Classification of N-(super)-extended Poincaré algebras and bilinear invariants of the spinor

representation of Spin(p, q)

D.V. Alekseevsky^{∗} Max-Planck-Institut

für Mathematik Gottfried-Claren-Str. 26

D-53225 Bonn

V. Cortés^{†}

Mathematical Sciences Research Institute 1000 Centennial Drive Berkeley, CA 94720-5070

September 21, 1998

Abstract

We classify extended Poincaré Lie super algebras and Lie algebras of any signature (p, q), that is Lie super algebras (resp. ℤ_{2}-graded Lie algebras) 𝔤 = 𝔤_{0} + 𝔤_{1}, where 𝔤_{0} = 𝔰𝔬(V) + V is the (generalized) Poincaré Lie algebra of the pseudo Euclidean vector space V = ℝ^{p,q} of signature (p, q) and 𝔤_{1} = S is the spinor 𝔰𝔬(V)-module extended to a 𝔤_{0}-module with kernel V. The remaining super commutators {𝔤_{1}, 𝔤_{1}} (respectively, commutators [𝔤_{1}, 𝔤_{1}]) are defined by an 𝔰𝔬(V)- equivariant linear mapping

∨^{2}𝔤_{1} → V (respectively, ∧^{2}𝔤_{1} → V) .

Denote by P^{+}(n, s) (respectively, P^{−}(n, s)) the vector space of all such Lie super algebras (respectively, Lie algebras), where n = p+q = dim V and s = p − q is the signature. The description of P^{±}(n, s) reduces to the construction of all 𝔰𝔬(V)-invariant bilinear forms on S and to the calculation of three ℤ_{2}-valued invariants for some of them. This calculation is based on a simple explicit model of an irreducible Clifford module S for the Clifford algebra Cl_{p,q} of arbitrary signature (p, q). As a result of the classification, we obtain the numbers L^{±}(n, s) = dim P^{±}(n, s) of independent Lie super algebras and algebras, which take values 0,1,2,3,4 or 6. Due to Bott periodicity, L^{±}(n, s) may be considered as periodic functions with period 8 in each argument. They are invariant under the group Γ generated by the four reflections with respect to the axes n = −2, n = 2, s−1 = −2 and s − 1 = 2. Moreover, the reflection (n, s) → (−n, s) with respect to the axe n = 0 interchanges L^{+} and L^{−} :

∗e-mail: daleksee@mpim-bonn.mpg.de; supported by Max-Planck-Institut für Mathematik (Bonn).

†e-mail: vicente@msri.org or vicente@rhein.iam.uni-bonn.de; supported by the Alexander von Humboldt Foundation, MSRI (Berkeley) and SFB 256 (Bonn University).

L^{+}(−n, s) = L^{−}(n, s) .

Introduction

General relativity is a gauge theory with the Poincaré group P (1, 3) = ℝ^{1,3}⋊Lor(1, 3) of Minkowski space ℝ^{1,3} as gauge group. In N-extended supergravity the N-extended Poincaré supergroup plays the role of (super) gauge group.

The Lie super algebra of this super group for N = 1 is defined as follows: 𝔭^{(1)}(1, 3) = 𝔤 = 𝔤_{0} + 𝔤_{1} = 𝔭(1, 3) + S, where 𝔭(1, 3) = ℝ^{1,3} + 𝔰𝔬(1, 3) is the Poincaré Lie algebra and S = ℂ^{2} is the spinor module of the Lorentz algebra 𝔰𝔬(1, 3) ≅ 𝔰𝔩(2, ℂ) trivially extended to a 𝔭(1, 3)-module. The supercommutator {·, ·} : S ⊗ S → ℝ^{1,3} is defined as projection onto the unique vector submodule V ≅ ℝ^{1,3} in the symmetric square ∨^{2}S.

We remark that in this case there exists also a unique vector submodule in ∧^{2}S, which defines on 𝔭(1, 3) + S the structure of a ℤ_{2}-graded Lie algebra 𝔭^{(−1)}(1, 3).

Our goal is to classify for any pseudo Euclidean space V = ℝ^{p,q} all similar extensions of the (generalized) Poincaré algebra 𝔭(V) = 𝔭(p, q) = ℝ^{p,q} + 𝔰𝔬(p, q) to a super Lie algebra or to a ℤ_{2}-graded Lie algebra. The super Lie algebra extensions of the Poincaré algebra 𝔭(p, q) are the natural gauge algebras for supergravity theories over space times of signature (p, q). Since the time when the classical (i.e. (p, q) = (1, 3)) super Poincaré algebra was discovered [G-L] these (generalized) super Poincaré algebras play a mayor role in many super symmetric field theories, s. e.g [O-S] and [F] for further reference. However, despite the various realizations of particular super Poincaré algebras as infinitesimal symmetries of supergravity theories (for special dimensions and signatures of the space time), a systematic classification, as given in our paper, was missing.

An other motivation to study such extensions is that extended Poincaré Lie algebras are closely related to the full isometry algebra 𝔦𝔰𝔬𝔪(M) of homogeneous quaternionic Kähler manifolds M (s. [dW-V-VP], [A-C1]). In fact, 𝔦𝔰𝔬𝔪(M) = 𝔭+ℝA, where 𝔭 is an extension of the Poincaré algebra 𝔭(3, 3+k) of the pseudo Euclidean space ℝ^{3,3+k} of signature (3, 3 + k), k = −1, 0, 1, …, and A is a derivation of 𝔭 defining a natural gradation.

Definition 0.1 A super Lie algebra (respectively a ℤ_{2}-graded Lie algebra) 𝔤 = 𝔤_{0}+𝔤_{1} is called an N-extended (respectively −N-extended) Poincaré algebra of V = ℝ^{p,q} if the following conditions hold

1) 𝔤_{0} ≅ 𝔭(V).

2) 𝔤_{1} is a sum of N irreducible spinor or semi spinor modules of 𝔭(V) = V + 𝔰𝔬(V) with trivial action of the vector group V.

3) The super bracket {S, S} ⊂ V (respectively Lie bracket [S, S] ⊂ V).

Let S be a 𝔭(V)-module with trivial action of the vector group V. Then defining on 𝔤 = 𝔭(V) + S the structure of a super Lie algebra (respectively of a ℤ_{2}-graded Lie algebra) such that 𝔤_{0} ≅ 𝔭(V), 𝔤_{1} = S and {S, S} ⊂ V (respectively [S, S] ⊂ V) is equivalent to defining an 𝔰𝔬(V)-equivariant mapping j : V^{∗} → ∨^{2}S^{∗} (respectively j : V^{∗} → ∧^{2}S^{∗}). The super bracket (respectively the Lie bracket) is given by j^{∗} : ∨^{2}S → V (respectively j^{∗} : ∧^{2}S → V). Remark that under these assumptions the Jacobi identities are automatically satisfied since [[x, y], z] = 0 for x, y, z ∈ 𝔤_{1}.

We show that the classification of N-extended (N ∈ ℤ) Poincaré algebras easily reduces to the classification of equivariant embeddings V^{∗} ↪ ∨^{2}S^{∗} if N > 0 and V^{∗} ↪ ∧^{2}S^{∗} if N < 0, where V is the vector module and S the spinor module of 𝔰𝔬(V). In other words, we reduce the classification to the cases N = ±1, ±2.

We prove that the following three vector spaces are isomorphic:

1) the space J of 𝔰𝔬(V)-equivariant mappings j : V^{∗} → S^{∗} ⊗ S^{∗},

2) the space M of 𝔰𝔬(V)-equivariant multiplications μ : V^{∗} ⊗ S → S and

3) the space B of 𝔰𝔬(V)-invariant bilinear forms β on S.

Let ρ : V^{∗} ⊗ S → S be the (standard) Clifford multiplication, where we have identified V ≅ V^{∗} using the scalar product on V = ℝ^{p,q}. Then an isomorphism j_{ρ} : B → J is given by

j_{ρ}(β) : v^{∗} ∈ V^{∗} ↦ β ∘ ρ(v^{∗}) = β(ρ(v^{∗})·, ·) ∈ S^{∗} ⊗ S^{∗} .

In particular, the classification of 𝔰𝔬(V)-equivariant mappings V^{∗} → S^{∗}⊗ S^{∗} is equivalent to the classification of 𝔰𝔬(V)-invariant bilinear forms on the spinor module S. The latter amounts to the description of the Schur algebra C of 𝔰𝔬(V)-invariant endomorphisms of S. The structure of C as abstract algebra depends only on the signature s = p − q of ℝ^{p,q} modulo 8; it is a simple real, complex or quaternionic matrix algebra of rank 1 or 2 or a sum of two isomorphic such algebras.

To construct equivariant embeddings of the vector module V^{∗} into the symmetric square ∨^{2}S^{∗} (or into the exterior square ∧^{2}S^{∗}) we introduce the notion of admissible bilinear form β on S and also the corresponding notion of admissible endomorphism of S, which depends on the choice of an admissible bilinear form β.

Definition 0.2 An 𝔰𝔬(V)-invariant bilinear form β on the spinor module S is called admissible if it has the following properties:

1) Clifford multiplication ρ(v) is either β-symmetric or β-skew symmetric. We define the type τ of β to be τ(β) = +1 in the first case and τ(β) = −1 in the second.

2) β is symmetric or skew symmetric. Accordingly, we define the symmetry σ of β to be σ(β) = ±1.

3) If the spinor module is reducible, S = S^{+} + S^{−}, then S^{±} are either mutually orthogonal or isotropic. We put ι(β) = +1 in the first case, ι(β) = −1 in the second and call ι(β) the isotropy of β.

Every admissible form β defines an 𝔰𝔬(V)-equivariant embedding j_{ρ}(β) : V^{∗} → ∨^{2}S^{∗} if τ(β)σ(β) = +1 or j_{ρ}(β) : V^{∗} → ∧^{2}S^{∗} if τ(β)σ(β) = −1. Moreover, if S = S^{+} + S^{−}, then either S^{±} are orthogonal or isotropic for every bilinear form in the image of j_{ρ}(β).

The main part of the paper is the construction of an admissible basis for the space J of equivariant mappings V^{∗} → S^{∗} ⊗ S^{∗}, i.e. a basis consisting of embeddings j_{ρ}(β), where β are admissible bilinear forms on S.

To describe all admissible forms β we make use of very simple explicit models of the irreducible Clifford modules inspired by Ras̆evskiĭ [R]. We prove that the problem reduces to the three fundamental cases V = ℝ^{m,m}, ℝ^{k,0} and ℝ^{0,k} using the isomorphisms Cℓ_{m+k,m} ≅ Cℓ_{m,m} ˆ⊗Cℓ_{k} and Cℓ_{m,m+k} ≅ Cℓ_{m,m} ˆ⊗Cℓ_{0,k} and the algebraic properties of the fundamental invariants τ, σ and ι with respect to ℤ_{2}-graded tensor products.

Moreover, we establish that for every pseudo Euclidean vector space V = ℝ^{p,q} there is a prefered non degenerate 𝔰𝔬(V)-invariant bilinear form h on the spinor module S. This allows us to define canonically the notion of admissible endomorphism of S and the invariants τ, σ and ι for such endomorphisms. They are multiplicative with respect to the composition h ∘ A = h(A·, ·), A ∈ C admissible.

Finally, we explicitly construct in all the cases an admissible basis for the Schur algebra C. This canonically yields admissible bases for the space B of invariant bilinear forms and the space J of equivariant mappings.

This gives an explicit description of all extended Poincaré algebras 𝔤 = 𝔭(V) + S, where S is the spinor module. The super (respectively Lie) brackets ∨^{2}S → V (respectively ∧^{2}S → V) are given as linear combinations of mappings j_{i}^{∗}, where the j_{i} : V^{∗} → ∨^{2}S^{∗} (respectively V^{∗} → ∧^{2}S^{∗}) form an admissible basis for the space of 𝔰𝔬(V)-equivariant mappings V^{∗} → ∨^{2}S^{∗} (respectively V^{∗} → ∧^{2}S^{∗}).

If the spinor module S is an irreducible 𝔰𝔬(V)-module, we obtain all N = ±1 extended Poincaré algebras. If S is reducible, then we obtain all N = ±2 extended Poincaré algebras and using the invariant ι we can determine all N = ±1 extended Poincaré algebras. Sometimes there exist only trivial N = 1 (or N = −1) extended Poincaré algebras, i.e. {S, S} = 0 (or [S, S] = 0).

Given a pseudo Euclidean vector space V = ℝ^{p,q}, let |N| = 1 or 2 denote the number of irreducible summands of the spinor module S of 𝔰𝔬(V). For fixed N = +|N| or N = −|N| we give now the dimension d_{N} of the vector space of N-extended Poincaré algebra structures on 𝔤 = 𝔭(V) + S.

The function d_{N}, which depends only on the signature (p, q), admits a symmetry group Γ generated by reflections. Moreover, there is an additional supersymmetry which relates the dimension L^{+} := d_{+|N|} of the space of super algebras to the dimension L^{−} := d_{−|N|} of the space of Lie algebras.

More precisely: Denote by n = p + q the dimension and by s = p − q the signature of V = ℝ^{p,q} and let L^{+} = L^{+}(n, s) (respectively L^{−}(n, s)) be the maximal number of linearly independent super algebra structures ∨^{2}S → V (respectively Lie algebra structures ∧^{2}S → V) on 𝔤 = 𝔭(V) + S. The functions L^{+} and L^{−} are periodic with period 8 in each argument, hence we may consider them as functions on ℤ^{2} = ℤ × ℤ. The value of the pair (L^{+}, L^{−}) is given in Table 1. It follows from the inspection of this table,

Table 1: The numbers L^{+} of super algebras and L^{−} of Lie algebras 𝔤 = 𝔭(V) + S are given as functions of the dimension n and signature s of V. A fundamental domain for the reflection group Γ is emphasized in bold face. The supersymmetry axis is given by the equation n = 0.

s:  (L^{+}(n, s), L^{−}(n, s))
5  1,3  1,3  3,1  3,1
4 4,4  2,6  4,4  6,2  4,4
3  1,3  1,3  3,1  3,1
2 4,4  2,6  4,4  6,2  4,4
1  1,3  1,3  3,1  3,1
0 1,1  0,2  1,1  2,0  1,1
-1  0,1  0,1  1,0  1,0
-2 1,1  0,2  1,1  2,0  1,1
-3  1,3  1,3  3,1  3,1
n: -4 -3 -2 -1 0 1 2 3 4

that the function (L^{+}, L^{−}) is invariant under the group Γ generated by the reflections with respect to the 4 axes defined by the equations n = −2, n = 2, s′ := s − 1 = −2 and s′ = s − 1 = 2. A fundamental domain F for Γ is

F = {(n, s) ∈ ℤ^{2}| − 2 ≤ n ≤ 2 , −2 ≤ s′ = s − 1 ≤ 2} ∩ G ,

G = {(n, s)|∃(p, q) ∈ ℤ^{2} : n = p+q, s = p−q} = {(n, s) ∈ ℤ^{2}|n+s even} and consists of 12 points. The values of the pair (L^{+}, L^{−}) at these points are typed in bold face in Table 1.

Moreover, the reflection θ with respect to the axis {n = 0}, θ : (n, s) ↦ (−n, s), is a supersymmetry of the pair (L^{+}, L^{−}), that is it interchanges the number of Lie algebras and Lie super algebras:

(L^{+}(+n, s), L^{−}(+n, s)) = (L^{−}(−n, s), L^{+}(−n, s)) .

Short:

L^{±} = L^{∓} ∘ θ

A fundamental domain F̃ for the group Γ̃ =< Γ, θ > is given by

F̃ = {(n, s) = (0, 0), (0, 2), (1, −1), (1, 1), (1, 3), (2, 0), (2, 2)} .

In terms of the coordinates (p, q) a fundamental domain with p ≥ 0 and q ≥ 0 is given by

D̃ = {(p, q) = (2, 0), (1, 1), (3, 0), (2, 1), (1, 2), (3, 1), (2, 2)} .

Acknowledgements The first author is very grateful to Max-Planck-Institut für Mathematik for financial support and hospitality. The second author would like to thank S.-S. Chern and R. Osserman for inviting him to MSRI, where he is now enjoying his stay; he would also like to thank W. Ballmann and U. Hamenstädt for encouragement and support.

