𝔬(p, q) is its largest semisimple subalgebra.

Apart from degenerate cases where n ≤ 2, any transalgebra is of semi-direct type.

Transalgebras are minimal cases of algebras of translational type in the sense that there are no proper subalgebras, see definition 1. In fact, any algebra of translational type can be written as 𝔤 = 𝔤′ + 𝔞, where 𝔞 ⊂ W_{0} is an 𝔰𝔬(p, q) representation, which is irrelevant in the sense that all its generators commute with all of W_{1} and W_{0} and do not appear in [W_{1}, W_{1}]. The algebra 𝔤′ is a transalgebra.

For any choice of W_{1} there is a unique transalgebra where W_{0} has all the 𝔰𝔬(p, q) representations that appear in the (anti)symmetric product of W_{1} with itself. The (anti)commutators of W_{1} are then

[W_{1}, W_{1}] = ∑_{all r} W_{0}^{(r)} ,  (B.22)

Here r labels all representations that appear in the symmetric product for ϵ = 1, i.e. superalgebras, and in the antisymmetric product for ϵ = −1, i.e. for Lie algebras.

Any other transalgebra can be obtained by removing an arbitrary number of terms in (B.22). We can consider these to be contractions of this basic transalgebra, where the representations to be removed are multiplied by some parameter t and the limit t → 0 is taken.

Any ϵ-extension of semi-direct type with W_{1} irreducible and of dimension at least 3, is of the following form

W_{0} = A + K ,  [K, W_{1}] = 0 ,  [A, W_{1}] = ρW_{1} ,
[𝔰𝔬(V), A] = 0 ,  [A, A] ⊂ K ,  [W_{1}, W_{1}] ⊂ K .  (B.23)

where ρ = 0 or ℝ·Id or ℂ·Id. This is thus a transalgebra iff A = 0. Also, if the algebra is minimal, then [W_{1}, W_{1}] = W_{0} and it is a transalgebra.

Also, if W_{0} and W_{1} are irreducible 𝔰𝔬(p, q) representations, then either the algebra is of translational type, i.e. [W_{0}, W] = 0, or W_{0} is an abelian generator and [W_{0}, W_{1}] = a W_{1}, where a is a number.

We now restrict ourselves to transalgebras where W_{1} = S, the irreducible spinor representation of 𝔰𝔬(p, q). Then the representations that appear in the right-hand side of (B.22) are either k-forms or, in the case that s = p − q is divisible by 4, also (anti- )selfdual (n/2)-forms. Thus, the unique maximal transalgebra has (anti)commutators

[S_{α}, S_{β}] = ∑_{k}(𝒞Γ^{a_{1}...a_{k}})_{αβ}W_{0}^{k} _{a_{1}...a_{k}} ,  (B.24)

where α, β denote spinor indices. The classification of transalgebras with W_{1} = S reduces to the description of all the charge conjugation matrices 𝒞 and the specification of the range of the summation over k. The relevant issue is the symmetry for a particular k, i.e. the σ(Γ_{𝒞}^{k}) of the previous subsection. When there are chiral spinors involved, the chirality should be respected, which is related to ι(Γ_{𝒞}^{k}).

First, in Section 5, the complex case is discussed. That means that there are no reality conditions on bosonic or fermionic generators. When the dimension is odd, the result is given in Theorem 2. There is only one charge conjugation matrix, and the result can be

Superalgebra: σ = +1 Lie algebra: σ = −1
n = 2m + 1 k = m − 4i  k = m − 1 − 4i
k = m − 3 − 4i  k = m − 2 − 4i
n = 2m
[S_{±}, S_{±}] k = m − 4 − 4i k = m − 2 − 4i

k = m (anti)selfdual

[S_{+}, S_{−}]  k = m − 1 − 2i  k = m − 1 − 2i

Table B1: The values of k in (B.24) for the case of complex spinors. n is the dimension of the vector space. i can be 0, 1, … limited by the fact that obviously k ≥ 0. For the even-dimensional case, we split the (anti)commutator between spinors of different and equal chirality. For equal chirality, the k = m generator is either selfdual or antiselfdual.

