Compositio Mathematica

Research Article

Complexity for modules over the classical Lie superalgebra

Brian D. Boea1, Jonathan R. Kujawaa2 and Daniel K. Nakanoa3

a1 Department of Mathematics, University of Georgia, Athens, GA 30602, USA (email: brian@math.uga.edu)

a2 Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA (email: kujawa@math.ou.edu)

a3 Department of Mathematics, University of Georgia, Athens, GA 30602, USA (email: nakano@math.uga.edu)

Abstract

Let be a classical Lie superalgebra and let ℱ be the category of finite-dimensional -supermodules which are completely reducible over the reductive Lie algebra . 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 . In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra . In both cases we show that the complexity is related to the atypicality of the block containing the module.

(Received July 15 2011)

(Accepted December 06 2011)

(Online publication July 25 2012)

2010 Mathematics Subject Classification

  • 17B56;
  • 17B10 (primary);
  • 13A50 (secondary)

Keywords

  • Lie superalgebras;
  • representation theory;
  • cohomology;
  • support varieties;
  • complexity

Footnotes

Research of the second author was partially supported by NSF grant DMS-0734226 and NSA grant H98230-11-1-0127. Research of the third author was partially supported by NSF grant DMS-1002135.