Representations of a class of infinite-dimensional primitive Lie superalgebras

TL;DR

This work extends the representation theory of the infinite-dimensional primitive Lie superalgebras ${\mathbf W}(m,n)$ by formulating and analyzing the BGG category ${\mathcal O}$ for these algebras. The authors construct costandard modules via mixed-product realizations and establish irreducibility criteria, classifying irreducible objects as ${L(\lambda)}$ with $\lambda$ in the dominant set, augmented by two exceptional weight families $\omega_k$ and $\theta_q$ that require refined analysis. They develop a Skryabin-type framework in the super setting to prove irreducibility of non-exceptional mixed-product modules and compute explicit irreducible characters, while also determining tilting modules and their characters through semi-infinite characters and Soergel reciprocity. The results deliver a complete description of irreducibles and tilting modules in ${\mathcal O}$ for ${\mathbf W}(m,n)$, including precise character formulas, thus advancing the super-analogue of Kac’s infinite-dimensional primitive Lie algebras and providing tools for further structural and categorical studies.

Abstract

In [Kac77, Section 5.4] and [Kac 98], V. G. Kac tried to raise, and finished a classification of infinite-dimensional primitive Lie superalgebras. The series $\mathbf{W}(m,n)$ with $m,n$ being positive integers are the fundamental ones. In this article, we introduce the BGG category $\mathcal{O}$ of modules over $\textbf{W}(m,n)$, and try to systematically investigate the representations of $\mathbf{W}(m,n)$ in this category, analogue of the study in [Duan-Shu-Yao2024} dealing with finite-dimensional Lie superalgebra case $\mathbf{W}(0,n)$, or analogue of the study in [Duan-Shu-Yao2020] dealing with infinite-dimensional Lie algebra case $\mathbf{W}(m,0)$. Beyond a compound of the arguments in [Duan-Shu-Yao2020} and in [Duan-Shu-Yao2024], it is nontrivial to understand irreducible modules in $\mathcal{O}$, which is the main goal of this article. We solve the question with aid of homological analysis on costandard modules along with extending Skryabin's theory on independence of operators for graded differential operator Lie algebras in [Skryabin] to the super case. After classifying irreducible modules in this category and describing their structure, we finally obtain irreducible characters. In the end, by confirming the semi-infinite character property, and applying Soergel's tilting module theory in [Soergel], we study indecomposable tilting modules in $\mathcal{O}$, obtaining their character formulas.

Theorems & Definitions (71)

• Theorem 2
• Theorem 3
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8