Superalgebra : σ = +1  Lie algebra : σ = −1

n = 2m + 1

s = 1, 7 (M) k = m − 4i  k = m − 1 − 4i
k = m − 3 − 4i  k = m − 2 − 4i
s = 3, 5 (SM) k = m − 4i  triplet k = m − 4i  singlet
k = m − 3 − 4i  triplet k = m − 3 − 4i  singlet
k = m − 1 − 4i  singlet k = m − 1 − 4i  triplet
k = m − 2 − 4i  singlet k = m − 2 − 4i  triplet

n = 2m s = 0 (MW)

[S_{±}, S_{±}] k = m − 4 − 4i k = m − 2 − 4i

k = m (anti)selfdual

[S_{+}, S_{−}] k = m − 1 − 2i k = m − 1 − 2i
s = 2, 6 (M) k = m − 4i, m + 4 + 4i  k = m − 1 − 4i, m + 3 + 4i
k = m − 3 − 4i, m + 1 + 4i k = m − 2 − 4i, m + 2 + 4i
s = 4 (SMW)
[S_{±}, S_{±}] k = m − 4 − 4i triplet k = m − 2 − 4i triplet
k = m (anti)selfdual triplet k = m (anti)selfdual singlet
k = m − 2 − 4i  singlet k = m − 4 − 4i  singlet
[S_{+}, S_{−}]  k = m − 1 − 2i  2 × 2 k = m − 1 − 2i  2 × 2

Table B2: The values of k in (B.24) for the case of real spinors. n is the dimension of the vector space. i can be 0, 1, … limited by the fact that obviously k ≥ 0 (and k ≤ n for s = 2, 6). In cases where there are real Weyl spinors, we split the (anti)commutator between spinors of different and equal chirality, and the k = m generator is either selfdual or antiselfdual. When there are symplectic spinors, the right-hand side of (B.24) contains for some k’s triplets of the automorphism group 𝔰𝔲(2), and singlets for other k’s. The types of real spinors, Majorana (M), symplectic-Majorana (SM), or symplectic Majorana-Weyl (SMW) are indicated. understood from (B.19) and (B.5). For even dimensions the result is given in Theorem 3. This depends mainly on (B.7) and (B.8). Here the spinors can be split into chiral spinors, and we can separately consider the commutators between spinor generators of the same and of opposite chirality. The result for allowed values of k in (B.24) can be found also in Table B1.

As an example we may check that in 11 dimensions we can indeed have P, Z^{ab} and Z^{a_{1}...a_{5}} generators in W_{0}, as is the case of the M-algebra, and the classification implies that we can consistently put any one of these to zero.

For the case of real generators, it is important to note that (anti)selfdual tensors in even dimensions are only consistent for s/2 even. We now discuss the algebras according to the 8 different values of s. The results are shown in Table B2.

\underline{s = 0} (Majorana-Weyl spinors). There are chiral spinors and we can split the commutators. The k values that appear in Tables A3– A6 with ι = 1 can appear in commutators of equal chirality. The value of σ indicates whether they appear in superalgebras (σ = 1) or in Lie algebras (σ = −1). Those with ι = −1 appear in the same way in commutators of different chirality. The (anti)selfdual tensors appear in the commutators between spinors of the same chirality.

\underline{s = 1} (Majorana spinors). The two projections to semispinors mentioned above (B.16), lead to equivalent spinors. We thus consider only the commutator between these irreducible spinors (including the others is contained in the ‘extended algebras’ discussed below). In Tables A3– A6 we thus consider the ι = 1 cases. We can check that ι = −1 always allows both σ = 1 and σ = −1 as this concerns commutators between unrelated but equivalent spinors.

\underline{s = 2} (Majorana spinors). The two projections to semispinors lead to equivalent spinors. We thus consider only the commutator between these irreducible spinors. Note that in the table we indicate here also forms with k > m. These are dual to k < m forms, and this duality has been used in the formulation of the s = 2 part of Theorem 6. The formulation here shows the gamma matrices completely, e.g. the appearance of Γ_{abc} = ε_{abcd}γ_{5}γ^{d} in 4 dimensions.

