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A sketch of Lie superalgebra theory. (English) Zbl 0359.17009

Graded Lie algebras have recently become of interest in physics, and the present article on Lie superalgebras is written to introduce these to physicists. Its content is then mathematical, but the style is not. Most proofs are only sketched or, more often, omitted entirely with a promise that details will be given in a forthcoming article in [Usp. Mat. Nauk].
A Lie superalgebra is a \(\mathbb Z_2\)-graded algebra \(G=G_0\oplus G_1\) with a product \([\cdot, \cdot]\) satisfying the identities
\[ [a,b]=-(-1)^{\alpha\beta}[b,a],\quad [a,[b,c]]=[[a,b],c]+(-1)^{\alpha\beta}[b,[a,c]] \]
for \(\alpha\in G_\alpha\) and \(\beta\in G_\beta\). Finite dimensional Lie superalgebras possess a unique maximal solvable ideal \(R\) such that \(G/R\) is semisimple. Theorem 7 classifies finite dimensional irreducible representations of solvable Lie superalgebras, and Theorem 6 describes semisimple Lie superalgebras in terms of simple ones. The main thrust of the paper is the classification of finite dimensional simple Lie superalgebras over an algebraically closed field of characteristic zero. (Theorem 9 treats the real field.)
The possible degeneracy of the Killing form prevents simple imitation of the Killing-Cartan approach to classification. There are two main families: (a) classical algebras and (b) nonclassical algebras. In case (a), one obtains four infinite families \(A(m,n)\), \(B(m,n)\), \(C(n)\), and \(D(m,n)\) resembling the corresponding classical Lie algebras in several respects. In addition there are two “strange” families denoted \(P(n)\) and \(Q(n)\), two exceptional superalgebras \(F(4)\) and \(G(3)\), and a family of seventeen-dimensional superalgebras related to one member of the family \(D(m,n)\). In case (b) there are four families of superalgebras of Cartan type, analogous to the corresponding families of infinite dimensional Lie algebras of Cartan type.
Theorem 8 states that finite dimensional irreducible representations of simple Lie superalgebras are determined by highest weights. (Finite dimensional representations of simple Lie superalgebras need not however be completely reducible as in the case of Lie algebras over the complex field.)
The paper closes with a partial analogue (Theorem 10) of the classification of infinite dimensional primitive Lie algebras, and with several open questions.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
17B70 Graded Lie (super)algebras
17B65 Infinite-dimensional Lie (super)algebras
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