## Supersymmetry algebras: Extensions of orthogonal Lie algebras.(English)Zbl 0639.17012

Abstract analysis, Proc. 14th Winter Sch., Srnî/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 77-92 (1987).
[For the entire collection see Zbl 0627.00012.]
Let $$\eta$$ be a nondegenerate quadratic form on a finite-dimensional real vector space and let $${\mathfrak so}(\eta)$$ denote the corresponding orthogonal Lie algebra. The author presents a classification of all (real) $${\mathbb{Z}}_ 2$$-graded Lie algebras and all Lie superalgebras $$L=L_{\bar 0}\oplus L_{\bar 1}$$ which have the following properties:
(1) Let f denote the adjoint representation of $$L_{\bar 0}$$ in $$L_{\bar 1}$$. Then there exists a Lie algebra homomorphism $$g: {\mathfrak so}(\eta)\to L_{\bar 0}$$ such that the representation $$f\circ g$$ of $${\mathfrak so}(\eta)$$ in $$L_{\bar 1}$$ is equivalent to one of the (at most two) irreducible spinor representations. (2) The graded algebra L is simple. (3) Let E denote the associative algebra of all endomorphisms of the real vector space $$L_{\bar 1}$$. If A is the associative subalgebra of E generated by fg($${\mathfrak so}(\eta))$$ and if A’ (isomorphic to $${\mathbb{R}}$$, $${\mathbb{C}}$$, or $${\mathbb{H}})$$ is the centralizer of A in E, then $$f(L_{\bar 0})$$ is contained in $$A+A'$$.
Reviewer: M.Scheunert

### MSC:

 17B70 Graded Lie (super)algebras 17A70 Superalgebras 15A66 Clifford algebras, spinors 17B05 Structure theory for Lie algebras and superalgebras

Zbl 0627.00012
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