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Complexity for modules over the classical Lie superalgebra.
- Source :
- Compositio Mathematica; Sep2012, Vol. 148 Issue 5, p1561-1592, 32p
- Publication Year :
- 2012
-
Abstract
- 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0010437X
- Volume :
- 148
- Issue :
- 5
- Database :
- Complementary Index
- Journal :
- Compositio Mathematica
- Publication Type :
- Academic Journal
- Accession number :
- 82070566
- Full Text :
- https://doi.org/10.1112/S0010437X12000231