Extension Quiver for Lie Superalgebra $\mathfrak{q}(3)$

TL;DR

This paper classifies the extension quivers of all blocks in the category of finite-dimensional modules over the Lie superalgebra (3), and provides new insights into the structure and representation theory of (n).

Contribution

It explicitly describes the extension quivers for (3) blocks and introduces a method to derive the Ext quiver of (n) from (n-1), along with a virtual BGG-reciprocity result.

Findings

Extension quivers for (3) blocks are fully described.

The Ext quiver of (n) is obtained from that of (n-1) by vertex identification.

A virtual BGG-reciprocity for (n) is established.

Abstract

We describe all blocks of the category of finite-dimensional $\mathfrak{q}(3)$-supermodules by providing their extension quivers. We also obtain two general results about the representation of $\mathfrak{q}(n)$: we show that the Ext quiver of the standard block of $\mathfrak{q}(n)$ is obtained from the principal block of $\mathfrak{q}(n-1)$ by identifying certain vertices of the quiver and prove a ''virtual'' BGG-reciprocity for $\mathfrak{q}(n)$. The latter result is used to compute the radical filtrations of $\mathfrak{q}(3)$ projective covers.