Simple supermodules over Lie superalgebras. (English) Zbl 1457.17010
The authors consider a classification problem in representation theory of Lie superalgebras. In general, for a Lie superalgebra \(\mathfrak{g}\), a classification of simple \(\mathfrak{g}\)-supermodules is, naturally, at least as hard as a classification of simple modules over the even Lie algebra part \(\mathfrak{g}_0\) (under usual notation of the area).
It is shown that, for many Lie superalgebras admitting a compatible \(\mathbb Z\)-grading, the Kac induction functor gives rise to a bijection between simple supermodules over a Lie superalgebra and simple supermodules over the even part of this Lie superalgebra, providing a reduction to this classification problem for some relevant types of Lie superalgebras. The authors also provide several criteria for simplicity of certain Kac modules and study the the rough structure of Kac supermodules.
It is shown that, for many Lie superalgebras admitting a compatible \(\mathbb Z\)-grading, the Kac induction functor gives rise to a bijection between simple supermodules over a Lie superalgebra and simple supermodules over the even part of this Lie superalgebra, providing a reduction to this classification problem for some relevant types of Lie superalgebras. The authors also provide several criteria for simplicity of certain Kac modules and study the the rough structure of Kac supermodules.
Reviewer: Angelo Bianchi (São José dos Campos)
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
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