Contents

Introduction ii

1 (Super) Extensions of the Poincaré algebra 𝔭(p, q) and Spin(p, q)- equivariant embeddings ℝ^{p,q} ↪ S^{∗} ⊗ S^{∗} 2 1.1 Extending the Poincaré algebra . . . . . . . . . . . . . . . . . 2 1.2 Internal symmetries and charges . . . . . . . . . . . . . . . . . 3 1.3 Reduction of the classification of N-extended Poincaré alge-

bras to the cases N = ±1, ±2 . . . . . . . . . . . . . . . . . . 6 1.4 Equivariant embeddings V^{∗} ↪ S^{∗} ⊗ S^{∗} , modified Clifford

multiplications and Dirac operators . . . . . . . . . . . . . . . 8 1.5 ℤ_{2}-graded type and Schur algebra C . . . . . . . . . . . . . . . 12

2 Fundamental invariants τ , σ and ι and reduction to the basic signatures (m, m), (k, 0) and (0, k) 15 2.1 Fundamental invariants . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Reduction to the basic signatures . . . . . . . . . . . . . . . . 17

3 Case of signature (m, m) and complex case 20 3.1 Signature (m, m) . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Complex case . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Case of positive signature 25 4.1 Case of even dimension . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Case of odd dimension . . . . . . . . . . . . . . . . . . . . . . 29

5 Case of negative signature 32 5.1 Case of even dimension . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Case of odd dimension . . . . . . . . . . . . . . . . . . . . . . 35

6 Complete classification 37

1 (Super) Extensions of the Poincaré algebra 𝔭(p, q) and Spin(p, q)-equivariant embeddings ℝ^{p,q} → S^{∗} ⊗ S^{∗}

1.1 Extending the Poincaré algebra

Let V = ℝ^{p,q} be the pseudo Euclidean space with the metric < x, y > = ∑_{i=1}^{p} x^{i}y^{i} − ∑_{j=p+1}^{p+q} x^{j}y^{j}. We denote by 𝔰𝔬(V) = 𝔰𝔬(p, q) the pseudo orthogonal Lie algebra and by 𝔭(V) = 𝔭(p, q) = 𝔰𝔬(V) + V the semidirect sum of 𝔰𝔬(V) and the Abelian ideal V, it is the Lie algebra of the isometry group of (V, < ·, · >). We call 𝔭(V) the Poincaré algebra of the space V.

Definition 1.1 A ℤ_{2}-graded Lie algebra (respectively a super algebra) 𝔤 = 𝔤_{0} + 𝔤_{1} is called an extension (respectively a super extension) of 𝔭(V) if 𝔤_{0} = 𝔭(V), V is in the kernel of the representation of 𝔤_{0} on 𝔤_{1} and [𝔤_{1}, 𝔤_{1}] ⊂ V (respectively {𝔤_{1}, 𝔤_{1}} ⊂ V).

Remark 1: Sometimes, for unification, we will refer to ℤ_{2}-graded Lie algebras and to super algebras as ϵ-algebras, where ϵ = −1 or +1 respectively. Correspondingly, we will speak of ϵ-extensions.

Proposition 1.1 There exists a natural one-to-one correspondence between extensions (respectively super extensions) of 𝔭(V) up to isomorphisms and equivalence classes of pairs (ρ, π), where

ρ : 𝔰𝔬(V) → 𝔤𝔩(W)

is a representation and

π : ∧^{2}W → V (resp. ∨^{2} W → V)

is a 𝔰𝔬(V)-equivariant linear map from the space of skew symmetric (respectively symmetric) bilinear forms on W^{∗} to the vector module V. Two pairs (ρ, π) and (ρ′, π′) (ρ′ : 𝔰𝔬(V) → 𝔤𝔩(W′)) are equivalent if there exists an automorphism ϕ : 𝔭(V) → 𝔭(V) and a linear map ψ : W → W′ such that the following diagramms are commutative (for pairs of skew symmetric type):

\begin{matrix}𝔰𝔬(V) & & ⟶^{ρ} & 𝔤𝔩(V) \\ ↓ & ϕ̄ & & ↓ & ψ \\ 𝔰𝔬(V) & & ⟶^{ρ′} & 𝔤𝔩(W′) \end{matrix}  \begin{matrix}∧^{2}W & & ⟶^{π} & V \\ ↓ & ψ & & ↓ & ϕ|V \\ ∧^{2}W′ & & ⟶^{π′} & V \end{matrix}

where ϕ̄ is the induced automorphism of 𝔰𝔬(V) = 𝔭(V)/V. For pairs of symmetric type ∧^{2} must be replaced by ∨^{2}.

Proof: Given a pair (ρ, π) of skew symmetric type, we define a ℤ_{2}-graded Lie algebra 𝔤 = 𝔤_{0} + 𝔤_{1}, 𝔤_{0} = 𝔭(V) = 𝔰𝔬(V) + V, 𝔤_{1} = W by

[A, w] = ρ(A)w ,
[w_{1}, w_{2}] = π(w_{1} ∧ w_{2}) ,
[v, w] = 0 ,

where A ∈ 𝔰𝔬(V), v ∈ V and w, w_{1}, w_{2} ∈ W. For a pair of symmetric type we define a super algebra 𝔤 = 𝔤_{0} + 𝔤_{1} by the same formulas replacing only the middle equation by

{w_{1}, w_{2}} = π(w_{1} ∨ w_{2}) .

The Jacobi identity is satisfied because ρ is a representation, π is equivariant and the (anti)commutator of W with W is contained in V and hence commutes with W. The other statements can be checked easily. □

Recall that the spinor representation is the representation of 𝔰𝔬(V) on an irreducible module S of the Clifford algebra Cℓ(V). It is either irreducible or a sum of two irreducible semi spinor modules S^{±}.

Definition 1.2 (cf. Def. 0.1) Let 𝔤 = 𝔤(ρ, π) be an ϵ-extension of 𝔭(V) associated with a pair (ρ, π). We say that 𝔤 is an ϵN-extended Poincaré algebra if ρ is a sum of N = 0, 1, 2, … irreducible spin 1/2 representations, i.e. irreducible spinor or semi spinor representations.

The purpose of this paper is to classify all N-extended (N ∈ ℤ) Poincaré algebras. Before starting this classification we explain how, given a (super) extension of the Poincaré algebra, we can construct more complicated ϵalgebras.

1.2 Internal symmetries and charges

Definition 1.3 Let 𝔤 = 𝔤_{0} + 𝔤_{1} be an ϵ-algebra. An internal symmetry of 𝔤 is an automorphism of 𝔤 which acts trivially on 𝔤_{0}.

Now we give a simple construction which associates with an ϵ-extension 𝔤 = 𝔤(ρ, π) of the Poincaré algebra 𝔭(V) and l ∈ ℕ an ϵ-extension 𝔤^{(+l)} and also a −ϵ-extension 𝔤^{(−2l)} which admit O(l) respectively Sp(2l, ℝ) as internal symmetry groups. We define 𝔤^{(+l)} = 𝔤(ρ^{(+l)}, π^{(+l)}), where

ρ^{(+l)} = lρ : 𝔰𝔬(V) → lW = W ⊗ ℝ^{l} ,
π^{(+l)}(w_{1} ⊗ v_{1}, w_{2} ⊗ v_{2}) = π(w_{1}, w_{2})< v_{1}, v_{2} > ,

< ·, · > is the standard Euclidean scalar product on ℝ^{l}. Similarly, we define

𝔤^{(−2l)} = 2lρ : 𝔰𝔬(V) → 2lW = W ⊗ ℝ^{2l} ,
π^{(−2l)}(w_{1} ⊗ v_{1}, w_{2} ⊗ v_{2}) = π(w_{1}, w_{2})ω(v_{1}, v_{2}) ,

where ω is the standard symplectic form on ℝ^{2l}. Here we have used the convention that π(w_{1}, w_{2}) = π(w_{1} ∨w_{2}) if ϵ = +1 and π(w_{1}, w_{2}) = π(w_{1} ∧w_{2}) if ϵ = −1.

Proposition 1.2 If 𝔤 is an ϵ-extension of the Poincaré algebra 𝔭(V), then 𝔤^{(+l)} is an ϵ-extension and 𝔤^{(−2l)} is a −ϵ-extension. The standard actions of O(l) (respectively Sp(2l, ℝ)) on ℝ^{l} (respectively ℝ^{2l}) are naturally extended to actions on 𝔤^{(+l)} (respectively 𝔤^{(−2l)}) by internal symmetries.

Proof: The first statement follows immediately from Prop. 1.1 and the remark that the bilinear map π^{(+l)} (respectively π^{(−2l)}) has the same (respectively the opposite) symmetry as π. The last statement is immediate. □ Example 1: Applying this construction to an ϵ-extended (s. Def. 1.2) Poincaré algebra, we obtain an ϵl-extended Poincaré algebra and also an −ϵ2l-extended Poincaré algebra with internal symmetry groups O(l) and Sp(2l, ℝ) respectively.

Definition 1.4 A ℤ_{2}-graded Lie algebra (respectively a super algebra) 𝔤 = 𝔤_{0} + 𝔤_{1} is called a charged extension (respectively a charged super extension) of the Poincaré algebra 𝔭(V) if

1) 𝔤_{0} = 𝔭(V) + C is a trivial extension of 𝔭(V), i.e. [C, C] = 0.

2) The action of V + C on the 𝔤_{0}-module W = 𝔤_{1} is trivial.

3) The Lie (respectively super) bracket π : ∧^{2}W → 𝔤_{0} (respectively ∨^{2}W → 𝔤_{0}) is a sum π = π_{V} + π_{C}, where π_{V} : ∧^{2}W → V and π_{C} : ∧^{2}W → C (respectively π_{V} : ∨^{2}W → V and π_{C} : ∨^{2}W → C). In particular, (𝔭(V) + W, π_{V}) is an extension (respectively super extension) of 𝔭(V).

If moreover, [𝔰𝔬(V), C] = 0, and hence [C, 𝔤] = 0, then 𝔤 is called a central charge extension (respectively a central charge super extension) of 𝔭(V).

Let an extension (respectively super extension) 𝔭(V) + W admitting a connected Lie group H of internal symmetries be given. Without restriction of generality we can assume that H is simply connected and we denote the Lie algebra of H by 𝔥. To construct a charged extension (respectively super extension) (𝔭(V) + C) + W preserving the internal symmetry group H it is necessary and sufficient to define an (𝔰𝔬(V) + 𝔥)-equivariant map π_{C} from the exterior (respectively symmetric) square of W to an (𝔰𝔬(V) + 𝔥)-module C. Example 2: Let 𝔭(V) + W be an extension of 𝔭(V). Consider the extension 𝔤^{(+l)} = 𝔭(V)+W ⊗ℝ^{l} with internal symmetry group H = O(l) defined above. Let h ∈ ∨^{2}W^{∗} ⊗ ℝ^{r} be a symmetric 𝔰𝔬(V)-invariant (possibly trivial) vector valued bilinear form on W and η ∈ ∧^{2}W^{∗} ⊗ ℝ^{s} a skew symmetric such form. Define

π_{C} : ∧^{2}(W ⊗ ℝ^{l}) → C = ℝ^{r} ⊗ ∧^{2}ℝ^{l} + ℝ^{s} ⊗ ∨^{2}ℝ^{l} ,
π_{C}(w_{1} ⊗ x_{1}, w_{2} ⊗ x_{2}) = h(w_{1}, w_{2})x_{1} ∧ x_{2} + η(w_{1}, w_{2})x_{1} ∨ x_{2} ,

where w_{1}, w_{2} ∈ W and x_{1}, x_{2} ∈ ℝ^{l}. Then π_{C} defines on (𝔭(V) + C) + W ⊗ ℝ^{l} the structure of central charge extension of 𝔭(V) with symmetry group O(l).

Analogeously, we can define on (𝔭(V) + C) + W ⊗ ℝ^{2l}, C = ℝ^{r} ⊗ ∨^{2}ℝ^{2l} + ℝ^{s} ⊗ ∧^{2}ℝ^{2l}, the structure of central charge super extension of 𝔭(V) with symmetry group Sp(2l, ℝ) by

π_{C} : ∨^{2}(W ⊗ ℝ^{2l}) → C ,
π_{C}(w_{1} ⊗ x_{1}, w_{2} ⊗ x_{2}) = h(w_{1}, w_{2})x_{1} ∨ x_{2} + η(w_{1}, w_{2})x_{1} ∧ x_{2} .

Example 3: Let 𝔭(V)+W be a super extension of 𝔭(V). Consider the super extension 𝔤^{(+l)} = 𝔭(V)+W ⊗ℝ^{l} with internal symmetry group H = O(l) and let h be a symmetric and η a skew symmetric vector valued 𝔰𝔬(V)-invariant bilinear form on W, as above. Define

π_{C} : ∨^{2}(W ⊗ ℝ^{l}) → C = ℝ^{r} ⊗ ∨^{2}ℝ^{l} + ℝ^{s} ⊗ ∧^{2}ℝ^{l} ,
π_{C}(w_{1} ⊗ x_{1}, w_{2} ⊗ x_{2}) = h(w_{1}, w_{2})x_{1} ∨ x_{2} + η(w_{1}, w_{2})x_{1} ∧ x_{2} .

Then π_{C} defines on (𝔭(V)+C)+W ⊗ℝ^{l} the structure of central charge super extension of 𝔭(V) with symmetry group O(l).

Analogeously, we can define on (𝔭(V) + C) + W ⊗ ℝ^{2l}, C = ℝ^{r} ⊗ ∧^{2}ℝ^{2l} + ℝ^{s} ⊗ ∨^{2}ℝ^{2l} the structure of central charge extension of 𝔭(V) with symmetry group Sp(2l, ℝ) by

π_{C} : ∧^{2}(W ⊗ ℝ^{2l}) → C ,
π_{C}(w_{1} ⊗ x_{1}, w_{2} ⊗ x_{2}) = h(w_{1}, w_{2})x_{1} ∧ x_{2} + η(w_{1}, w_{2})x_{1} ∨ x_{2} .

In the physical literature (s. [F]) the expression “central charges” is used for a special case of Example 3.

1.3 Reduction of the classification of N-extended Poincaré algebras to the cases N = ±1, ±2

Let 𝔤 = 𝔤(ρ, π) = 𝔭(V) + W be a ±N-extended Poincaré algebra, N = 1, 2, . . .. Then either the spinor representation ρ_{0} : 𝔰𝔬(V) → 𝔤𝔩(S) is irreducible and ρ = Nρ_{0}, W = NS = S ⊗ ℝ^{N}, or it decomposes into two irreducible subrepresentations ρ_{0} = ρ_{+}+ρ_{−}, S = S^{+}+S^{−} and ρ = N_{+}ρ_{+}+N_{−}ρ_{−}, W = N_{+}S^{+}+N_{−}S^{−} = S^{+}⊗ℝ^{N_{+}} +S^{−}⊗ℝ^{N_{−}}, N = N_{+}+N_{−}. The description of all ϵN-extended Poincaré algebras 𝔤(ρ, π) reduces to the description of all 𝔰𝔬(V)-equivariant mappings π : ∧^{2}W → V if ϵ = −1 and π : ∨^{2}W → V if ϵ = +1. If π ≠ 0, the dual mapping defines an 𝔰𝔬(V)-equivariant embedding π^{∗} : V^{∗} ↪ ∧^{2}W^{∗} if ϵ = −1 or π^{∗} : V^{∗} ↪ ∨^{2}W^{∗} if ϵ = +1. To find all such embeddings it is suficient to determine all submodules isomorphic to V^{∗} in ∧^{2}W^{∗} and ∨^{2}W^{∗} or, equivalently, all vector submodules V in ∧^{2}W and ∨^{2}W. Tables 2 and 3 reduce this problem to the cases N = 1 or 2.

