Ext^1-quivers for the Witt algebra W(1,1)

TL;DR

This paper computes the Ext^1 groups between simple modules of the reduced enveloping algebra u(g,χ) for the Witt algebra W(1,1) over an algebraically closed field of characteristic p > 3, revealing extension structures in modular representation theory.

Contribution

It explicitly determines Ext^1 between simple modules of u(g,χ) for the Witt algebra, providing new insights into its modular representation theory.

Findings

Explicit Ext^1 computations for simple modules

Extension structures characterized for u(g,χ)

Advances understanding of Witt algebra representations

Abstract

Let g be the finite dimensional Witt algebra W(1,1) over an algebraically closed field of characteristic p > 3. It is well known that all simple W(1,1)-modules are finite dimensional. Each simple module admits a character \chi in g^*. Given such a \chi one can form the (finite dimensional) reduced enveloping algebra u(g,\chi). The simple modules for u(g,\chi) are precisely those simple W(1,1)-modules admitting the character \chi. In this paper the authors compute Ext^1 between pairs of simple modules for u(g,\chi).