Cohomology and support varieties for Lie superalgebras II

Boe, Brian D.; Kujawa, Jonathan R.; Nakano, Daniel K.

doi:10.1112/plms/pdn019

Cited by 28 publications

(28 citation statements)

References 41 publications

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“…Proof. The second equality is immediate from [BKN3, Theorem 2.9.1(c)] and [BKN1,Proposition 5.2.2]. We now consider the first equality.…”

Section: A Categorical Invariantmentioning

confidence: 95%

Complexity for modules over the classical Lie superalgebra

Boe¹,

Kujawa²,

Nakano³

2012

Compositio Math.

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Let g = g0 ⊕ g1 be a classical Lie superalgebra and let F be the category of finitedimensional g-supermodules which are completely reducible over the reductive Lie algebra g0. 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 g1. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra 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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“…Proof. The second equality is immediate from [BKN3, Theorem 2.9.1(c)] and [BKN1,Proposition 5.2.2]. We now consider the first equality.…”

Section: A Categorical Invariantmentioning

confidence: 95%

Complexity for modules over the classical Lie superalgebra

Boe¹,

Kujawa²,

Nakano³

2012

Compositio Math.

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“…Supergroups, Lie superalgebras, and related Z 2 -graded structures (including Z-graded Hopf algebras as defined by Milnor and Moore [39]) thus provide another natural setting for geometric methods. With Boe and Nakano, the second author initiated a study of support varieties for complex Lie superalgebras and showed that they capture information about the representation theory of these algebras, including atypicality, complexity, and the thick tensor ideals of the category [12][13][14][15]. In independent work, Duflo and Serganova [20] also defined associated varieties for Lie superalgebras in characteristic zero and showed they too provide representation theoretic information.…”

Section: Introductionmentioning

confidence: 99%

On support varieties for Lie superalgebras and finite supergroup schemes

Drupieski

Kujawa

2019

Journal of Algebra

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We study the spectrum of the cohomology rings of cocommutative Hopf superalgebras, restricted and non-restricted Lie superalgebras, and finite supergroup schemes. We also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory. We completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero, while for non-restricted Lie superalgebras we obtain results in positive characteristic that are strikingly similar to results of Duflo and Serganova in characteristic zero. Our computations for restricted Lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of Friedlander and Parshall and of Bendel, Friedlander, and Suslin in the classical setting.

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“…The category F has enough projectives ( [3]), is self-injective ( [4]), meaning that a module is projective if and only if it is injective, and for Type I classical Lie superalgebras, e.g. gl(m|n), F is a highest weight category ( [2]).…”

Section: Introductionmentioning

confidence: 99%

On endotrivial modules for Lie superalgebras

Talian

2015

Journal of Algebra

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Let $\mathfrak{g} = \mathfrak{g}_{\overline{0}} \oplus \mathfrak{g}_{\overline{1}}$ be a Lie superalgebra over an algebraically closed field, $k$, of characteristic 0. An endotrivial $\mathfrak{g}$-module, $M$, is a $\mathfrak{g}$-supermodule such that $\operatorname{Hom}_k(M,M) \cong k_{ev} \oplus P$ as $\mathfrak{g}$-supermodules, where $k_{ev}$ is the trivial module concentrated in degree $\overline{0}$ and $P$ is a projective $\mathfrak{g}$-supermodule. In the stable module category, these modules form a group under the operation of the tensor product. We show that for an endotrivial module $M$, the syzygies $\Omega^n(M)$ are also endotrivial, and for certain Lie superalgebras of particular interest, we show that $\Omega^1(k_{ev})$ and the parity change functor actually generate the group of endotrivials. Additionally, for a broader class of Lie superalgebras, for a fixed $n$, we show that there are finitely many endotrivial modules of dimension $n$.Comment: 27 pages; updates to section

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Cohomology and support varieties for Lie superalgebras II

Cited by 28 publications

References 41 publications

Complexity for modules over the classical Lie superalgebra

Complexity for modules over the classical Lie superalgebra

On support varieties for Lie superalgebras and finite supergroup schemes

On endotrivial modules for Lie superalgebras

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