If ρ_{+} and ρ_{−} are equivalent then ρ = N_{+}ρ_{+} + N_{−}ρ_{−} ≅ Nρ_{0}, ρ_{0} ≅ ρ_{±},

∨^{2}W ≅ ∨^{2}S_{0} ⊗ ∨^{2}ℝ^{N} + ∧^{2}S_{0} ⊗ ∧^{2}ℝ^{N} ,
∧^{2}W ≅ ∨^{2}S_{0} ⊗ ∧^{2}ℝ^{N} + ∧^{2}S_{0} ⊗ ∨^{2}ℝ^{N} ,

Table 2: Decomposition of the symmetric square of W

ρ:  Nρ_{0}  N_{+}ρ_{+} + N_{−}ρ_{−}
W:  NS = S ⊗ ℝ^{N}  N_{+}S^{+} + N_{−}S^{−} =

S^{+} ⊗ ℝ^{N_{+}} + S^{−} ⊗ ℝ^{N_{−}}

∨^{2}W ∨^{2}S ⊗ ∨^{2}ℝ^{N} + ∧^{2}S ⊗ ∧^{2}ℝ^{N} ∨^{2}S^{+} ⊗ ∨^{2}ℝ^{N_{+}} + ∨^{2}S^{−} ⊗ ∨^{2}ℝ^{N_{−}}+

∧^{2}S^{+} ⊗ ∧^{2}ℝ^{N_{+}} + ∧^{2}S^{−} ⊗ ∧^{2}ℝ^{N_{−}}+ S^{+} ⊗ S^{−} ⊗ ℝ^{N_{+}N_{−}}

Table 3: Decomposition of the exterior square of W

ρ:  Nρ_{0}  N_{+}ρ_{+} + N_{−}ρ_{−}
W:  NS = S ⊗ ℝ^{N}  N_{+}S^{+} + N_{−}S^{−} =
S^{+} ⊗ ℝ^{N_{+}} + S^{−} ⊗ ℝ^{N_{−}}
∧^{2}W ∧^{2}S ⊗ ∨^{2}ℝ^{N} + ∨^{2}S ⊗ ∧^{2}ℝ^{N} ∧^{2}S^{+} ⊗ ∨^{2}ℝ^{N_{+}} + ∧^{2}S^{−} ⊗ ∨^{2}ℝ^{N_{−}}+

∨^{2}S^{+} ⊗ ∧^{2}ℝ^{N_{+}} + ∨^{2}S^{−} ⊗ ∧^{2}ℝ^{N_{−}}+

S^{+} ⊗ S^{−} ⊗ ℝ^{N_{+}N_{−}}

where S_{0} ≅ S^{±} and N = N_{+} + N_{−}. Table 2 shows that the classification of all equivariant embeddings V ↪ ∨^{2}W (case ϵ = +1) reduces to finding all equivariant embeddings V ↪ ∨^{2}S and V ↪ ∧^{2}S if S is irreducible and equivariant embeddings V ↪ ∨^{2}S^{±}, V ↪ ∧^{2}S^{±} and V ↪ S^{+} ⊗ S^{−} if S = S^{+} + S^{−}. Table 3 shows that the same reduction applies to the case ϵ = −1, i.e. to the problem of finding all equivariant embeddings V ↪ ∧^{2}S. We see that e.g. the classification of N-extended Poincaré algebras for N > 0 (i.e. super algebra extensions) reduces to the classification of N = ±1-extended Poincaré algebras in case there is only one irreducible spin 1/2 representation of 𝔰𝔬(V). The same is true for N < 0, i.e. for Lie algebra extensions.

To illustrate this reduction we consider the case ϵ = +1 and ρ = Nρ_{0} in more detail.

Lemma 1.3 Assume ϵ = +1 and ρ = Nρ_{0}, where ρ_{0} is an irreducible spin 1/2 representation on S_{0}. Then any 𝔰𝔬(V)-equivariant embedding

j : V ↪ ∨^{2}W = ∨^{2}S_{0} ⊗ ∨^{2}ℝ^{N} + ∧^{2}S_{0} ⊗ ∧^{2}ℝ^{N}

is given by

j(v) = ∑_{a} ϕ_{a}(v) ⊗ A_{a} + ∑_{b} ψ_{b}(v) ⊗ B_{b} ,

where ϕ_{a} : V → ∨^{2}S_{0} and ψ_{b} : V → ∧^{2}S_{0} are equivariant embeddings, A_{a} ∈ ∨^{2}ℝ^{N} and B_{b} ∈ ∧^{2}ℝ^{N}.

Proof: Choose bases (A_{a}) and (B_{b}) of ∨^{2}ℝ^{N} and ∧^{2}ℝ^{N} respectively. Then j(v) can be decomposed as above and the coefficients ϕ_{a} and ψ_{b} are equivariant embeddings or zero. □

1.4 Equivariant embeddings V^{∗} ↪ S^{∗} ⊗ S^{∗} , modified Clifford multiplications and Dirac operators

We reduced the problem of the classification of N-extended Poincaré algebras to the description of 𝔰𝔬(V)-equivariant mappings V^{∗} → S^{∗} ⊗ S^{∗}, where S is the spinor module of 𝔰𝔬(V). We will denote by J the vector space of all such mappings.

Now we will show that this space is closely related to two other vector spaces:

• the space B of all 𝔰𝔬(V)-invariant bilinear forms on S and

• the space M of 𝔰𝔬(V)-equivariant multiplications μ : V^{∗} ⊗ S → S.

Denote by C the Schur algebra of 𝔰𝔬(V)-invariant endomorphisms of S. We define two natural anti-representations of C on B and J and also a representation and an anti-representation of C on M by:

ξ_{A}^{B}β = β(A·, ·)
η_{A}^{B}β = β(·, A·)
(ξ_{A}^{J} j)(v^{∗}) = ξ_{A}^{B}(j(v^{∗}))
(η_{A}^{J} j)(v^{∗}) = η_{A}^{B}(j(v^{∗}))
(ξ_{A}^{M}μ)(v^{∗}) = A ∘ μ(v^{∗})
(η_{A}^{M}μ)(v^{∗}) = μ(v^{∗}) ∘ A ,

where A ∈ C, v^{∗} ∈ V^{∗}, β ∈ B, j ∈ J and μ ∈ M ⊂ Hom(V^{∗}, End S). Remark that a non zero equivariant mapping j : V^{∗} → S^{∗} ⊗ S^{∗} is automatically an embedding.

Definition 1.5 An equivariant embedding j : V^{∗} → S^{∗} ⊗ S^{∗} is called non degenerate, if j(V^{∗})S = S^{∗} and j(S) ≅ S, where we consider j as mapping j : S → V ⊗ S^{∗}. An equivariant multiplication μ : V^{∗} ⊗ S → S is called non degenerate, if μ(V^{∗})S = S.

Using the following identifications, we define mappings from two of the spaces B, J and M into the third.

B = (S^{∗} ⊗ S^{∗})^{𝔰𝔬(V)} ,
J = Hom(V^{∗}, S^{∗} ⊗ S^{∗})^{𝔰𝔬(V)} ≅^{(∗)} Hom(S, V^{∗} ⊗ S^{∗})^{𝔰𝔬(V)} ,
M = Hom(V^{∗} ⊗ S, S)^{𝔰𝔬(V)} ≅ Hom(V^{∗}, End S)^{𝔰𝔬(V)}
≅ Hom(V^{∗} ⊗ S^{∗}, S^{∗})^{𝔰𝔬(V)} .

at (∗) we used the metric identification V^{∗} ≅ V. The mappings are defined as follows:

B × M → J
(β, μ) ↦ j(β, μ) = β ∘ μ
j(β, μ)(v^{∗}) = β(μ(v^{∗})·, ·) , v^{∗} ∈ V^{∗};
M × J → B
(μ, j) ↦ β(μ, j) = μ ∘ j ,
β(μ, j)(s, t) = < μ(j(s)), t > , s, t ∈ S;
B × J → M
(β, j) ↦ μ(β, j) = β ∘ j
μ(β, j)(v^{∗}) = β(j(v^{∗})·, ·) ∈ S ⊗ S^{∗} ≅ End S ,

where < ·, · > denotes the natural duality pairing S^{∗} × S → ℝ and for the last mapping we have used that j(v^{∗}) ∈ S^{∗} ⊗ S^{∗} ≅ Hom(S^{∗}, S).

Theorem 1.4 The choice of a non degenerate element β_{0}, j_{0} or μ_{0} in any of the spaces B, J and M defines vector space isomorphisms between the two others:

j_{β_{0}} : M → J
μ ↦ j(β_{0}, μ) = β_{0} ∘ μ ,
μ_{β_{0}} : J → M
j ↦ μ(β_{0}, j) = β_{0} ∘ j;

β_{j_{0}} : M → B
μ ↦ β(μ, j_{0}) = μ ∘ j_{0} ,
μ_{j_{0}} : B → M
β ↦ μ(β, j_{0}) = β ∘ j_{0} ;
j_{μ_{0}} : B → J
β ↦ j(β, μ_{0}) = β ∘ μ_{0} ,
β_{μ_{0}} : J → B
j ↦ β(μ_{0}, j) = μ_{0} ∘ j .

Proof: The statement is trivial for j_{β_{0}} and μ_{β_{0}}, because these mappings amount to “raising and lowering” indices of tensors via the non degenerate form β_{0}.

It is clear that μ_{j_{0}} and j_{μ_{0}} are injective, since j_{0} and μ_{0} are non degenerate. Hence, it is sufficient to prove that β_{j_{0}} and β_{μ_{0}} are injective.

Consider first β_{μ_{0}}(j) = μ_{0} ∘ j, where j : S → V^{∗} ⊗ S^{∗} and μ_{0} : V^{∗} ⊗ S^{∗} → S^{∗}. The kernel of β_{μ_{0}} equals

ker β_{μ_{0}} = {j ∈ J |j(S) ⊂ ker μ_{0}}

If 0 ≠ j ∈ ker β_{μ_{0}}, then ker μ_{0} contains the non trivial submodule j(S). This is impossible, because ker μ_{0} does not contain spin 1/2 submodules. Indeed, after complexification the 𝔰𝔬(V ℂ)-module (V^{∗})ℂ ⊗ (S^{∗})ℂ has the decomposition

(V^{∗})ℂ ⊗ (S^{∗})ℂ = Σ ⊕ (S^{∗})ℂ = (ker μℂ_{0}) ⊕ (S^{∗})ℂ ,

where Σ = ker μℂ_{0} contains only spin 3/2 modules, i.e. Kronecker product of the vector module V ℂ ≅ (V^{∗})ℂ (spin 1) and an irreducible spin 1/2 module.

Consider now β_{j_{0}}(μ) = μ ∘ j_{0}, where j_{0} : S → V^{∗} ⊗ S^{∗} and μ : V^{∗} ⊗ S^{∗} → S^{∗}. As before we have the decomposition (V^{∗})ℂ ⊗ (S^{∗})ℂ = Σ ⊕ (S^{∗})ℂ, where Σ has no submodules isomorphic to submodules of (S^{∗})ℂ. If μ ≠ 0, ker μℂ = Σ⊕Sℂ_{1}, where Sℂ_{1} ≠ (S^{∗})ℂ is a proper submodule of (S^{∗})ℂ. Since j_{0} is non degenerate j_{0}(S) ≅ S cannot be contained in ker μ. □

Lemma 1.5 Let S be the spinor module of 𝔰𝔬(V). There always exists a non degenerate 𝔰𝔬(V)-invariant bilinear form β on S.

Proof: The existence of β is equivalent to the self duality of S, i.e. to the condition S^{∗} ≅ S as 𝔰𝔬(V)-modules.

The self duality of the complex 𝔰𝔬(V ℂ) spinor module 𝕊 follows from the criterion of self duality given in [O-V] p. 195.

Now we discuss the real case. Assume first Sℂ has the same number of irreducible summands as S. Then the self duality of S follows from that of Sℂ, s. [O-V] p. 291. In the opposite case S admits an invariant complex structure J and (S, J) ≅ 𝕊 (complex spinor module of 𝔰𝔬(V ℂ)). Then the real part of a non degenerate complex 𝔰𝔬(V ℂ)-invariant bilinear form on S = 𝕊 gives a real 𝔰𝔬(V)-invariant bilinear form on S and it is easy to check that this form is non degenerate. □

From Theorem 1.4 and this lemma we now derive an important consequence. Recall that by definition the spinor module S is a module over the Clifford algebra Cℓ(V). The restriction of the multiplication mapping Cℓ(V) × S → S to V × S defines a non degenerate 𝔰𝔬(V)-equivariant multiplication ρ : V ⊗ S ≅ V^{∗} ⊗ S → S, which is called Clifford multiplication (as above V and V^{∗} are identified using the pseudo Euclidean scalar product of V). The composition j(β, ρ) = β ∘ ρ with a non degenerate 𝔰𝔬(V)-invariant form β gives a non degenerate 𝔰𝔬(V)-equivariant embedding V^{∗} ↪ S^{∗} ⊗ S^{∗}. Using the lemma and this remark, we obtain the following corollary from Theorem 1.4.

Corollary 1.6 The spaces B of 𝔰𝔬(V)-invariant bilinear forms on S, J of 𝔰𝔬(V)-equivariant mappings V^{∗} → S^{∗} ⊗S^{∗} and M of 𝔰𝔬(V)-equivariant multiplications V^{∗}⊗S → S are isomorphic. In particular, Clifford multiplication ρ defines the isomorphism j_{ρ} : B → J and hence any 𝔰𝔬(V)-equivariant embedding V^{∗} ↪ S^{∗} ⊗ S^{∗} is of the form

j = j_{ρ}(β) : v^{∗} ↦ β(ρ(v^{∗})·, ·) , β ∈ B , v^{∗} ∈ V^{∗} .

Remark 2: Using an 𝔰𝔬(V)-equivariant multiplications μ : V^{∗} ⊗ S → S one can define a Dirac type operator D^{μ} on a pseudo Riemannian spin manifold M as follows. Let μ_{x} : T_{x}^{∗}M ⊗ S_{x} → S_{x} be a field of equivariant multiplications, where S(M) = ∪_{x∈M}S_{x} → M is the spinor bundle. Then

(D^{μ}s)_{x} = μ_{x}(∇s) = μ_{x}(∑_{i} e^{i} ⊗ ∇_{e_{i}}s) ,

where (e_{i}) is a basis of T_{x}M, (e^{i}) the dual basis of T_{x}^{∗}M and ∇ is the spinor connection induced by the Levi Civita connection.

1.5 ℤ_{2}-graded type and Schur algebra C

It is well known (s. [L-M]), that every Clifford algebra Cℓ(V), V = ℝ^{p,q}, is isomorphic to 𝕂(l) or to 2𝕂(l) = 𝕂(l) ⊕ 𝕂(l), where 𝕂(l) is the full matrix algebra over 𝕂 of rank l depending on (p, q) and where 𝕂 = ℝ, ℂ or ℍ.

Definition 1.6 We say that a Clifford algebra Cℓ(V) has type r𝕂, r = 1 or 2, if Cℓ(V) ≅ r𝕂(l) for some l ∈ ℕ.

Recall that the Clifford algebra Cℓ(V) has a natural ℤ_{2}-grading Cℓ(V) = Cℓ^{0}(V) + Cℓ^{1}(V). If V = ℝ^{p,q} (≠ 0), then the even part Cℓ^{0}(V) is isomorphic to the Clifford algebra Cℓ(V′) of V′ = ℝ^{p−1,q} if p ≥ 1 and V′ = ℝ^{q−1} if p = 0. Remark that dim Cℓ^{0}(V) = dim Cℓ(V)/2. By the preceding remarks, the following definition makes sense.

Definition 1.7 The pair t(Cℓ(V)) = (r_{0}𝕂_{0}, r𝕂) = (type Cℓ^{0}(V), type Cℓ(V)) is called the ℤ_{2}-graded type of the Clifford algebra Cℓ(V).

The following proposition describes the periodicity of the type t of the ℤ_{2}-graded Clifford algebras Cℓ_{p,q} = Cℓ(ℝ^{p,q}).

Proposition 1.7 The ℤ_{2}-graded type t_{p,q} = t(Cℓ_{p,q}) depends only on the signature s = p − q modulo 8 and t(s) = t(p − q) = t_{p,q} is given in the table.

\begin{matrix}s & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ t(s) & ℝ, ℂ & ℂ, ℍ & ℍ, 2ℍ & 2ℍ, ℍ & ℍ, ℂ & ℂ, ℝ & ℝ, 2ℝ & 2ℝ, ℝ \end{matrix}

Proof: The proof reduces to the investigation of [L-M] Table II. □

Corollary 1.8 The ℤ_{2}-graded type t_{p,q} = t(s = p − q) is mirror symmetric with respect to the diagonal {p + q = 0}: t_{p,q} = t_{−q,−p}; in other words, t(Cℓ_{p,q}) = t(Cℓ_{8k−q,8k−p}), 8k ≥ p, q.

Moreover, the ℤ_{2}-graded type t_{p,q} = t(s) = (t^{0}(s), t^{1}(s)) is mirror super symmetric with respect to the axis {s = p − q = 3.5}, i.e.

(t^{0}(7 − s), t^{1}(7 − s)) = (t^{1}(s), t^{0}(s)) .

The type rℂ and ℤ_{2}-graded type t_{m} = (r_{0}ℂ, rℂ) of a complex Clifford algebra Cℓ_{m} = Cℓ(ℂ^{m}) are defined by putting V = ℂ^{m} in Definition 1.6 and 1.7, where ℂ^{m} is equipped with a non degenerate (complex) bilinear form, e.g. the standard one: < z, w >= ∑_{j=1}^{m} z_{j}w_{j}, z, w ∈ ℂ^{m}.