\underline{s = 3, s = 5} (Symplectic-Majorana spinors). The symplectic spinors are in a doublet of 𝔰𝔲(2). According to the value of σ for a particular k we find either a triplet or a singlet of generators in the superalgebra or in the Lie algebra.

\underline{s = 4} (Symplectic Majorana-Weyl spinors). In the commutators between generators of equal chirality (which are again doublets of 𝔰𝔲(2)), we find either triplets (symmetric) or singlet (antisymmetric) generators. For commutators between generators of different chirality no symmetry or antisymmetry can be defined, and the generators allowed by the chirality (ι = −1) appear in the superalgebra as well as in the Lie algebra.

\underline{s = 6} (Majorana spinors). This case is straightforward from the tables and the spinors are just real and not projected. Remark that the result is then the same as for the projected ones of s = 2. The same remark about showing tensors with k > m holds here too. These are dualized in the formulation in Corollary 4.

\underline{s = 7} (Majorana spinors). Here also, the tables straightforwardly lead to the same result as for the projected spinors of s = 1.

We remark that the result is the same for s and for −s, which shows that the conventional choices discussed at the beginning of Section B.1 do not influence the final algebras.

Finally, in Section 7, results are obtained for N-extended polyvector Poincaré algebras. This means that W_{1} consists of N copies of the irreducible spinor S. In cases where there are two inequivalent copies (complex even dimensional, or real with s = 0 or s = 4) we have (N_{+}, N_{−})(N_{+}, N_{−})-extended polyvector Poincaré algebras.

The results are straightforward from the above tables and this shows why it has been useful to include the Lie algebra case. The generators in W_{1} are in an N-representation of the automorphism algebra that acts on the copies of S.

For the complex odd-dimensional case and real s = 1, 2, 6, 7 (Majorana): We just have to split the N × N representations into the symmetric and antisymmetric ones.

for superalgebras:  \frac{N(N+1)}{2} copies of the σ = 1 generators
+\frac{N(N−1)}{2} copies of the σ = −1 generators
for Lie algebras:  \frac{N(N−1)}{2} copies of the σ = 1 generators
+\frac{N(N+1)}{2} copies of the σ = −1 generators (B.25)

For the complex even-dimensional case and real s = 0 (Weyl): We have (N_{+}, N_{−}) algebras. We use the above rule separately for the commutators between the N_{+} chiral generators and between the N_{−} antichiral ones. Furthermore there are N_{+}N_{−} copies of the generators that appear in [S_{+}, S_{−}] in Tables B1 and B2. As an example, the (2, 1) superalgebra in 8-dimensional (4,4) space contains: three selfdual 4-forms, and one antiselfdual 4-form, four 0-forms (three in [2S_{+}, 2S_{+}] and one in [S_{−}, S_{−}], one 2-form (in [2S_{+}, 2S_{+}]) and two 3-forms and 1-forms in [2S_{+}, S_{−}].

For the symplectic real case s = 3, 5: The automorphism algebra is already 𝔰𝔭(2) = 𝔰𝔲(2) for the simple algebras discussed above. For the extended algebras it is 𝔰𝔭(N) where N is even. The simple case is thus similar to (B.25) with N = 2, and the ‘triplet’ and ‘singlet’ indications in Table B2 reflect this. Therefore for higher N (always even) we replace in Table B2 the ‘triplet’ by N(N + 1)/2 and the ‘singlet’ by N(N − 1)/2.

For the symplectic Majorana-Weyl case s = 4: We merely need to combine the remarks above for the symplectic case and the Weyl case. Extended algebras are of the form (N_{+}, N_{−}) where both numbers are even. The ‘triplet’ indication in Table B2 is replaced by N_{+}(N_{+} + 1)/2 and N_{−}(N_{−} + 1)/2 and ‘singlet’ is replaced by N_{+}(N_{+} − 1)/2 and N_{−}(N_{−} − 1)/2. The mixed commutators are multiplied by N_{+}N_{−}.

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