Reduction method for representations of queer Lie superalgebras

TL;DR

This paper introduces a reduction procedure that simplifies the study of blocks in the BGG category for queer Lie superalgebras by establishing equivalences with blocks of other superalgebra categories, aiding classification.

Contribution

It develops a novel reduction method to relate blocks of queer Lie superalgebras to simpler categories, facilitating their analysis and classification.

Findings

Established an equivalence between blocks of $ ext{q}(n)$ and $ ext{q}(m)$ with $ ext{Z} extpm s$-weights.

Described the structure of blocks in the BGG category for $ ext{q}(n)$.

Linked certain parabolic subcategories of $ ext{q}(n)$ to blocks of $ ext{gl}( ext{l}|n- ext{l})$.

Abstract

We develop a reduction procedure which provides an equivalence from an arbitrary block of the BGG category for the queer Lie superalgebra $\mathfrak{q}(n)$ to a "$\mathbb{Z}\pm s$-weights" ($s\in \mathbb{C}$) block of a BGG category for finite direct sum of queer Lie superalgebras. We give descriptions of blocks. We also establish equivalences between certain maximal parabolic subcategories for $\mathfrak{q}(n)$ and blocks of atypicality-one of the category of finite-dimensional modules for $\mathfrak{gl}(\ell|n-\ell)$.