K{\"u}lshammer ideals of algebras of quaternion type

TL;DR

This paper computes K{"u}lshammer ideals for quaternion type algebras, enabling distinction of these algebras based on parameters, thus advancing classification in modular representation theory.

Contribution

It extends the computation of K{"u}lshammer ideals to quaternion type algebras, aiding their classification by parameters.

Findings

K{"u}lshammer ideals are explicitly determined for quaternion type algebras.

The invariants distinguish algebras with different parameters within quaternion type.

The results support classification efforts in modular representation theory.

Abstract

For a symmetric algebra A over a field K of characteristic p > 0 K{\"u}lshammer constructed a descending sequence of ideals of the centre of A. If K is perfect this sequence was shown to be an invariant under derived equivalence and for algebraically closed K under stable equivalence of Morita type. Erdmann classified algebras of tame representation type which may be blocks of group algebras, and Holm classified Erdmann's list up to derived equivalence. In both classifications certain parameters occur in the classification, and it was unclear if different parameters lead to different algebras. Erdmann's algebras fall into three classes, namely of dihedral, semidihedral and of quaternion type. In previous joint work with Holm we used K{\"u}lshammer ideals to distinguish classes with respect to these parameters in case of algebras of dihedral and semidihedral type. In the present paper we determine the K{\"u}lshammer ideals for algebras of quaternion type and distinguish again algebras with respect to certain parameters.