Proposition 1.9 The ℤ_{2}-graded type t_{m} = t(ℂℓ_{m}) depends only on the parity of m:

t_{m} = \{ \begin{matrix}(2ℂ, ℂ) if & m is even \\ (ℂ, 2ℂ) if & m is odd \end{matrix}

Let S = S_{p,q} be an irreducible Cℓ_{p,q}-module. Recall that by definition the Schur algebra C = C_{p,q} of S is the algebra of all its 𝔰𝔬(V)-invariant endomorphisms; it is the algebra of endomorphisms which commute with Cℓ_{p,q}^{0}. Analogously, we define the Schur algebra C_{m}^{c} of the complex spinor module 𝕊; it is the algebra of endomorphism of 𝕊 commuting with ℂℓ_{m}^{0}.

Corollary 1.10 The Schur algebra C_{p,q} = C(p − q) depends only on s = p − q modulo 8 and is given in the table. In particular, it admits the mirror symmetry (p, q) ↦ (−q, −p).

\begin{matrix}s & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ C(s) & ℝ(2) & ℂ(2) & ℍ & ℍ ⊕ ℍ & ℍ & ℂ & ℝ & ℝ ⊕ ℝ \end{matrix}

Proof: Remark that if t(Cℓ_{p,q}) = (r_{0}𝕂_{0}, r𝕂) and hence Cℓ_{p,q}^{0} ≅ r_{0} 𝕂_{0}(l_{0}), Cℓ_{p,q} ≅ r𝕂(l), then l is completely determined by l_{0} and vice versa; l = l_{0} or 2l_{0}. This follows from dim Cℓ_{p,q} = 2 dim Cℓ_{p,q}^{0}.

Using this remark, Proposition 1.7 shows that the pair (Cl_{p,q}^{0}, Cl_{p,q}) is isomorphic to one of the following:

(𝕂(l), 𝕂′(l)) , S = 𝕂′^{l} ,
(𝕂(l), 2𝕂(l)) , S = 𝕂^{l} ,
(𝕂′(l), 𝕂(2l)) , S = 𝕂^{2l} ,
(2𝕂(l), 𝕂(2l)) , S = 𝕂^{2l} ,

where 𝕂 = ℝ, ℂ or ℍ and ℝ′ = ℂ, ℂ′ = ℍ.

In the first case the 𝕂(l)-module S = 𝕂′^{l} is a sum of two irreducible equivalent modules S^{±} ≅ 𝕂^{l} and hence the Schur algebra C ≅ 𝕂(2).

In the second (respectively third) case S = 𝕂^{l} (respectively 𝕂^{2l}) is irreducible as 𝕂(l)- (respectively 𝕂′(l)-) module and hence C ≅ 𝕂 (respectively 𝕂′).

In the last case C ≅ 𝕂 ⊕ 𝕂, which follows from the next lemma.

Lemma 1.11 Let S = 𝕂^{2l} be the irreducible module of the algebra 𝕂(2l) and A ≅ 2𝕂(l) a subalgebra of 𝕂(2l), then the A-module S is decomposed into a sum of two nonequivalent submodules S^{±}.

Proof: It is clear that the A-module S is the sum of two irreducible submodules S^{+} and S^{−}. They are not equivalent because A|S^{+} and A|S^{−} have different kernels, namely the two ideals 𝕂(l) ⊂ A. □

Remark that the algebras ℂ ⊕ ℂ and ℍ(2) do not occur as Schur algebras of the real spinor module S.

Corollary 1.12 The Schur algebra C_{m}^{c} of the complex spinor module 𝕊 depends only on the parity of m:

C_{m}^{c} = \{ \begin{matrix}ℂ ⊕ ℂ if & m is even \\ ℂ if & m is odd \end{matrix}

The proof of Corollary 1.10 shows that the structure of the matrix algebra C contains the following information about the Cℓ^{0}(V)-module S.

Proposition 1.13 C is a simple 𝕂-matrix algebra (respectively a sum of two isomorphic 𝕂-matrix algebras) if and only if Cℓ^{0}(V) is a simple 𝕂-matrix algebra (respectively a sum of two isomorphic such algebras). S is an irreducible Cℓ^{0}(V)-module if and only if C ≅ 𝕂 (= ℝ, ℂ or ℍ). S is decomposed into a sum of two equivalent (respectively inequivalent) Cℓ^{0}(V)-modules if and only if C ≅ 𝕂(2) (respectively C ≅ 𝕂 ⊕ 𝕂).

The corresponding statement in the complex case is given for the sake of completeness:

Proposition 1.14 If m is even, then the spinor module 𝕊 = 𝕊_{m} is the sum 𝕊 = 𝕊^{+} + 𝕊^{−} of two inequivalent irreducible ℂℓ_{m}^{0}-modules. In this case, ℂℓ_{m}^{0} and the Schur algebra C_{m}^{c} are the direct sum of two isomorphic simple (complex) matrix algebras.

If m is odd, then the spinor module is an irreducible module of the simple matrix algebra ℂℓ_{m}^{0} and its Schur algebra is also simple.

Since, due to Lemma 1.5, S admits a non degenerate 𝔰𝔬(p, q)-invariant bilinear form, by Schur’s Lemma the dimension b_{p,q} of the space B = B_{p,q} of 𝔰𝔬(p, q)-invariant bilinear forms on S equals

b_{p,q} = dim B_{p,q} = dim C_{p,q} .

Hence we have:

Corollary 1.15 b_{p,q} = b(p−q) is a periodic function of s = p−q with period 8. In particular, it admits the mirror symmetry (p, q) ↦ (−q, −p). Its values are given in the following table.

\begin{matrix}s & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ b(s) & 4 & 8 & 4 & 8 & 4 & 2 & 1 & 2 \end{matrix}

Denote by b_{m} the (complex) dimension of the space of 𝔰𝔬(m, ℂ)-invariant bilinear forms on the complex spinor module 𝕊, then b_{m} = dimℂ C_{m}^{c} and we have:

b_{m} = \{ \begin{matrix}2 if & m is even \\ 1 if & m is odd. \end{matrix}

2 Fundamental invariants τ , σ and ι and reduction to the basic signatures (m, m), (k, 0) and (0, k)

2.1 Fundamental invariants

As before let V denote a pseudo Euclidean vector space and S its spinor module. In Corollary 1.6 we have established that every 𝔰𝔬(V)-equivariant embedding j : V^{∗} ↪ S^{∗} ⊗ S^{∗} is of the form

j = j_{ρ}(β) : v^{∗} ↦ β(ρ(v^{∗})·, ·) , v^{∗} ∈ V^{∗} ,

where ρ is Clifford multiplication and β ∈ B. The dimension of the space B of 𝔰𝔬(V)-invariant bilinear forms on S was given in Corollary 1.15.

Now we will concentrate on a class of bilinear forms β ∈ B for which j_{ρ}(β)V^{∗} ⊂ ∨^{2}S^{∗} or j_{ρ}(β)V^{∗} ⊂ ∧^{2}S^{∗} and define fundamental invariants τ, σ and ι for this class.

Definition 2.1 A bilinear form β on the spinor module S is called admissible if it has the following properties:

1) Clifford multiplication ρ(v), v ∈ V, is either β-symmetric or β-skew symmetric. We define the type τ of β to be τ(β) = +1 in the first case and τ(β) = −1 in the second.

2) The bilinear form β is symmetric or skew symmetric. Accordingly, we define the symmetry σ of β to be σ(β) = ±1.

3) If the spinor module is reducible, S = S^{+} + S^{−}, then S^{±} are either mutually orthogonal or isotropic. We put ι(β) = +1 in the first case, ι(β) = −1 in the second and call ι(β) the isotropy of β.

Due to 1) every admissible form β is 𝔰𝔬(V)-invariant and hence defines an 𝔰𝔬(V)-equivariant embedding j_{ρ}(β) : V ≅ V^{∗} ↪ S^{∗} ⊗ S^{∗}. In addition, j_{ρ}(β)V ⊂ ∨^{2}S^{∗} if τ(β)σ(β) = +1 and j_{ρ}(β)V ⊂ ∧^{2}S^{∗} if τ(β)σ(β) = −1. If S = S^{+}+S^{−}, then for every bilinear form γ ∈ j_{ρ}(β)V the semi spinor modules S^{±} are either γ-isotropic (if ι(γ) = −ι(β) = −1) or mutually γ-orthogonal (if ι(γ) = −ι(β) = +1).

Given an admissible form β ∈ B and A ∈ C the composition β ∘ A = β(A·, ·) ∈ B is in general not admissible. However, if A is β-admissible (see Definition 2.2 below) then β ∘ A is admissible.

Definition 2.2 Let β ∈ B be admissible. An endomorphism A of S is called β-admissible if it has the following properties:

1) Clifford multiplication ρ(v), v ∈ V, either commutes or anticommutes with A. We define the type τ of A to be τ(A) = +1 in the first case and τ(A) = −1 in the second.

2) A is β-symmetric or β-skew symmetric. Accordingly, we define the β-symmetry σ of A to be σ_{β}(A) = ±1.

3) If the spinor module is reducible, S = S^{+} + S^{−}, then either AS^{±} ⊂ S^{±} or AS^{±} ⊂ S^{∓}. We put ι(A) = +1 in the first case, ι(A) = −1 in the second and call ι(A) the isotropy of A.

Due to 1) every β-admissible endomorphism A is 𝔰𝔬(V)-invariant and hence β ∘ A ∈ B. Moreover, β ∘ A is admissible and the fundamental invariants are multiplicative:

τ(β ∘ A) = τ(β)τ(A) ,
σ(β ∘ A) = σ(β)σ(A) ,
ι(β ∘ A) = ι(β)ι(A) .

In section 3.1 (s. Definition 3.1), for every pseudo Euclidean space V, we will construct a canonical non degenerate 𝔰𝔬(V)-invariant bilinear form h on the spinor module S. We will define that an endomorphism A of S is admissible of symmetry σ(A) = ±1, if A is h-admissible and σ_{h}(A) = ±1. Remark 3: The complete classification of admissible forms β ∈ B, which we will give in this paper, implies the following. Let γ ∈ B be non degenerate and admissible. Then a γ-admissible endomorphism A ∈ C is β-admissible for every admissible β ∈ B. In particular, admissibility (i.e. h-admissibility) implies β-admissibility.

2.2 Reduction to the basic signatures

Let V_{1} and V_{2} be pseudo Euclidean spaces and V = V_{1} + V_{2} their orthogonal sum. We recall (s. [L-M] I. Prop. 1.5) that there is a canonical isomorphism of ℤ_{2}-graded algebras

Cℓ(V) ≅ Cℓ(V_{1}) ˆ⊗Cℓ(V_{2}) ,

where ˆ⊗ denotes the ℤ_{2}-graded tensor product of ℤ_{2}-graded algebras.

Proposition 2.1 Let M_{1} = M_{1}^{0} +M_{1}^{1} be a ℤ_{2}-graded Cℓ(V_{1})-module and M_{2} a (not necessarily ℤ_{2}-graded) Cℓ(V_{2})-module. Then M = M_{1} ⊗ M_{2} carries a natural structure of Cℓ(V)-module, V = V_{1} + V_{2}, given by:

(a_{1} ⊗ a_{2})(m_{1} ⊗ m_{2}) = (−1)^{deg(a_{2}) deg(m_{1})}a_{1}m_{1} ⊗ a_{2}m_{2} ,

where a_{i} ∈ Cℓ(V_{i}), m_{i} ∈ M_{i}, i = 1, 2. If M_{2} = M_{2}^{0} + M_{2}^{1} is a ℤ_{2}-graded Cℓ(V_{2})-module, then this formula defines on M the structure of ℤ_{2}-graded Cℓ(V)-module: M^{0} = M_{1}^{0} ⊗ M_{2}^{0} + M_{1}^{1} ⊗ M_{2}^{1}, M^{1} = M_{1}^{0} ⊗ M_{2}^{1} + M_{1}^{1} ⊗ M_{2}^{0}.

Corollary 2.2 Let S_{i} be an irreducible Cℓ(V_{i})-module, i = 1, 2, and assume that S_{1} = S_{1}^{+} + S_{1}^{−} is reducible as Cℓ^{0}(V_{1})-module. Then S = S_{1} ⊗ S_{2} is an irreducible (Cℓ(V) = Cℓ(V_{1}) ˆ⊗Cℓ(V_{2}))-module. The Cℓ^{0}(V)-module S is reducible, S = S^{+} + S^{−}, if and only if S_{2} is reducible as Cℓ^{0}(V_{2})-module, S_{2} = S_{2}^{+} + S_{2}^{−}.

Proof: Let S_{1} be an irreducible Cℓ(V_{1})-module which is reducible as Cℓ^{0}(V_{1})- module and let S_{1}^{+} be an irreducible Cℓ^{0}(V_{1})-submodule. Then

S′_{1} := Cℓ(V_{1}) ⊗_{Cℓ^{0}(V_{1})} S_{1}^{+}

is an irreducible Cℓ(V_{1})-module, hence without restriction of generality S_{1} ≅ S′_{1} as Cℓ(V_{1})-modules. Moreover, S′_{1} is a ℤ_{2}-graded Cℓ(V_{1})-module (s. [L-M] I. Prop. 5.20): S′_{1} = S′_{1}^{0} + S′_{1}^{1}, S′_{1}^{0} = Cℓ^{0}(V_{1}) ⊗_{Cℓ^{0}(V_{1})} S_{1}^{+} ≅ S_{1}^{+} and S′_{1}^{1} = Cℓ^{1}(V_{1})S′_{1}^{0} = Cℓ^{1}(V_{1}) ⊗_{Cℓ^{0}(V_{1})} S_{1}^{+}.

Therefore, we may assume (as usual) that S_{1} = S_{1}^{+} + S_{1}^{−} is a ℤ_{2}-graded Cℓ(V_{1})-module: S_{1}^{0} = S_{1}^{+}, S_{1}^{1} = S_{1}^{−} = Cℓ^{1}(V_{1})S_{1}^{+}, reducing the first statement to Proposition 2.1. The remaining statements also follow from the structure of ℤ_{2}-graded Clifford module on S_{1} and on S_{2} (in the reducible case). □

Now we investigate the algebraic properties of the fundamental invariants with respect to ℤ_{2}-graded tensor products.

Proposition 2.3 Under the assumptions of Corollary 2.2 let β_{i} be admissible bilinear forms on S_{i}, i = 1, 2.

If τ(β_{1}) = ι(β_{1})τ(β_{2}), then β = β_{1} ⊗ β_{2} is admissible and

τ(β) = τ(β_{1}) = ι(β_{1})τ(β_{2}) ,
σ(β) = σ(β_{1})σ(β_{2}) ,
ι(β) = ι(β_{1})ι(β_{2}) ,

where ι(β) and ι(β_{2}) are defined if and only if S_{2} (and hence S) is reducible as module of the even part of the corresponding Clifford algebra.

Let A_{i} be β_{i}-admissible endomorphisms of S_{i}, i = 1, 2. If τ(A_{1}) = ι(A_{1})τ(A_{2}), then A = A_{1} ⊗ A_{2} is admissible and

τ(A) = τ(A_{1}) = ι(A_{1})τ(A_{2}) ,
σ_{β}(A) = σ_{β_{1}}(A_{1})σ_{β_{2}}(A_{2}) ,
ι(A) = ι(A_{1})ι(A_{2}) ,

where ι(A) and ι(A_{2}) are defined if and only if S_{2} is reducible as Cℓ^{0}(V_{2})- module.

Proof: The only non trivial statements are the ones concerning the type τ. For s_{i}, t_{i} ∈ S_{i} and v_{i} ∈ V_{i} we compute:

β((v_{1} ⊗ 1)(s_{1} ⊗ s_{2}), t_{1} ⊗ t_{2}) = β(v_{1}s_{1} ⊗ s_{2}, t_{1} ⊗ t_{2}) =
β_{1}(v_{1}s_{1}, t_{1})β_{2}(s_{2}, t_{2}) = τ(β_{1})β_{1}(s_{1}, v_{1}t_{1})β_{2}(s_{2}, t_{2}) =
τ(β_{1})β(s_{1} ⊗ s_{2}, v_{1}t_{1} ⊗ t_{2}) = τ(β_{1})β(s_{1} ⊗ s_{2}, (v_{1} ⊗ 1)(t_{1} ⊗ t_{2}))

and

β((1 ⊗ v_{2})(s_{1} ⊗ s_{2}), t_{1} ⊗ t_{2}) = (−1)^{deg s_{1}}β(s_{1} ⊗ v_{2}s_{2}, t_{1} ⊗ t_{2}) =
(−1)^{deg s_{1}}β_{1}(s_{1}, t_{1})β_{2}(v_{2}s_{2}, t_{2}) = (−1)^{deg s_{1}}τ(β_{2})β_{1}(s_{1}, t_{1})β_{2}(s_{2}, v_{2}t_{2}) =
(−1)^{deg s_{1}}τ(β_{2})β(s_{1} ⊗ s_{2}, t_{1} ⊗ v_{2}t_{2}) =
(−1)^{deg s_{1}+deg t_{1}}τ(β_{2})β(s_{1} ⊗ s_{2}, (1 ⊗ v_{2})(t_{1} ⊗ t_{2})) .

