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Complexity for modules over the classical Lie superalgebra $\Xmathfrak {gl}(m|n)$

Part of: Lie algebras and Lie superalgebras General commutative ring theory

Published online by Cambridge University Press:  25 July 2012

Brian D. Boe ,
Jonathan R. Kujawa  and
Daniel K. Nakano
Brian D. Boe
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA (email: brian@math.uga.edu)
Jonathan R. Kujawa
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA (email: kujawa@math.ou.edu)
Daniel K. Nakano
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA (email: nakano@math.uga.edu)
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Abstract

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Let ${\Xmathfrak g}={\Xmathfrak g}_{\zerox }\oplus {\Xmathfrak g}_{\onex }$ be a classical Lie superalgebra and let ℱ be the category of finite-dimensional ${\Xmathfrak g}$-supermodules which are completely reducible over the reductive Lie algebra ${\Xmathfrak g}_{\zerox }$. In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of ${\Xmathfrak g}_{\onex }$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\Xmathfrak {gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

Keywords

Lie superalgebrasrepresentation theorycohomologysupport varietiescomplexity

MSC classification

Secondary: 17B56: Cohomology of Lie (super)algebras 17B10: Representations, algebraic theory (weights) 13A50: Actions of groups on commutative rings; invariant theory

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 5 , September 2012 , pp. 1561 - 1592
Copyright
Copyright © Foundation Compositio Mathematica 2012

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