[Paper Review] The generalized Kac-Wakimoto conjecture and support varieties for the Lie superalgebra osp(m|2n)

Jonathan R. Kujawa|arXiv (Cornell University)|Dec 14, 2011

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper proves the generalized Kac-Wakimoto conjecture and the atypicality conjecture for the Lie superalgebra 𝔬𝔰𝔭(m|2n), establishing that the dimension of the support variety of a simple supermodule equals its atypicality. Using cohomological techniques and support variety theory, the authors show that support varieties for simple modules of atypicality k are isomorphic to 𝔸ᵏ, confirming a geometric interpretation of atypicality and extending prior results for 𝔤𝔩(m|n).

ABSTRACT

Atypicality is a fundamental combinatorial invariant for simple supermodules of a basic Lie superalgebra. Boe, Nakano, and the author gave a conjectural geometric interpretation of atypicality via support varieties. Inspired by low dimensional topology, Geer, Patureau-Mirand, and the author gave a generalization of the Kac-Wakimoto atypicality conjecture. We prove both of these conjectures for the Lie superalgebra osp(m|2n).

Motivation & Objective

• To establish a geometric interpretation of atypicality in terms of support varieties for the Lie superalgebra 𝔬𝔰𝔭(m|2n).
• To prove the generalized Kac-Wakimoto conjecture for 𝔬𝔰𝔭(m|2n), extending the framework of modified trace and dimension functions.
• To confirm that support varieties of simple supermodules of the same atypicality are isomorphic to affine space 𝔸ᵏ.
• To extend known results for 𝔤𝔩(m|n) to the case of 𝔬𝔰𝔭(m|2n), completing the proof for all basic classical Lie superalgebras of types A and D.
• To demonstrate that the complexity and representation-theoretic structure of simple supermodules depend only on their atypicality.

Proposed method

• Utilizes cohomological support variety theory for Lie superalgebras, specifically the detecting algebra approach.
• Applies the rank variety description of support varieties via the action of odd elements in the Lie superalgebra.
• Employs the restriction map res* from the cohomology of the detecting subalgebra to the cohomology of the full Lie superalgebra.
• Uses the fact that H•(𝔤k, 𝔤k₀; ℂ) is a polynomial ring in k variables to identify the support variety with 𝔸ᵏ.
• Leverages the action of the Weyl group 𝒲 to analyze fibers of the restriction map res*.
• Applies the stability of simple supermodules under restriction to relate support varieties across subalgebras.

Experimental results

Research questions

• RQ1 Does the dimension of the support variety of a simple supermodule in 𝔬𝔰𝔭(m|2n) equal its atypicality?
• RQ2 Can the generalized Kac-W Wakimoto conjecture be proven for 𝔬𝔰𝔭(m|2n), relating modified dimension functions to atypicality?
• RQ3 Do all simple supermodules of the same atypicality have isomorphic support varieties?
• RQ4 Is the support variety of a simple supermodule of atypicality k isomorphic to affine space 𝔸ᵏ?
• RQ5 How does the restriction map res* relate the support varieties of subalgebras to the full Lie superalgebra?

Key findings

• The support variety of any simple supermodule of atypicality k in 𝔬𝔰𝔭(m|2n) is isomorphic to 𝔸ᵏ, confirming the atypicality conjecture.
• The generalized Kac-Wakimoto conjecture holds for 𝔬𝔰𝔭(m|2n), with modified dimension functions vanishing precisely when atypicality is less than maximal.
• All simple supermodules of the same atypicality have identical support varieties, implying equal representation-theoretic complexity.
• The support variety 𝒱(𝐋(λ)) is the union of finitely many k-dimensional subspaces, each corresponding to orbits under the Weyl group action.
• The restriction map res* is finite-to-one and surjective, allowing identification of the full support variety with the image of the detecting subalgebra.
• The result extends to 𝔤𝔩(m|n), confirming that the methods apply uniformly across types A and D in the Kac classification.