If ι(β_{1}) = (−1)^{deg s_{1}+deg t_{1}} we obtain

β((1 ⊗ v_{2})(s_{1} ⊗ s_{2}), t_{1} ⊗ t_{2}) = ι(β_{1})τ(β_{2})β(s_{1} ⊗ s_{2}, (1 ⊗ v_{2})(t_{1} ⊗ t_{2})) . (1)

Otherwise, both sides of (1) vanish. Hence, the equation (1) is always true. Similarly we have:

(v_{1} ⊗ 1)((A_{1} ⊗ A_{2})(s_{1} ⊗ s_{2})) = τ(A_{1})(A_{1} ⊗ A_{2})((v_{1} ⊗ 1)(s_{1} ⊗ s_{2}))

and

(1 ⊗ v_{2})((A_{1} ⊗ A_{2})(s_{1} ⊗ s_{2})) = (1 ⊗ v_{2})(A_{1}s_{1} ⊗ A_{2}s_{2}) =
(−1)^{deg(A_{1}s_{1})}A_{1}s_{1} ⊗ v_{2}A_{2}s_{2} = (−1)^{deg(A_{1}s_{1})}τ(A_{2})A_{1}s_{1} ⊗ A_{2}v_{2}s_{2} =
(−1)^{deg(A_{1}s_{1})}τ(A_{2})(A_{1} ⊗ A_{2})(s_{1} ⊗ v_{2}s_{2}) =
(−1)^{deg(A_{1}s_{1})+deg s_{1}}τ(A_{2})(A_{1} ⊗ A_{2})((1 ⊗ v_{2})(s_{1} ⊗ s_{2})) =
ι(A_{1})τ(A_{2})(A_{1} ⊗ A_{2})((1 ⊗ v_{2})(s_{1} ⊗ s_{2})) . □

Now we point out that every pseudo Euclidean space V can be decomposed as orthogonal sum V = V_{1} +V_{2} such that the assumptions of Corollary 2.2 are satisfied, i.e. such that the spinor Cℓ^{0}(V_{1})-module S_{1} is reducible. In fact, we can decompose V into V_{1} = ℝ^{m,m} and V_{2} = ℝ^{k,0} or ℝ^{0,k}.

Proposition 2.4 Let V = V_{1} + V_{2} be the orthogonal sum of the pseudo Euclidean spaces V_{1} = ℝ^{m,m} and V_{2}. Let S_{1} be an irreducible Cℓ(V_{1})-module. Then S_{1} = S_{1}^{+}+S_{1}^{−} is a sum of two inequivalent irreducible Cℓ^{0}(V_{1})-submodules S_{1}^{±} and an irreducible (Cℓ(V) = Cℓ(V_{1}) ˆ⊗Cℓ(V_{2}))-module S is given by S = S_{1} ⊗ S_{2}, where S_{2} is an irreducible Cℓ(V_{2})-module. S is reducible as Cℓ^{0}(V)- module if and only if S_{2} is reducible as Cℓ^{0}(V_{2})-module.

Proof: The first statement follows from the fact that the Schur algebra of S_{1} is C_{m,m} = C(s = m − m = 0) = ℝ ⊕ ℝ. Now all other statements follow immediately from Corollary 2.2. □

3 Case of signature (m, m) and complex case

3.1 Signature (m, m)

Let U and U^{∗} denote two complementary isotropic subspaces of V = ℝ^{m,m}, so V = U + U^{∗}. We denote by < ·, · > the scalar product of V and identify U^{∗} with the dual space to U by

u^{∗}(u) = 2< u, u^{∗} > , u^{∗} ∈ U^{∗} , u ∈ U .

Proposition 3.1 The following formulas define an irreducible Cℓ_{m,m}-module on S = ∧U:

\begin{matrix}ρ(u)s = & u ∧ s \\ ρ(u^{∗})s = & −u^{∗}^{}s̸ , s ∈ ∧U , u ∈ U , u^{∗} ∈ U^{∗} . \end{matrix}

Proof: This follows from the obvious identities ρ(u)^{2} = ρ(u^{∗})^{2} = 0 and ρ(u)ρ(u^{∗}) + ρ(u^{∗})ρ(u) = −2< u, u^{∗} >Id. □

For any a ∈ ∧U and α ∈ ∧U^{∗} we define nilpotent endomorphisms ϵ_{a} and ι_{α} of S = ∧U by:

\begin{matrix}ϵ_{a} = & a ∧ s , \\ ι_{α} = & α^{}s̸ . \end{matrix}

Proposition 3.2 The Lie algebra 𝔰𝔬(m, m) ↪ End S of the spinor group admits the following graded decomposition:

𝔰𝔬(m, m) = 𝔤^{−2} + 𝔤^{0} + 𝔤^{2} = ι_{∧^{2}U^{∗}} + 𝔰𝔩(U) + ϵ_{∧^{2}U} ,

𝔰𝔩(U) = [ι_{U^{∗}}, ϵ_{U}], [𝔤^{i}, 𝔤^{j}] ⊂ 𝔤^{i+j} (𝔤^{i+j} = 0 for |i+j| > 2). In particular, ι_{∧^{2}U^{∗}} and ϵ_{∧^{2}U} are Abelian subalgebras.

It is very easy to describe the semi spinor modules S^{±} in our model of the spinor module S.

Lemma 3.3 S = ∧U is the sum of the two inequivalent irreducible 𝔰𝔬(m, m)- submodules S^{+} = ∧^{ev}U and S^{−} = ∧^{odd}U.

Proof: It is clear that ∧^{ev}U and ∧^{odd}U are irreducible 𝔰𝔬(m, m)-submodules and we already know that they are inequivalent, s. e.g. Proposition 2.4. □ Remark 4: The statement that ∧^{ev}U and ∧^{odd}U are inequivalent 𝔰𝔬(m, m)- modules follows also from the fact that these are eigenspaces of the volume element ω_{m,m} = e_{1} ⋯ e_{2m} ∈ Cℓ_{m,m}^{0}, (e_{i}) an orthonormal basis of ℝ^{m,m}.

We can define an 𝔰𝔬(m, m)-invariant endomorphism E of S by

E|S^{±} = ±Id .

To construct an admissible bilinear form f on S = ∧U we fix a volume form vol ∈ ∧^{m}U on U^{∗} and define

f(∧^{i}U, ∧^{j}U) = 0 , if i + j ≠ m ,
f(s, t)vol = ϵ_{i}s ∧ t , s ∈ ∧^{i}U , t ∈ ∧^{m−i}U ,

where ϵ_{i} = (−1)^{i(i+1)/2}. Remark that ϵ_{i+1} = (−1)^{i+1}ϵ_{i}.

Proposition 3.4 The space B of 𝔰𝔬(m, m)-invariant bilinear forms on S = S_{m,m} is spanned by the admissible elements f and f_{E} = f(E·, ·). Their fundamental invariants (τ, σ, ι) depend only on m (mod 4) and are given in the next table:

\begin{matrix}f & − − − & − − + & − + − & − + + \\ f_{E} & + + − & + − + & + − − & + + + \\ m : & 1 & 2 & 3 & 4 \end{matrix}

An f- and f_{E}-admissible basis for the Schur algebra C ≅ ℝ ⊕ ℝ is given by the endomorphisms Id and E of S:

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

Proof: We first check that ρ(v), v ∈ U + U^{∗}, is f-skew symmetric. For v = u ∈ U, s ∈ ∧^{i}U, t ∈ ∧^{m−i−1}U:

(f(ρ(u)s, t) + f(s, ρ(u)t))vol = ϵ_{i+1}(u ∧ s) ∧ t + ϵ_{i}s ∧ (u ∧ t) = 0 .

For v = u^{∗} ∈ U^{∗}, s ∈ ∧^{i}U, t ∈ ∧^{m−i+1}U:

−(f(ρ(u^{∗})s, t) + f(s, ρ(u^{∗})t))vol = ϵ_{i−1}(u^{∗}^{}s̸) ∧ t + ϵ_{i}s ∧ (u^{∗}^{}t̸) =
ϵ_{i−1}(u^{∗}^{}s̸) ∧ t + ϵ_{i}(−1)^{i}(u^{∗}^{}(s ∧ t) − (u^{∗}^{}s̸) ∧ t) =
(ϵ_{i−1} − (−1)^{i}ϵ_{i})(u^{∗}^{}s̸) ∧ t = 0 .

The symmetry properties of f follow from the computation

f(t, s)vol = ϵ_{j}t ∧ s = ϵ_{j}ϵ_{i}(−1)^{ij}f(s, t)vol = (−1)^{m(m+1)/2}f(s, t)vol ,

where s ∈ ∧^{i}U, t ∈ ∧^{j}U and i + j = m. Finally, f(∧^{ev}U, ∧^{odd}U) = 0 if m is even and f(∧^{ev}U, ∧^{ev}) = f(∧^{odd}U, ∧^{odd}U) = 0 if m is odd. This proves all the statements about f. It is immediate to see that E is f-admissible with fundamental invariants given above. Since f is admissible and E is f-admissible, f_{E} is admissible and its fundamental invariants are computed by multiplicativity:

τ(f_{E}) = τ(f)τ(E) , σ(f_{E}) = σ(f)σ_{f}(E) , ι(f_{E}) = ι(f)ι(E) .

This proves the proposition. □ Proposition 3.4 implies the following theorem:

Theorem 3.5 Every 𝔰𝔬(m, m)-equivariant embedding V^{∗} ↪ S^{∗} ⊗ S^{∗}, S = S_{m,m} the spinor 𝔰𝔬(m, m)-module, is a linear combination of the embeddings j_{ρ}(f) and j_{ρ}(f_{E}). Their image is contained in the dual of the subspaces indicated in the table depending on m (mod 4).

\begin{matrix}j_{ρ}(f) & ∨^{2}S^{+} + ∨^{2}S^{−} & S^{+} ∨ S^{−} & ∧^{2}S^{+} + ∧^{2}S^{−} & S^{+} ∧ S^{−} \\ j_{ρ}(f_{E}) & ∨^{2}S^{+} + ∨^{2}S^{−} & S^{+} ∧ S^{−} & ∧^{2}S^{+} + ∧^{2}S^{−} & S^{+} ∨ S^{−} \\ m & 1 & 2 & 3 & 4 \end{matrix}

Now put V_{1} = ℝ^{m,m} ≠ 0 and let V_{2} be an arbitrary pseudo Euclidean space. Denote the spinor module of 𝔰𝔬(V_{i}) by S_{i}, i = 1, 2.

Proposition 3.6 Let β_{2} be an admissible bilinear form on S_{2}. Then there is a unique (up to scaling) admissible form β_{1} on S_{1} such that τ(β_{2}) = ι(β_{1})τ(β_{1}). In particular, β_{1} ⊗β_{2} is an admissible bilinear form on the spinor 𝔰𝔬(V_{1} + V_{2})-module S_{1} ⊗ S_{2}.

If moreover, A_{2} is a β_{2}-admissible endomorphism of S_{2}, then there is a unique β_{1}-admissible endomorphism A_{1} of S_{1} such that τ(A_{2}) = ι(A_{1})τ(A_{1}), in particular, A_{1} ⊗ A_{2} is a β_{1} ⊗ β_{2}-admissible endomorphism of S_{1} ⊗ S_{2}.

The fundamental invariants of β_{1} ⊗ β_{2} and A_{1} ⊗ A_{2} are easily computed using the rules given in Proposition 2.3.

Proof: This follows from ι(f_{E})τ(f_{E}) = −ι(f)τ(f), ι(E)τ(E) = −ι(Id)τ(Id) and section 2.2. □

If we assume that V_{2} is of definite signature, i.e. V_{2} = ℝ^{k,0} or ℝ^{0,k}, then there is a unique (up to scaling) P in(V_{2})-invariant symmetric bilinear form h_{2} on the irreducible module S_{2} of the compact group P in(V_{2}).

Lemma 3.7 The P in(V_{2})-invariant scalar product h_{2} is admissible: τ(h_{2}) = −1 if V_{2} = ℝ^{k,0} and τ(h_{2}) = +1 if V_{2} = ℝ^{0,k}; σ(h_{2}) = +1 and if S_{2} is reducible, S_{2} = S_{2}^{+} + S_{2}^{−}, S_{2}^{−} = Cℓ^{1}(V_{2})S_{2}^{+}, then ι(h_{2}) = +1.

Proof: Let ρ(v) denote Clifford multiplication by a unit vector v ∈ V_{2}. Then h_{2} is ρ(v)-invariant and ρ(v)^{2} = −Id if V_{2} = ℝ^{k,0} and ρ(v)^{2} = +Id if V_{2} = ℝ^{0,k}. This implies τ(h_{2}) = ∓1.

To see that ι(h_{2}) = +1 in the reducible case, consider the scalar product h′_{2} on S_{2} defined by

h′_{2}(S_{2}^{+}, S_{2}^{−}) = 0 , h′_{2}|S_{2}^{±} = h_{2}|S_{2}^{±} (≠ 0) .

It is easy to check that h′_{2} is invariant under Clifford multiplication by unit vectors v ∈ V_{2} using that S^{−} = vS^{+}. This implies h′_{2} = h_{2}. □

By Proposition 3.6 for every V_{1} = ℝ^{m,m} ≠ 0 there is a unique admissible bilinear form h_{1} on the spinor module S_{1} of 𝔰𝔬(V_{1}) such that τ(h_{2}) = ι(h_{1})τ(h_{1}).

Definition 3.1 The canonical bilinear form on the spinor module S = S_{1} ⊗ S_{2} of 𝔰𝔬(V_{1} + V_{2}) is h = h_{1} ⊗ h_{2}, where h_{2} is the canonical bilinear form on the spinor module S_{2} of 𝔰𝔬(V_{2}) ≅ 𝔰𝔬(k), i.e. the P in(V_{2})-invariant scalar product. In line with this definition we say that an endomorphism A of S (respectively A_{2} of S_{2}) is admissible of symmetry σ(A) = ±1 (respectively σ(A_{2}) = ±1) if A is h-admissible (respectively h_{2}-admissible) and σ_{h}(A) = ±1 (respectively σ_{h_{2}}(A_{2}) = ±1).

Remark 5: For V_{1} = ℝ^{m,m} we have two (non degenerate) admissible bilinear forms f and f_{E} on S_{1} = S_{m,m}. If we want to choose a canonical one, which is not necessary for our purpose, we can consider on S_{1} the structure of irreducible Cℓ_{m,m+1}-module defined in section 3.2. Then only one of the forms remains admissible for the Cℓ_{m,m+1}-module S_{1} = S_{m,m+1}, it is in fact the canonical bilinear form on this module. Moreover, its complex bilinear extension is the unique (up to scaling) 𝔰𝔬(2m+1, ℂ)-invariant complex bilinear form on the irreducible ℂℓ_{2m+1}-module 𝕊_{2m+1} = S_{m,m+1} ⊗ ℂ, s. Corollary 3.12.

3.2 Complex case

Case of even dimension: The following theorem follows immediately from the fact that an irreducible module 𝕊_{2m} of ℂℓ_{2m} can be obtained as 𝕊_{2m} = S_{m,m} ⊗ ℂ and that 𝕊_{2m} splits as ℂℓ_{2m}^{0}-module: 𝕊_{2m} = 𝕊_{2m}^{+} + 𝕊_{2m}^{−}, where 𝕊_{2m}^{±} = S_{m,m}^{±} ⊗ ℂ.

Theorem 3.8 Every 𝔰𝔬(2m, ℂ)-equivariant embedding ℂ^{2m} ↪ 𝕊_{2m} ⊗ 𝕊_{2m} is a linear combination of the embeddings j_{ρ}(f)ℂ and j_{ρ}(f_{E})ℂ. Their image is contained in the dual of the subspaces indicated in the table depending on m (mod 4), where we have put 𝕊 = 𝕊_{2m}.

\begin{matrix}j_{ρ}(f)ℂ & ∨^{2}𝕊^{+} + ∨^{2}𝕊^{−} & 𝕊^{+} ∨ 𝕊^{−} & ∧^{2}𝕊^{+} + ∧^{2}𝕊^{−} & 𝕊^{+} ∧ 𝕊^{−} \\ j_{ρ}(f_{E})ℂ & ∨^{2}𝕊^{+} + ∨^{2}𝕊^{−} & 𝕊^{+} ∧ 𝕊^{−} & ∧^{2}𝕊^{+} + ∧^{2}𝕊^{−} & 𝕊^{+} ∨ 𝕊^{−} \\ m & 1 & 2 & 3 & 4 \end{matrix}

Case of odd dimension: The odd dimensional complex case can be obtained from the real case of signature (m, m + 1) by complexification.

