Families Of Finite-dimensional Algebras Research Articles

Nearly Frobenius structures in some families of algebras

In this article we continue with the study started in Artenstein et al. (Algebras Represent Theory 18:339–367, 2015) of nearly Frobenius structures in some representative families of finite dimensional algebras, as radical square zero algebras, string algebras and toupie algebras. We prove that such radical square zero algebras with at least one path of length two are nearly Frobenius. As for the string algebras, those who are not gentle can be endowed with at least one non-trivial nearly Frobenius structure. Finally, in the case of toupie algebras, we prove that the existence of monomial relations is a sufficient condition to have non-trivial nearly Frobenius structures. Using the technics developed for the previous families of algebras we prove sufficient conditions for the existence of non-trivial Frobenius structures in quotients of path algebras in general.

Higher spherical algebras

We introduce and study higher spherical algebras, an exotic family of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher tetrahedral algebra studied in Erdmann and Skowroński (Algebras Represent Theory 22:387–406, 2019), and hence that it is a tame symmetric periodic algebra of period 4.

Canonical idempotents of multiplicity-free families of algebras

Any multiplicity-free family of finite dimensional algebras has a canonical complete set of pairwise orthogonal primitive idempotents in each level. We give various methods to compute these idempotents. In the case of symmetric group algebras over a field of characteristic zero, the set of canonical idempotents is precisely the set of seminormal idempotents constructed by Young. As an example, we calculate the canonical idempotents for semisimple Brauer algebras.

Hochschild cohomology of a generalisation of canonical algebras

We compute the Hochschild cohomology groups of a family of finite dimensional algebras, the toupie algebras, characterized by having the same associated ordinary quiver as canonical algebras and any admissible ideal. Moreover, we determine the simple connectedness of them and we give necessary and sufficient conditions for rigidity.