# Extensions of modules for twisted current algebras

## TL;DR

This paper investigates the extension properties of finite-dimensional simple modules over twisted current algebras, revealing their block structure and providing explicit computations of extensions, especially for twisted forms.

## Contribution

It establishes the absence of nontrivial extensions between evaluation and non-evaluation modules and computes all extensions, leading to a detailed block decomposition for twisted current algebras.

## Key findings

- No nontrivial extensions between evaluation and non-evaluation modules
- Explicit extension calculations for all simple module pairs
- Block decomposition described via fundamental group maps in twisted forms

## Abstract

Twisted current algebras are fixed point subalgebras of current algebras under a finite group action. Special cases include equivariant map algebras and twisted forms of current algebras. Their finite-dimensional simple modules fall into two categories, those which factor through an evaluation map and those which do not. We show that there are no nontrivial extensions between finite-dimensional simple evaluation and non-evaluation modules. We then compute extensions between any pair of finite-dimensional simple modules for twisted current algebras, and use this information to determine the block decomposition for the category. In the special case of twisted forms, this decomposition can be described in terms of maps to the fundamental group of the underlying root system.