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Serre functors for Lie superalgebras and tensoring with $S^{\mathrm{top}}(\mathfrak{g}_{\overline{1}})$

Chih-Whi Chen, Volodymyr Mazorchuk

TL;DR

The paper provides a general realization of the Nakayama and Serre functors on the parabolic category $ ext{O}^{\frak p}$ for quasi-reductive Lie superalgebras, showing that on the projective-injective subcategory the Nakayama functor is implemented by tensoring with the top odd-symmetric power via $E_{\frak g}$, i.e. $\mathbf N(P)\cong E_{\frak g}\otimes P$. This yields a practical criterion for symmetry of $\mathcal{PI}^{\frak p}_{\text{int}}$: it is symmetric precisely when $E_{\frak g}$ is trivial, and more generally the Serre functor on the corresponding perfect subcategory is isomorphic to a parity twist or a fixed tensoring functor, depending on the Lie superalgebra type. The authors develop three complementary realizations of $\mathbf N$ via Harish-Chandra bimodules, twisting functors, and Joseph’s Enright completion, enabling explicit calculations and extensions to strange algebras, including queer and P-type cases. Specializing to type Q and type P strangeness, they determine when the integral projective-injective subcategories are symmetric and identify the corresponding Serre functors as parity shifts $\Pi^n$ or related parity-twisted completions. Overall, the work unifies Serre- and Nakayama-theoretic phenomena across a broad class of Lie superalgebras and provides concrete symmetry criteria for important families of modules in parabolic categories.

Abstract

We show that the action of the Serre functor on the subcategory of projective-injective modules in a parabolic BGG category $\mathcal O$ of a quasi-reductive finite dimensional Lie superalgebra is given by tensoring with the top component of the symmetric power of the odd part of our superalgebra. As an application, we determine, for all strange Lie suepralgebras, when the subcategory of projective injective modules in the parabolic category $\mathcal O$ is symmetric.

Serre functors for Lie superalgebras and tensoring with $S^{\mathrm{top}}(\mathfrak{g}_{\overline{1}})$

TL;DR

The paper provides a general realization of the Nakayama and Serre functors on the parabolic category for quasi-reductive Lie superalgebras, showing that on the projective-injective subcategory the Nakayama functor is implemented by tensoring with the top odd-symmetric power via , i.e. . This yields a practical criterion for symmetry of : it is symmetric precisely when is trivial, and more generally the Serre functor on the corresponding perfect subcategory is isomorphic to a parity twist or a fixed tensoring functor, depending on the Lie superalgebra type. The authors develop three complementary realizations of via Harish-Chandra bimodules, twisting functors, and Joseph’s Enright completion, enabling explicit calculations and extensions to strange algebras, including queer and P-type cases. Specializing to type Q and type P strangeness, they determine when the integral projective-injective subcategories are symmetric and identify the corresponding Serre functors as parity shifts or related parity-twisted completions. Overall, the work unifies Serre- and Nakayama-theoretic phenomena across a broad class of Lie superalgebras and provides concrete symmetry criteria for important families of modules in parabolic categories.

Abstract

We show that the action of the Serre functor on the subcategory of projective-injective modules in a parabolic BGG category of a quasi-reductive finite dimensional Lie superalgebra is given by tensoring with the top component of the symmetric power of the odd part of our superalgebra. As an application, we determine, for all strange Lie suepralgebras, when the subcategory of projective injective modules in the parabolic category is symmetric.
Paper Structure (16 sections, 17 theorems, 46 equations)

This paper contains 16 sections, 17 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Description of main results
  4. Structure of the paper.
  5. Preliminaries
  6. Induction, coinduction and restriction functors
  7. Parabolic category $\mathcal{O}^{\mathfrak p}$
  8. Hairsh-Chandra bimodules
  9. Symmetric algebras and Serre functors
  10. Realizations of the Nakayama functor $\mathbf{N}$
  11. Realization via Harish-Chandra $(\mathfrak{g},\mathfrak{g}_{\bar{0}})$-bimodules
  12. Realization via twisting functors
  13. Realization via Joseph's Enright completion functors
  14. Examples: strange Lie superalgebras
  15. Lie superalgebras of type P
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathfrak{g}$ be a quasi-reductive Lie superalgebra with a parabolic subalgebra $\mathfrak p$ of $\mathfrak g$. Let $\mathbf{N}$ be the Nakayama functor on $\mathcal{O}^{\mathfrak p}$. Then, we have for any projective-injective module $P\in \mathcal{O}^{\mathfrak p}$. In particular, the following are equivalent:

Theorems & Definitions (29)

  • Theorem 1
  • Conjecture 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • proof
  • Lemma 7
  • Theorem 8
  • Lemma 9
  • ...and 19 more