Dominant Dimension and Almost Relatively True Versions of Schur’s Theorem

Abstract

Perhaps the most fundamental problems of representation theory are to classify and to describe irreducible (=simple) representations and to determine cohomology. It is crucial to develop techniques that allow to transfer information from some (known) cases to other (unknown) cases. A classical result of this kind, due to Schur, recently has been extended widely, and put into a general context. These modern 'relative' versions of Schur's result will be presented. Moreover, the theoretical background behind these results, and the crucial invariant controlling the existence and strength of such equivalences, will be explained, and illustrated by an explicit example. Finally, some open problems will be stated and discussed.

Dominant dimensions of two classes of finite dimensional algebras

“…In [10] the dominant dimension has been used to prove several Schur-Weyl-dualities. Though the theory of dominant dimension is growing rapidly in applied context, see [6,9], the precise value of dominant dimension for many well-known classes of algebras is still unknown. Algebras of infinite dominant dimension also have been of interest of many ( e.g.…”