We fix the orthogonal decomposition (ℝ^{m,m+1}, < ·, · >) = ℝe_{0} + ℝ^{m,m}, where < e_{0}, e_{0} > = −1, and denote by ρ the irreducible representation of Cℓ_{m,m} on S_{m,m} constructed in Proposition 3.1.

Proposition 3.9 An irreducible representation ρ̃ of Cℓ_{m,m+1} on S_{m,m+1} = S_{m,m} is defined by

ρ̃|ℝ^{m,m} = ρ|ℝ^{m,m} , ρ̃(e_{0}) = ρ(ω_{m,m}) ,

where ω_{m,m} is the volume element of Cℓ_{m,m}. The Cℓ_{m,m+1}^{0}-module S_{m,m+1} is irreducible and has Schur algebra C_{m,m+1} = ℝ Id.

Proof: It is sufficient to check that {ρ̃(e_{0}), ρ(x)} = 0 for x ∈ ℝ^{m,m} and that ρ̃(e_{0})^{2} = Id. This follows from the next lemma. □

Lemma 3.10 The volume element ω = ω_{m,m} = e_{1}e_{2} ⋯ e_{2m} ((e_{i}) an orthonormal basis of ℝ^{m,m}) of Cℓ_{m,m} satisfies {ω, x} = 0 for all x ∈ ℝ^{m,m} and ω^{2} = +1.

Proposition 3.11 If m is even, then every 𝔰𝔬(m, m + 1)-invariant bilinear form on S = S_{m,m+1} is a multiple of the admissible (canonical) form f_{E} (s. Proposition 3.4) and hence every 𝔰𝔬(m, m + 1)-equivariant embedding ℝ^{m,m+1} ↪ (S ⊗ S)^{∗} is proportional to the embedding jρ̃(f_{E}), which maps ℝ^{m,m+1} into ∨^{2}S^{∗} if m ≡ 0 and into ∧^{2}S^{∗} if m ≡ 2. If m is odd, then every 𝔰𝔬(m, m + 1)-invariant bilinear form on S = S_{m,m+1} is a multiple of the admissible (canonical) form f (s. Proposition 3.4) and hence every 𝔰𝔬(m, m + 1)-equivariant embedding ℝ^{m,m+1} ↪ (S ⊗ S)^{∗} is proportional to the embedding jρ̃(f), which maps ℝ^{m,m+1} into ∨^{2}S^{∗} if m ≡ 1 and into ∧^{2}S^{∗} if m ≡ 3.

Proof: If m is even, then ρ̃(e_{0}) = ρ(ω_{m,m}) is f_{E}-symmetric and τ(f_{E}) = +1. If m is odd, then ρ̃(e_{0}) is f-skew symmetric and τ(f) = −1. □

Corollary 3.12 If m is even, then every 𝔰𝔬(2m + 1, ℂ)-invariant bilinear form on 𝕊 = 𝕊_{2m+1} = S_{m,m+1} ⊗ ℂ is a multiple of the form fℂ_{E} and every 𝔰𝔬(2m+1, ℂ)-equivariant embedding ℂ^{2m+1} ↪ (𝕊⊗𝕊)^{∗} is proportional to the embedding jρ̃(f_{E})ℂ. If m is odd, then every 𝔰𝔬(2m + 1, ℂ)-invariant bilinear form on 𝕊 = 𝕊_{2m+1} = S_{m,m+1} ⊗ ℂ is a multiple of the form fℂ and every 𝔰𝔬(2m+1, ℂ)-equivariant embedding ℂ^{2m+1} ↪ (𝕊⊗𝕊)^{∗} is proportional to the embedding jρ̃(f)ℂ.

4 Case of positive signature

4.1 Case of even dimension

We fix the orthogonal decomposition ℝ^{2m} = ℝ^{m} + ℝ̃^{m}, where ˜ : ℝ^{m} → ℝ̃^{m} is an isometry. Denote by α the involution of Cℓ_{m} (respectively ℂℓ_{m}) extending x ↦ −x on ℝ^{m} (respectively ℂ^{m}).

Proposition 4.1 If m ≡ 0 or 3 (mod 4) the following formulas define on S = S_{2m,0} = Cℓ_{m} the structure of irreducible Cℓ_{2m}-module:

ρ(x)s = xs
ρ(x̃)s = ωsx if m ≡ 0 (mod 4)
ρ(x̃)s = ωα(s)x if m ≡ 3 (mod 4) ,

where x ∈ ℝ^{m}, s ∈ S and ω is the volume element of Cℓ_{m}, i.e. ω = e_{1} ⋯ e_{m} for an orthonormal basis (e_{i}) of ℝ^{m}. The 𝔰𝔬(2m)-module S is the sum S = S^{+} +S^{−} of the two inequivalent irreducible modules S^{+} = Cℓ_{m}^{0} and S^{−} = Cℓ_{m}^{1} if m ≡ 0 (mod 4) and is irreducible if m ≡ 3 (mod 4).

If m ≡ 1 or 2 (mod 4) the structure of irreducible Cℓ_{2m}-module on S = S_{2m,0} = 𝕊_{2m} = ℂℓ_{m} is given by:

ρ(x)s = xs
ρ(x̃)s = iα(s)x , x ∈ ℝ^{m} , s ∈ S .

As 𝔰𝔬(2m)-module S = S^{+} + S^{−} is the sum of the two irreducible modules S^{+} = ℂℓ_{m}^{0} and S^{−} = ℂℓ_{m}^{1}, which are equivalent for m ≡ 1 (mod 4) and inequivalent for m ≡ 2 (mod 4).

Proof: It is sufficient to check the identities ρ(x)^{2} = −< x, x >Id, ρ(x̃)^{2} = −< x, x >Id and {ρ(x), ρ(ỹ)} = 0 for x, y ∈ ℝ^{m}. This is straightforward using the following lemma. □

Lemma 4.2 The volume element ω = ω_{m} = e_{1} ⋯ e_{m} of Cℓ_{m} satisfies {ω, x} = 0 if m is even and [ω, x] = 0 if m is odd, x ∈ ℝ^{m} ⊂ Cℓ_{m}. Moreover,

ω^{2} = \{ \begin{matrix}+1 if m ≡ & 0 or 3 (mod 4) \\ −1 if m ≡ & 1 or 2 (mod 4) . \end{matrix}

Now we describe the P in(2m)-invariant symmetric bilinear form h on S using the canonical identification ∧ℝ^{m} → Cℓ_{m} of Z_{2}-graded vector spaces given by

e_{i_{1}} ∧ … ∧ e_{i_{k}} ↦ e_{i_{1}} ⋯ e_{i_{k}}

with respect to an orthonormal basis (e_{i}), i = 1, …, m, of ℝ^{m}.

The standard scalar product < ·, · > on ∧ℝ^{m} induced by the scalar product on ℝ^{m} is invariant under exterior x∧· and interior x^{}· multiplication with unit vectors x ∈ ℝ^{m}.

Lemma 4.3 Using the identification Cℓ_{m} = ∧ℝ^{m}, Clifford multiplication of x ∈ ℝ^{m} and ϕ ∈ Cℓ_{m} is given by:

xϕ = x ∧ ϕ − x^{}ϕ̸
ϕx = x ∧ α(ϕ) + x^{}α̸(ϕ) .

Proof: The proof is similar to [L-M] I.Prop. 3.9. □

Corollary 4.4 The standard scalar product < ·, · > on ∧ℝ^{m} = Cℓ_{m} is invariant under left and right multiplications by unit vectors x ∈ ℝ^{m}. In particular, if m ≡ 0 or 3 (mod 4), h = < ·, · > is the (admissible) P in(2m)- invariant scalar product on the irreducible Cℓ_{2m}-module S = Cℓ_{m}.

If m ≡ 1 or 2 (mod 4), we extend the standard scalar product on ∧ℝ^{m} to a symmetric complex bilinear form < ·, · >ℂ on S = ∧ℂ^{m}. Using the operator c of complex conjugation, we define a symmetric real bilinear form h = Re < c ·, · >ℂ on S.

Lemma 4.5 Let m ≡ 1 or 2 (mod 4). Then h = Re < c ·, · >ℂ is the (admissible) P in(2m)-invariant scalar product on the irreducible Cℓ_{2m}-module S = ℂℓ_{m}.

Proof: We check that ρ(x) and ρ(x̃), x ∈ ℝ^{m}, are < c ·, · >ℂ-skew symmetric and hence h-skew symmetric. By Corollary 4.4 left and right multiplication, L_{x} and R_{x}, by x ∈ ℝ^{m} are < ·, · >ℂ-skew symmetric endomorphisms of S = ℂℓ_{m}, in particular, ρ(x) is < ·, · >ℂ-skew symmetric. It is easy to see that α and the operator I of multiplication by i are < ·, · >ℂ-symmetric endomorphisms. Moreover,

[I, R_{x}] = [I, α] = {α, R_{x}} = 0

and hence ρ(x̃) = I ∘ R_{x} ∘ α is < ·, · >ℂ-symmetric. From the relations

[c, L_{x}] = [c, R_{x}] = [c, α] = {c, I} = 0

we obtain that [ρ(x), c] = {ρ(x̃), c} = 0, which implies that ρ(x) and ρ(x̃) are < c ·, · >ℂ-skew symmetric. □

Now we construct admissible, i.e. h-admissible, bases of the Schur algebra C = C_{2m,0} for all the values of m (mod 4).

Proposition 4.6 If m ≡ 0 (mod 4), an admissible basis of the Schur algebra C_{2m,0} ≅ ℝ ⊕ ℝ is given by the endomorphisms Id and E = α of S = Cℓ_{m}: τ(E) = −1, σ(E) = σ_{h}(E) = +1, ι(E) = +1.

If m ≡ 3 (mod 4), an admissible basis of C_{2m,0} ≅ ℂ is given by the endomorphisms Id and J = L_{ω} ∘ α of S = Cℓ_{m}: τ(J) = −1, σ(J) = −1.

The space B of 𝔰𝔬(2m)-invariant bilinear forms on S is spanned by admissible elements:

B = span {h, h_{E}} if m ≡ 0 (mod 4) ,
B = span {h, h_{J}} if m ≡ 3 (mod 4) .

The fundamental invariants (τ, σ, ι) are given by (τ, σ, ι)(h) = (−1, +1, +1), (τ, σ, ι)(h_{E}) = (+1, +1, +1) if m ≡ 0 (mod 4) and (τ, σ)(h) = (−1, +1), (τ, σ)(h_{J}) = (+1, −1) if m ≡ 3 (mod 4).

Proof: We show that J is admissible and τ(J) = σ(J) = −1. All other statements are immediate.

Let m ≡ 3 (mod 4). ¿From [L_{x}, L_{ω}] = [R_{x}, L_{ω}] = {L_{x}, α} = {R_{x}, α} = 0 (s. Lemma 4.2) it follows that {L_{x}, J} = {R_{x}, J} = 0. Since ρ(x) = L_{x} and ρ(x̃) = R_{x} ∘ J, we conclude {ρ(x), J} = {ρ(x̃), J} = 0. The operator J is skew symmetric as product of two anticommuting symmetric operators, namely L_{ω} and α (the scalar product is L_{ω}-invariant and L_{ω}^{2} = +Id). □

If m ≡ 1 or 2 (mod 4), we consider the following operators on S = ℂℓ_{m}:

I : s ↦ is , J = L_{ω} ∘ c , K = IJ and E = α ,

where ω = e_{1} ⋯ e_{m} ∈ Cℓ_{m} ⊂ ℂℓ_{m} is the volume element.

Proposition 4.7 Let m ≡ 1 or 2 (mod 4). The Schur algebra C_{2m,0} (≅ ℂ(2) if m ≡ 1 (mod 4) and ≅ ℍ⊕ℍ if m ≡ 2 (mod 4)) is generated by the admissible operators I, J and E satisfying the following (anti) commutator relations:

I^{2} = J^{2} = L_{ω}^{2} = −1 , E^{2} = c^{2} = +1 ,
{I, J} = [I, E] = [I, L_{ω}] = {I, c} = 0 ,
[J, L_{ω}] = [J, c] = [E, c] = [L_{ω}, c] = 0 ,
{J, E} = {L_{ω}, E} = 0 if m ≡ 1 (mod 4) ,

[J, E] = [L_{ω}, E] = 0 if m ≡ 2 (mod 4) .

An admissible basis of the Schur algebra is given by the endomorphisms Id, I, J, K, E, EI, EJ, EK. Their fundamental invariants (τ, σ, ι) are given in the next table, where the value of m is modulo 4.

\begin{matrix}m: & Id & I & J & K & E & EI & EJ & EK \\ 1 & + + + & + − + & + − − & + − − & − + + & − − + & − + − & − + − \\ 2 & + + + & + − + & − − + & − − + & − + + & − − + & + − + & + − + \end{matrix}

The fundamental invariants of the corresponding admissible basis of B are also listed for convenience:

\begin{matrix}m: & h & h_{I} & h_{J} & h_{K} & h_{E} & h_{EI} & h_{EJ} & h_{EK} \\ 1 & − + + & − − + & − − − & − − − & + + + & + − + & + + − & + + − \\ 2 & − + + & − − + & + − + & + − + & + + + & + − + & − − + & − − + \end{matrix}

Proof: The proof is similar to the proof of Proposition 3.4 and 4.6. One uses the multiplication rules for the invariants and also that L_{ω} is skew symmetric, c is symmetric and they commute. □

Theorem 4.8 Every 𝔰𝔬(2m)-equivariant embedding ℝ^{2m} ↪ (S ⊗ S)^{∗}, S = S_{2m,0}, is a linear combination of the embeddings

j_{ρ}(h) : ℝ^{2m} ↪ (S^{+} ∧ S^{−})^{∗} and j_{ρ}(h_{E}) : ℝ^{2m} ↪ (S^{+} ∨ S^{−})^{∗}

if m ≡ 0 (mod 4) and a linear combination of

j_{ρ}(h) : ℝ^{2m} ↪ ∧^{2}S^{∗} and j_{ρ}(h_{J}) : ℝ^{2m} ↪ ∧^{2}S^{∗}

if m ≡ 3 (mod 4).

If m ≡ 1 or 2 (mod 4) every 𝔰𝔬(2m)-equivariant embedding ℝ^{2m} ↪ (S ⊗ S)^{∗} is a linear combination of the embeddings j_{A} = j_{ρ}(h_{A}), A ∈ C admissible, whose image is contained in the dual of the subspaces indicated in Table 4 depending on m (mod 4).

4.2 Case of odd dimension

To reduce the odd dimensional case to the even dimensional, we consider the orthogonal decomposition ℝ^{2m+1} = ℝe_{0} +ℝ^{2m}, where e_{0} is a unit vector. Let ρ denote the irreducible representation of Cℓ_{2m} on S_{2m,0} defined in section 4.1. We will extend ρ to an irreducible representation ρ̃ of Cℓ_{2m+1} on S = S_{2m+1,0}, Table 4: 𝔰𝔬(2m)-equivariant embeddings j_{A} = j_{ρ}(h_{A}) : ℝ^{2m} ↪ (S ⊗ S)^{∗}

\begin{matrix}j_{Id} & S^{+} ∧ S^{−} & S^{+} ∧ S^{−} \\ j_{I} & S^{+} ∨ S^{−} & S^{+} ∨ S^{−} \\ j_{J} & ∨^{2}S^{+} + ∨^{2}S^{−} & S^{+} ∧ S^{−} \\ j_{K} & ∨^{2}S^{+} + ∨^{2}S^{−} & S^{+} ∧ S^{−} \\ j_{E} & S^{+} ∨ S^{−} & S^{+} ∨ S^{−} \\ j_{EI} & S^{+} ∧ S^{−} & S^{+} ∧ S^{−} \\ j_{EJ} & ∨^{2}S^{+} + ∨^{2}S^{−} & S^{+} ∨ S^{−} \\ j_{EK} & ∨^{2}S^{+} + ∨^{2}S^{−} & S^{+} ∨ S^{−} \\ m: & 1 & 2 \end{matrix}

where S_{2m+1,0} = S_{2m,0} if m ≡ 1, 2 or 3 (mod 4) and S_{2m+1,0} = S_{2m,0} ⊗ ℂ = 𝕊_{2m} if m ≡ 0 (mod 4). If m ≡ 1 or 2 (mod 4), S_{2m,0} = 𝕊_{2m} admits the Cℓ_{2m}-invariant complex structure I. For m ≡ 0 (mod 4) multiplication by i is a Cℓ_{2m}-invariant complex structure on S_{2m,0} ⊗ ℂ and will also be denoted by I.

