Hypercubic Decomposition of Verma Supermodules and Semibricks Realizing the Khovanov Algebra of Defect One

TL;DR

This paper explores the structure of Verma supermodules in Lie superalgebras, revealing how certain blocks of category O relate to the Khovanov algebra of defect one, with implications for representation theory.

Contribution

It introduces variants of Verma modules via Borel subalgebra changes, connecting the principal block of gl(1|1) to atypical blocks of category O in basic Lie superalgebras.

Findings

Principal block of gl(1|1) is realized as a full subcategory of any atypical block of category O.

Variants of Verma modules demonstrate new structural insights into Lie superalgebra representations.

The work links Khovanov algebra structures to supermodule categories.

Abstract

We study some variants of Verma modules of basic Lie superalgebras obtained via changing Borel subalgebras. These allow us to demonstrate that the principal block of \(\mathfrak{gl}(1|1)\) is realized as (non-Serre) full subcategories of any atypical block of BGG category \( \mathcal{O} \) of basic Lie superalgebras.