On the Gabriel quiver of extensions of Leibniz algebras

TL;DR

This paper calculates the Gabriel quiver for simple bimodule categories over simple Leibniz algebras, revealing the structure of extensions between simple objects through Ext groups.

Contribution

It provides the first explicit computation of the Gabriel quiver for bimodule categories over simple Leibniz algebras, including the trivial case.

Findings

Vertices correspond to simple bimodule classes.

Arrows are determined by Ext^1 group dimensions.

Results clarify the extension structure in these categories.

Abstract

We compute the Gabriel quiver of simple objects in the category of bimodules over a simple Leibniz algebra and over the trivial $1$-dimensional Leibniz algebra. Vertices of the quiver are the classes of simple objects, arrows are given by the dimensions of Ext$^1$-groups.