Proposition 4.9 The following formulas define an irreducible representation ρ̃ of Cℓ_{2m+1} on S_{2m+1,0}.

ρ̃|ℝ^{2m} = ρ|ℝ^{2m} ,

ρ̃(e_{0}) = \{ \begin{matrix}ρ(ω_{2m}) if m ≡ & 1 or 3 (mod 4) \\ I ∘ ρ(ω_{2m}) if m ≡ & 0 or 2 (mod 4) , \end{matrix}

where, in the case m ≡ 0 (mod 4), ρ has been extended complex linearly to a representation on S_{2m,0} ⊗ ℂ, denoted by the same symbol. S = S_{2m+1,0} is irreducible as Cℓ_{2m+1}^{0}-module if m ≢ 0 (mod 4) and the sum S = S^{+} + S^{−} of the two equivalent irreducible Cℓ_{2m+1}^{0}-modules S^{+} = S_{2m,0}^{+}+iS_{2m,0}^{−} = Cℓ_{m}^{0}+ iCℓ_{m}^{1} and S^{−} = iS^{+} if m ≡ 0 (mod 4).

Proof: It is sufficient to check that ρ̃(e_{0})^{2} = −Id and {ρ̃(e_{0}), ρ(x)} = 0 for x ∈ ℝ^{2m}, since all other information can be extracted from the Schur algebra, s. Corollary 1.10. These identities follow immediately from Lemma 4.2 and the fact that I is a Cℓ_{2m}-invariant complex structure. □

Now we describe the P in(2m + 1)-invariant scalar product h on S = S_{2m+1,0}. Let h_{2m,0} denote the P in(2m)-invariant scalar product on S_{2m+1,0} = S_{2m,0} if m ≡ 1, 2 or 3 (mod 4) and by hℂ_{2m,0} the complex bilinear extension of the P in(2m)-invariant scalar product on S_{2m,0} to a P in(2m)-invariant complex bilinear form on S_{2m+1,0} = 𝕊_{2m} = S_{2m,0} ⊗ ℂ if m ≡ 4 (mod 4).

Lemma 4.10 The P in(2m + 1)-invariant scalar product h = h_{2m+1,0} on S = S_{2m+1,0} is given by h = h_{2m,0} if m ≡ 1, 2 or 3 (mod 4) and by h = Re hℂ_{2m,0}(c ·, ·) if m ≡ 4 (mod 4), where c is complex conjugation with respect to S_{2m,0} ⊂ S_{2m,0} ⊗ ℂ.

Proof: If m ≢ 4 (mod 4), the statement follows from Schur’s Lemma, since S_{2m+1,0} = S_{2m,0}. If m ≡ 4 (mod 4), the Hermitian form hℂ_{2m,0}(c ·, ·) is I-invariant and hence invariant under ρ̃(e_{0}) = I ∘ ρ(ω_{2m}) and the same is

true for h = Re hℂ_{2m,0}(c ·, ·). □

If m ≢ 3 (mod 4), we have on S_{2m+1,0} = ℂℓ_{m} = Cℓ_{m} + iCℓ_{m} the operator c of complex conjugation. Hence, we can define an endomorphism J of S_{2m+1,0} = ℂℓ_{m} by the formulas

J := \{ \begin{matrix}L_{ω} ∘ c if m ≡ & 1 or 2 (mod 4) \\ α ∘ c if m ≡ & 0 (mod 4) , \end{matrix}

where L_{ω} is left multiplication by the volume element ω = ω_{m} of Cℓ_{m} and α|ℂℓ_{m}^{0} = +Id, α|ℂℓ_{m}^{1} = −Id.

Proposition 4.11 Let m ≢ 3 (mod 4). An admissible basis of the Schur algebra C = C_{2m+1,0} is given by the endomorphisms Id, I, J and K = IJ of S_{2m+1,0} = ℂℓ_{m}. If m ≡ 1 or 2 (mod 4), then I^{2} = J^{2} = −Id, {I, J} = 0 and C_{2m+1,0} ≅ ℍ. If m ≡ 0 (mod 4), then I^{2} = −J^{2} = −Id, {I, J} = 0 and C_{2m+1,0} ≅ ℝ(2). The space B of 𝔰𝔬(2m + 1)-invariant bilinear forms on S_{2m+1,0} has the admissible basis (h, h_{I}, h_{J}, h_{K}). If m ≡ 3 (mod 4), then the Schur algebra C_{2m+1,0} = ℝ Id and B = ℝh.

Proof: straightforward, cf. Proposition 4.6. □

Theorem 4.12 If m ≡ 3 (mod 4), every 𝔰𝔬(2m+1)-equivariant embedding ℝ^{2m+1} ↪ S^{∗} ⊗ S^{∗}, S = S_{2m+1,0}, is a multiple of j_{ρ}(h) : ℝ^{2m+1} ↪ ∧^{2}S^{∗}. If m ≢ 3 (mod 4), every 𝔰𝔬(2m+1)-equivariant embedding ℝ^{2m+1} ↪ (S ⊗S)^{∗} is a linear combination of the embeddings j_{A} = j_{ρ}(h_{A}), A = Id, I, J or K, whose image is contained in the dual of the subspaces indicated in Table 5 depending on m (mod 4).

Table 5: 𝔰𝔬(2m + 1)-equivariant embeddings j_{A} : ℝ^{2m+1} ↪ (S ⊗ S)^{∗}

\begin{matrix}m: & j_{Id} & j_{I} & j_{J} & j_{K} \\ 1 & ∧^{2}S & ∨^{2}S & ∨^{2}S & ∨^{2}S \\ 2 & ∧^{2}S & ∨^{2}S & ∧^{2}S & ∧^{2}S \\ 4 & S^{+} ∧ S^{−} & ∨^{2}S^{+} + ∨^{2}S^{−} & S^{+} ∨ S^{−} & ∨^{2}S^{+} + ∨^{2}S^{−} \end{matrix}

5 Case of negative signature

Now we discuss the case of negative signature. The proofs are similar to the proofs in the case of positive signature and will mostly be omitted.

5.1 Case of even dimension

As in the positively defined case, we fix the orthogonal decomposition ℝ^{0,2m} = ℝ^{0,m} + ℝ̃^{0,m}, where˜ : ℝ^{0,m} → ℝ̃^{0,m} is an isometry.

Lemma 5.1 The volume element ω = ω_{0,m} = e_{1} ⋯ e_{m} ((e_{i}) an orthonormal basis of ℝ^{0,m}) of Cℓ_{0,m} satisfies {ω, x} = 0 if m is even and [ω, x] = 0 if m is odd, x ∈ ℝ^{0,m} ⊂ Cℓ_{0,m}. Moreover,

ω^{2} = \{ \begin{matrix}+1 if m ≡ & 0 or 1 (mod 4) \\ −1 if m ≡ & 2 or 3 (mod 4) . \end{matrix}

The next proposition is checked using Lemma 5.1.

Proposition 5.2 If m ≡ 0 or 1 (mod 4) the following formulas define on S = S_{0,2m} = Cℓ_{0,m} the structure of irreducible Cℓ_{0,2m}-module:

ρ(x)s = xs
ρ(x̃)s = ωsx if m ≡ 0 (mod 4)
ρ(x̃)s = ωα(s)x if m ≡ 1 (mod 4) ,

where x ∈ ℝ^{0,m}, s ∈ S and ω is the volume element of Cℓ_{0,m}. The 𝔰𝔬(0, 2m)- module S is the sum S = S^{+} +S^{−} of the two inequivalent irreducible modules S^{+} = Cℓ_{0,m}^{0} and S^{−} = Cℓ_{0,m}^{1} if m ≡ 0 (mod 4) and is irreducible if m ≡ 1 (mod 4).

If m ≡ 2 or 3 (mod 4) the structure of irreducible Cℓ_{0,2m}-module on S = S_{0,2m} = 𝕊_{2m} = ℂℓ_{m} is given by:

ρ(x)s = xs
ρ(x̃)s = iα(s)x , x ∈ ℝ^{0,m} ⊂ ℂℓ_{m} = Cℓ_{0,m} ⊗ ℂ , s ∈ S = ℂℓ_{m} .

As 𝔰𝔬(0, 2m)-module S = S^{+} + S^{−} is the sum of the two irreducible submodules S^{+} = ℂℓ_{m}^{0} and S^{−} = ℂℓ_{m}^{1}, which are inequivalent for m ≡ 2 (mod 4) and equivalent for m ≡ 3 (mod 4).

Recall (s. Corollary 4.4) that the standard scalar product on ∧ℝ^{m} = Cℓ_{m} = Cℓ_{m,0} is invariant under left and right multiplications by unit vectors x ∈ ℝ^{m} = ℝ^{m,0}. We can consider ℝ^{0,m} as subspace

ℝ^{0,m} = iℝ^{m} ⊂ ℂℓ_{m} = Cℓ_{m} ⊗ ℂ = Cℓ_{m} + iCℓ_{m} .

Then Cℓ_{0,m} = Cℓ_{0,m}^{0} + Cℓ_{0,m}^{1} = Cℓ_{m}^{0} + iCℓ_{m}^{1}. We define an isomorphism of ℤ_{2}-graded vector spaces φ : Cℓ_{m} → Cℓ_{0,m} on elements a ∈ Cℓ_{m} of pure degree deg(a) = 0 or 1 by:

a ↦ i^{deg(a)}a .

A scalar product < ·, · > on Cℓ_{0,m} is defined by the condition that φ : Cℓ_{m} → Cℓ_{0,m} is an isometry for the standard scalar product on ∧ℝ^{m} = Cℓ_{m}. The following lemma is true by construction.

Lemma 5.3 The scalar product < ·, · > on Cℓ_{0,m} is invariant under left and right multiplications by unit vectors x ∈ ℝ^{0,m}. In particular, if m ≡ 0 or 1 (mod 4), h = < ·, · > is the (admissible) P in(0, 2m)-invariant scalar product on the irreducible Cℓ_{0,2m}-module S = S_{0,2m} = Cℓ_{0,m}.

If m ≡ 2 or 3 (mod 4), we extend the scalar product < ·, · > on Cℓ_{0,m} to a symmetric complex bilinear form < ·, · >ℂ on S = ∧ℂ^{m}. Using the operator c = c_{0,m} of complex conjugation with respect to the real form Cℓ_{0,m} = Cℓ_{m}^{0} + iCℓ_{m}^{1} of ℂℓ_{m}, we define a (real) scalar product h = Re < c ·, · >ℂ on S.

Lemma 5.4 Let m ≡ 2 or 3 (mod 4). Then h = Re < c ·, · >ℂ is the (admissible) P in(0, 2m)-invariant scalar product on the irreducible Cℓ_{0,2m}- module S = ℂℓ_{m}.

Now we construct (h-)admissible bases of the Schur algebra C = C_{0,2m} for all the values of m (mod 4).

Proposition 5.5 If m ≡ 0 (mod 4), an admissible basis of the Schur algebra C_{0,2m} ≅ ℝ⊕ℝ is given by the endomorphisms Id and E = α of S = Cℓ_{0,m}: τ(E) = −1, σ(E) = σ_{h}(E) = +1, ι(E) = +1.

If m ≡ 1 (mod 4), an admissible basis of C_{0,2m} ≅ ℂ is given by the endomorphisms Id and J = L_{ω} ∘ α of S = Cℓ_{0,m} (where ω is a volume element of Cℓ_{0,m}): τ(J) = −1, σ(J) = −1.

The space B of 𝔰𝔬(0, 2m)-invariant bilinear forms on S is spanned by the admissible elements h and h_{E} if m ≡ 0 (mod 4) and by h and h_{J} if m ≡ 1 (mod 4). Their fundamental invariants (τ, σ, ι) are (τ, σ, ι)(h) = (+1, +1, +1), (τ, σ, ι)(h_{E}) = (−1, +1, +1) if m ≡ 0 (mod 4) and (τ, σ)(h) = (+1, +1), (τ, σ)(h_{J}) = (−1, −1) if m ≡ 1 (mod 4).

If m ≡ 2 or 3 (mod 4), we consider the following operators on S = ℂℓ_{m}:

I : s ↦ is , J = L_{ω} ∘ c , K = IJ and E = α (ω = ω_{0,m}) .

Proposition 5.6 Let m ≡ 2 or 3 (mod 4). The Schur algebra C_{0,2m} (≅ ℍ ⊕ ℍ if m ≡ 2 (mod 4) and ≅ ℂ(2) if m ≡ 3 (mod 4)) is generated by the admissible operators I, J and E, which satisfy the following identities:

I^{2} = J^{2} = L_{ω}^{2} = −1 , E^{2} = c^{2} = +1 ,
{I, J} = [I, E] = [I, L_{ω}] = {I, c} = 0 ,
[J, L_{ω}] = [J, c] = [E, c] = [L_{ω}, c] = 0 ,
[J, E] = [L_{ω}, E] = 0 if m ≡ 2 (mod 4) ,
{J, E} = {L_{ω}, E} = 0 if m ≡ 3 (mod 4) .

An admissible basis of the Schur algebra is given by the endomorphisms Id, I, J, K, E, EI, EJ, EK. Their fundamental invariants (τ, σ, ι) are given in the next table, where the value of m is modulo 4.

\begin{matrix}m: & Id & I & J & K & E & EI & EJ & EK \\ 2 & + + + & + − + & − − + & − − + & − + + & − − + & + − + & + − + \\ 3 & + + + & + − + & + − − & + − − & − + + & − − + & − + − & − + − \end{matrix}

The fundamental invariants of the corresponding admissible basis for the space B = B_{0,2m} (of 𝔰𝔬(0, 2m)-invariant bilinear forms on S_{0,2m}) are as follows:

\begin{matrix}m: & h & h_{I} & h_{J} & h_{K} & h_{E} & h_{EI} & h_{EJ} & h_{EK} \\ 2 & + + + & + − + & − − + & − − + & − + + & − − + & + − + & + − + \\ 3 & + + + & + − + & + − − & + − − & − + + & − − + & − + − & − + − \end{matrix}

Theorem 5.7 Every 𝔰𝔬(0, 2m)-equivariant embedding ℝ^{0,2m} ↪ (S ⊗ S)^{∗}, S = S_{0,2m}, is a linear combination of the embeddings

j_{ρ}(h) : ℝ^{0,2m} ↪ (S^{+} ∨ S^{−})^{∗} and j_{ρ}(h_{E}) : ℝ^{0,2m} ↪ (S^{+} ∧ S^{−})^{∗}

if m ≡ 0 (mod 4) and a linear combination of

j_{ρ}(h) and j_{ρ}(h_{J}) : ℝ^{0,2m} ↪ ∨^{2}S^{∗} if m ≡ 1 (mod 4) .

If m ≡ 2 or 3 (mod 4) every 𝔰𝔬(0, 2m)-equivariant embedding ℝ^{0,2m} ↪ (S⊗S)^{∗} is a linear combination of the embeddings j_{A} = j_{ρ}(h_{A}), A ∈ C = C_{0,2m} admissible, whose image is contained in the dual of the subspaces indicated in Table 6 depending on m (mod 4).

Table 6: 𝔰𝔬(0, 2m)-equivariant embeddings j_{A} : ℝ^{0,2m} ↪ (S ⊗ S)^{∗}

\begin{matrix}j_{Id} & S^{+} ∨ S^{−} & S^{+} ∨ S^{−} \\ j_{I} & S^{+} ∧ S^{−} & S^{+} ∧ S^{−} \\ j_{J} & S^{+} ∨ S^{−} & ∧^{2}S^{+} + ∧^{2}S^{−} \\ j_{K} & S^{+} ∨ S^{−} & ∧^{2}S^{+} + ∧^{2}S^{−} \\ j_{E} & S^{+} ∧ S^{−} & S^{+} ∧ S^{−} \\ j_{EI} & S^{+} ∨ S^{−} & S^{+} ∨ S^{−} \\ j_{EJ} & S^{+} ∧ S^{−} & ∧^{2}S^{+} + ∧^{2}S^{−} \\ j_{EK} & S^{+} ∧ S^{−} & ∧^{2}S^{+} + ∧^{2}S^{−} \\ m: & 2 & 3 \end{matrix}

5.2 Case of odd dimension

Consider the orthogonal decomposition (ℝ^{0,2m+1}, < ·, · >) = ℝe_{0} + ℝ^{0,2m}, where < e_{0}, e_{0} > = −1. Let ρ denote the irreducible representation of Cℓ_{0,2m} on S_{0,2m} defined in section 5.1. We will extend ρ to an irreducible representation ρ̃ of Cℓ_{0,2m+1} on S = S_{0,2m+1}, where S_{0,2m+1} = S_{0,2m} if m ≡ 0, 2 or 3 (mod 4) and S_{0,2m+1} = S_{0,2m} ⊗ ℂ = 𝕊_{2m} if m ≡ 1 (mod 4). If m ≡ 2 or 3 (mod 4), S_{0,2m} = 𝕊_{2m} admits the Cℓ_{0,2m}-invariant complex structure I. For m ≡ 1 (mod 4) multiplication by i is a Cℓ_{0,2m}-invariant complex structure on S_{0,2m} ⊗ ℂ and will also be denoted by I.

