Lie superalgebras. (English. Russian original) Zbl 0567.17003
J. Sov. Math. 30, 2481-2512 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 25, 3-49 (1984).
This is a survey of results on the representation theory of “classical” Lie superalgebras. The author reviews 131 papers. In §1 (classical Lie superalgebras over \(\mathbb C)\) the matrix Lie superalgebras, Lie superalgebras of (formal) vector fields, exceptional Lie superalgebras, Lie superalgebras of string theories, Kac superalgebras \(g_{\phi}^{(m)}\) and Kac-Moody superalgebras are considered. The modules over Lie superalgebras of vector fields are studied in §2. In §3 it is established that every irreducible topological module over the Lie superalgebra \({\mathcal L}\) of vector fields is realized as a submodule in some topological \({\mathcal L}\)-module \(T(V)\) (here \(T(V)\) is the superspace of formal tensor fields of type \(V\) which are constructed by means of a finite-dimensional \(L_ 0\)-module \(V\)). The author’s purpose is the description of this submodule. The characters of irreducible modules are investigated in §4. Finally other results on Lie superalgebras are enumerated in §5.
Reviewer: Aleksandr K. Guts (Omsk)
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
Keywords:
representations; bibliography; survey; classical Lie superalgebras; matrix Lie superalgebras; (formal) vector fields; exceptional Lie superalgebras; string theories; Kac superalgebras; Kac-Moody superalgebras; characters of irreducible modulesReferences:
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