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

Abstract analysis, Proc. 14th Winter Sch., SrnĂ®/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

Citations:

Zbl 0627.00012
Full Text: EuDML