On the use of Külshammer type invariants in representation theory

Alexander Zimmermann (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

Since 2005 a new powerful invariant of an algebra emerged using earlier work of Horváth, Héthelyi, Külshammer and Murray. The authors studied Morita invariance of a sequence of ideals of the centre of a finite dimensional algebra over a field of finite characteristic. It was shown that the sequence of ideals is actually a derived invariant, and most recently a slightly modified version of it an invariant under stable equivalences of Morita type. The invariant was used in various contexts to distinguish derived and stable equivalence classes of pairs of algebras in very subtle situations. Generalisations to non symmetric algebras and to higher Hochschild (co-)homology was given. This article surveys the results and gives some of the constructions in more detail.

Mots clés: Hochschild homology; Reynolds ideal; Külshammer ideal; domestic representation type; polynomial growth representation type; derived equivalences; stable equivalences of Morita type; periodic algebras; tame blocks

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

Alexander Zimmermann. On the use of Külshammer type invariants in representation theory. Bulletin of the Iranian Mathematical Society, 2011. ⟨hal-00530266⟩