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Strongly typical representations of the basic classical Lie superalgebras. (English) Zbl 0985.17011

Let \(\mathfrak g\) be a basic classical complex Lie superalgebra with a universal enveloping superalgebra \(U\). Denote by \(Z(\mathfrak g)\) the center of \(U\). Let \(T\in Z(\mathfrak g)\) be a special even element as constructed in the author’s paper [Ann. Inst. Fourier 50, 1745-1764 (2000; Zbl 1063.17006)]. A maximal ideal \(\chi\) in \(Z(\mathfrak g)\) is strongly typical if \(T^2\) does not belong to \(\chi\). For a maximal ideal \(\chi\) in \(Z(\mathfrak g)\) denote by \(\text{gr} C_{\infty}\) the category of graded \(\mathfrak g\)-modules \(N\) such that each element of \(N\) is annihilated by \(\chi^r\) for some \(r\). It is shown that for a strongly typical maximal ideal \(\chi\) in \(Z(\mathfrak g)\) the category \(\text{gr} C_{\infty}\) is equivalent to a similar category \(\text{gr} C_{\infty}\) defined in terms of maximal ideal of \(Z(\mathfrak g_0)\).

MSC:

17B35 Universal enveloping (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Citations:

Zbl 1063.17006
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References:

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