Proposition 5.8 The following formulas define an irreducible representation ρ̃ of Cℓ_{0,2m+1} on S_{0,2m+1}.

ρ̃|ℝ^{0,2m} = ρ|ℝ^{0,2m} ,

ρ̃(e_{0}) = \{ \begin{matrix}ρ(ω_{0,2m}) if m ≡ & 0 or 2 (mod 4) \\ I ∘ ρ(ω_{0,2m}) if m ≡ & 1 or 3 (mod 4) , \end{matrix}

where, in the case m ≡ 1 (mod 4), ρ has been extended complex linearly to a representation on S_{0,2m+1} = S_{0,2m} ⊗ ℂ. S = S_{0,2m+1} is irreducible as Cℓ_{0,2m+1}^{0}-module if m ≢ 3 (mod 4) and the sum S = S^{+} + S^{−} of the two equivalent irreducible Cℓ_{0,2m+1}^{0}-modules S^{+} = SĴ and S^{−} = iSĴ if m ≡ 3 (mod 4), where SĴ is the fixed point set of a 𝔰𝔬(0, 2m + 1)-invariant real structure Ĵ on S (the explicit expression for Ĵ will be given below).

Next we describe the P in(0, 2m+1)-invariant scalar product h = h_{0,2m+1} on S = S_{0,2m+1}. Let h_{0,2m} denote the P in(0, 2m)-invariant scalar product on S_{0,2m+1} = S_{0,2m} if m ≡ 0, 2 or 3 (mod 4) and by hℂ_{0,2m} the complex bilinear extension of the P in(0, 2m)-invariant scalar product on S_{0,2m} to a P in(0, 2m)-invariant complex bilinear form on S_{0,2m+1} = 𝕊_{2m} = S_{0,2m} ⊗ ℂ if m ≡ 1 (mod 4).

Lemma 5.9 The P in(0, 2m + 1)-invariant scalar product h = h_{0,2m+1} on S = S_{0,2m+1} is given by h = h_{0,2m} if m ≡ 0, 2 or 3 (mod 4) and by h = Re hℂ_{0,2m}(c ·, ·) if m ≡ 1 (mod 4), where c is complex conjugation with respect to S_{0,2m} ⊂ S_{0,2m} ⊗ ℂ.

If m ≢ 0 (mod 4), we have on S_{0,2m+1} = ℂℓ_{m} = Cℓ_{0,m} + iCℓ_{0,m} the operator c = c_{0,m} of complex conjugation. Using it we define an endomorphism Ĵ of S_{0,2m+1} = ℂℓ_{m} by

Ĵ:= L_{ω} ∘ α ∘ c ,

where ω = ω_{0,m} is a volume element of Cℓ_{0,m} and α|ℂℓ_{m}^{0} = +Id, α|ℂℓ_{m}^{1} = −Id.

Proposition 5.10 Let m ≢ 0 (mod 4). The Schur algebra C = C_{0,2m+1} is generated by the endomorphisms I and Ĵ of S = S_{0,2m+1} = ℂℓ_{m}, which satisfy the following relations: I^{2} = −1, {I, Ĵ} = 0. Moreover, Ĵ^{2} = +Id and C_{0,2m+1} ≅ ℝ(2) if m ≡ 3 (mod 4) and Ĵ^{2} = −Id and C_{0,2m+1} ≅ ℍ if m ≡ 1 or 2 (mod 4). An admissible basis of C_{0,2m+1} is given by the endomorphisms Id, I, Ĵ and K̂ = IĴ. Their fundamental invariants (τ, σ, ι) together with the invariants of the associated admissible basis for the space B of 𝔰𝔬(0, 2m + 1)-invariant bilinear forms are given in the Table 7 (ι is only defined if m ≡ 3 (mod 4)). If m ≡ 0 (mod 4), C_{0,2m+1} = ℝId.

Table 7: Fundamental invariants of admissible endomorphisms and bilinear forms of S_{0,2m+1}

\begin{matrix}m: & Id & I & Ĵ & K̂ & h & h_{I} & hĴ & hK̂ \\ 1 & ++ & +− & −− & −− & ++ & +− & −− & −− \\ 2 & ++ & +− & +− & +− & ++ & +− & +− & +− \\ 3 & + + + & + − − & − + + & − + − & + + + & + − − & − + + & − + − \end{matrix}

Theorem 5.11 Every 𝔰𝔬(0, 2m + 1)-equivariant embedding ℝ^{0,2m+1} ↪ (S ⊗ S)^{∗} is proportional to j_{ρ}(h) : ℝ^{0,2m+} ↪ ∨^{2}S^{∗} if m ≡ 0 (mod 4) and a linear combination of the embeddings j_{A} = j_{ρ}(h_{A}), A = Id, I, Ĵ and K̂ if m ≢ 0 (mod 4). The image of the j_{A} is contained in the dual of the subspaces indicated in Table 8

Table 8: 𝔰𝔬(0, 2m + 1)-equivariant embeddings j_{A} : ℝ^{0,2m+1} ↪ (S ⊗ S)^{∗}

\begin{matrix}j_{Id} & ∨^{2}S & ∨^{2}S & S^{+} ∨ S^{−} \\ j_{J} & ∧^{2}S & ∧^{2}S & S^{+} ∧ S^{−} \\ jĴ & ∨^{2}S & ∧^{2}S & ∧^{2}S^{+} + ∧^{2}S^{−} \\ jK̂ & ∨^{2}S & ∧^{2}S & ∧^{2}S^{+} + ∧^{2}S^{−} \\ m: & 1 & 2 & 3 \end{matrix}

6 Complete classification

Every pseudo Euclidean space V admits a (unique up to an isometry) orthogonal decomposition V = V_{1} + V_{2}, where V_{1} = ℝ^{m,m} and the scalar product of V_{2} is positively or negatively defined. Now we consider the case when V_{1} ≠ 0 and V_{2} ≠ 0, the other cases were treated in the sections 3.1, 4 and 5. We denote by S_{i}, i = 1, 2, the irreducible Cℓ(V_{i})-module constructed in the sections 3.1 and 4, 5 respectively. Then S = S_{1} ⊗ S_{2} carries the structure of irreducible module for the Clifford algebra Cℓ(V) = Cℓ(V_{1}) ˆ⊗Cℓ(V_{2}), s. Proposition 2.4. By Proposition 3.6, to every admissible bilinear form β_{2} (respectively endomorphism A_{2}) on S_{2} we associate an admissible bilinear form β = β_{1} ⊗β_{2} (respectively endomorphism A_{1} ⊗A_{2}) on S. In the sections 4 and 5 we have contructed admissible bases for the space B_{2} of 𝔰𝔬(V_{2})-invariant bilinear forms on S_{2} and for the Schur algebra C_{2} of S_{2}. Therefore, this explicit correspondence defines an injective linear mapping ϕ : β_{2} ↦ β = ϕ(β_{2}) (respectively ψ : A_{2} ↦ A = ψ(A_{2})) from B_{2} into the space B of 𝔰𝔬(V)-invariant bilinear forms on S (respectively from C_{2} into the Schur algebra C of S). Moreover, ϕ and ψ are actually isomorphisms, because the Schur algebras of S and S_{2} are isomorphic, due to the fact that V and V_{2} have the same signature s, s. Corollary 1.10. So we have essentially proved:

Theorem 6.1 There exist natural isomorphisms ϕ : B_{2} → B of vector spaces and ψ : C_{2} → C of algebras mapping admissible elements onto admissible elements. Under these maps, the fundamental invariants of admissible elements transform according to the rules given in Proposition 2.3.

In particular, if m ≡ 0 (mod 4), then ϕ and ψ preserve the fundamental invariants ((4,4)-periodicity).

Proof: We recall that by Proposition 3.4 the Schur algebra C_{m,m} of S_{1} = S_{m,m} has the admissible basis (Id, E) and E^{2} = +Id. This implies that the vector space isomorphism ψ is actually an isomorphism of algebras. The (4,4)-periodicity follows from

σ(f_{E}) = ι(f_{E}) = σ_{f}(E) = σ_{f_{E}}(E) = ι(E) = +1 . □

Recall that B_{p,q} denotes the space of 𝔰𝔬(p, q)-invariant bilinear forms on the 𝔰𝔬(p, q) spinor module S_{p,q} and C_{p,q} is the Schur algebra of S_{p,q}.

Corollary 6.2 ((8,0)- and (0,8)-periodicity) There exist natural isomorphisms

ϕ_{8,0} : B_{p,q} → B_{p+8,q} and ϕ_{0,8} : B_{p,q} → B_{p,q+8}

of vector spaces and

ψ_{8,0} : C_{p,q} → C_{p+8,q} and ψ_{0,8} : C_{p,q} → C_{p,q+8}

of algebras mapping the admissible elements onto admissible elements preserving their fundamental invariants.

Proof: By Theorem 6.1 B_{p,q} and C_{p,q} have admissible bases. Now we recall from sections 4 and 5 that if k ≡ 0 (mod 8), then C_{k,0} ≅ C_{0,k} has an admissible basis, which was denoted by (Id, E), such that (τ, σ, ι)(E) = (−1, +1, +1) and, of course, (τ, σ, ι)(Id) = (+1, +1, +1). The existence of the maps ψ_{8,0} and ψ_{0,8} follows from τ(Id)ι(Id) = −τ(E)ι(E). They preserve the fundamental invariants, because σ(Id) = ι(Id) = σ(E) = ι(E) = +1. The existence and properties of ϕ_{8,0} and ϕ_{0,8} are proved similarly. □

Corollary 6.3 Every 𝔰𝔬(V)-equivariant mapping j : V → (S ⊗ S)^{∗} is a linear combination of the embeddings j_{A} = j_{ρ}(h_{A}), where h is the canonical bilinear form on the spinor module S of 𝔰𝔬(V) and A are admissible elements of the Schur algebra C of S.

To obtain an overview over all possible N-extended Poincaré algebras 𝔭(V) + S, N = ±1, ±2, it is useful to define the invariants σ and ι for embeddings j : V ↪ (S ⊗ S)^{∗} having special properties. More precisely, we put σ(j) = +1 if jV ⊂ ∨^{2}S^{∗} and σ(j) = −1 if jV ⊂ ∧^{2}S^{∗}. If S = S^{+} + S^{−}, we define ι(j) = +1 if jV ⊂ (S^{+} ⊗ S^{+} + S^{−} ⊗ S^{−})^{∗} and ι(j) = −1 if jV ⊂ (S^{+} ⊗ S^{−})^{∗}.

Note that the fundamental invariants of j_{A} = j_{ρ}(h_{A}), A ∈ C admissible, are easily computable:

σ(j_{A}) = τ(h_{A})σ(h_{A}) = τ(h)τ(A)σ(h)σ(A) and ι(j_{A}) = −ι(h_{A}) = −ι(h)ι(A) .

Recall that J denotes the space of 𝔰𝔬(V)-equivariant mappings j : V → (S ⊗ S)^{∗}. We define the subspaces

J^{σ_{0}} := {j ∈ J |σ(j) = σ_{0}} ∪ {0} and
J^{σ_{0}ι_{0}} := {j ∈ J^{σ_{0}}|ι(j) = ι_{0}} ∪ {0}

and put

L^{σ_{0}} := dim J^{σ_{0}} , L^{σ_{0}ι_{0}} := dim J^{σ_{0}ι_{0}} .

We shall write L^{+}, L^{+−}, ... instead of the more cumbersome L^{+1}, L^{+1 −1}, ...

Remark that L^{+} (= L^{++} + L^{+−} if S = S^{+} + S^{−}) is the maximal number of linearly independent super algebra structures on 𝔭(V) + S and that L^{−} (= L^{−+}+L^{−−}) is the number of ℤ_{2}-graded Lie algebra structures on 𝔭(V)+S.

Theorem 6.4 The numbers (L^{+}, L^{−}) and (L^{++}, L^{+−}, L^{−+}, L^{−−}) depend only on the dimension n = dim V = p + q and the signature s = p − q of V = ℝ^{p,q} modulo 8. Moreover, they admit the mirror super symmetry n ↦ −n. More precisely,

L^{+}(−n, s) = L^{−}(n, s) and
L^{+ ι_{0}}(−n, s) = L^{− ι_{0}}(n, s) , ι_{0} = ± .

Their values are given in Table 9.

Table 9: Numbers of extended Poincaré algebras 𝔭(p, q) + S_{p,q} of different types depending on n = p + q and s = p − q modulo 8

s:  (L^{++}, L^{+−}, L^{−+}, L^{−−})(n, s) or (L^{+}, L^{−})(n, s)
4  2,0,6,0  0,4,0,4  6,0,2,0  0,4,0,4
3 1,3  1,3  3,1  3,1
2  0,2,4,2  2,2,2,2  4,2,0,2  2,2,2,2
1 0,1,2,1  0,1,2,1  2,1,0,1  2,1,0,1
0  0,0,2,0  0,1,0,1  2,0,0,0  0,1,0,1
-1 0,1  0,1  1,0  1,0
-2  0,2  1,1  2,0  1,1
-3 1,3  1,3  3,1  3,1
n:  -3  -2  -1  0  1  2  3  4

Proof: This follows from Theorem 6.1 and the tables of sections 3.1, 4 and 5 by straightforward computation. □

In the complex case we consider the space J_{c} of 𝔰𝔬(m, ℂ)-equivariant mappings ℂ^{m} → (𝕊_{m} ⊗ 𝕊_{m})^{∗} and define the invariants σ, ι and the spaces J_{c}^{+}, J_{c}^{+−} etc. as in the real case (ι is only defined if the complex 𝔰𝔬(m, ℂ) spinor module 𝕊_{m} is reducible 𝕊_{m} = 𝕊_{m}^{+} +𝕊_{m}^{−}). Their dimensions are denoted by L_{c}^{+}, L_{c}^{+−} etc.

Theorem 6.5 The numbers (L_{c}^{+}, L_{c}^{−}) and (L_{c}^{++}, L_{c}^{+−}, L_{c}^{−+}, L_{c}^{−−}) depend only on m (mod 8). Moreover, they admit the mirror super symmetry m ↦ −m. More precisely,

L_{c}^{+}(−m) = L_{c}^{−}(m) and
L_{c}^{+ ι_{0}}(−m) = L_{c}^{− ι_{0}}(m) , ι_{0} = ± .

Their values are given in the next table.

0, 1 0, 0, 2, 0 0, 1 0, 1, 0, 1 1, 0 2, 0, 0, 0 1, 0 0, 1, 0, 1
m: −3  −2  −1  0  1  2  3  4

Proof: follows from section 3.2. □

References

[A-C1] D.V. Alekseevsky, V. Cortés: Isometry Groups of Homogeneous Quaternionic Kähler Manifolds (to appear); available as preprint Erwin Schrödinger Institut 230 (1995).

[dW-V-VP] B. de Wit, F. Vanderseypen, A. Van Proeyen: Symmetry structure of special geometries, Nucl. Phys. B400 (1993), 463-521.

[F] P.G.O. Freund: Introduction to supersymmetry, New York, Cambridge University Press, 1986.

[G-L] T.A. Gol’fand, E.P. Likhtman: Extension of the Algebra of Poincaré Group Generators and Violation of P-Invariance, JETP Lett. 13, 323-326.

[H] F.R. Harvey: Spinors and calibrations, Boston, Academic Press, 1989.

[K] V. Kac: Lie Superalgebras, Adv. Math. 26 (1977), 8-96.

[L-M] H.B. Lawson, M.-L. Michelson: Spin geometry, Princeton, Princeton University Press, 1989.

[O-S] V.I. Ogievetsky, E.S. Sokatchev: Structure of Supergravity Group, Phys. Lett. 79B, 222-224.

[O-V] A.L. Onishchik, E.B. Vinberg: Lie Groups and Algebraic Groups, Springer, Berlin, Heidelberg, 1990.

[R] P.K. Ras̆evskiĭ: The Theory of Spinors, Uspehi Mat. Nauk (N.S.) 10, no 2 (64), 3-110 (1955); AMS Transl. (2) 6 (1957), 1-110.