Complexity for Modules Over the Classical Lie Superalgebra gl(m|n)

Details Complexity for Modules Over the Classical Lie Superalgebra gl(m|n) Complexity for Modules Over the Classical Lie Superalgebra gl(m|n)

2011-07-13

arxiv.org/abs/1107.2579v1

Mathematics

Representation Theory

32 pages

Scientific paper

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus \mathfrak{g}_{\bar{1}}$ be a classical Lie superalgebra and $\mathcal{F}$ be the category of finite dimensional $\mathfrak{g}$-supermodules which are completely reducible over the reductive Lie algebra $\mathfrak{g}_{\bar{0}}$. In an earlier paper the authors demonstrated that for any module $M$ in $\mathcal{F}$ the rate of growth of the minimal projective resolution (i.e., the complexity of $M$) is bounded by the dimension of $\mathfrak{g}_{\bar{1}}$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\mathfrak{gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

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