Külshammer ideals of algebras of quaternion type

Alexander Zimmermann (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

For a symmetric algebra A over a field K of characteristic p > 0 Külshammer 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 Külshammer 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 Külshammer ideals for algebras of quaternion type and distinguish again algebras with respect to certain parameters.

Mots clés: derived equivalences; stable equivalences of Morita type; algebras of quaternion type; tame blocks; K\"ulshammer ideals; socle deformation

Domaines: Théorie des représentations [math.RT]; Anneaux et algèbres [math.RA]

Alexander Zimmermann. Külshammer ideals of algebras of quaternion type. Journal of Algebra and Its Applications, 2017, 17 (8), pp.1850157. ⟨hal-01320761v3⟩