Torsion theories induced from commutative subalgebras

We begin a study of torsion theories for representations of an important class of associative algebras over a field which includes all finite W-algebras of type A, in particular the universal enveloping algebra of gl(n) (or sl(n)) for all n. If U is such and algebra which contains a finitely generated commutative subalgebra A, then we show that any A-torsion theory defined by the coheight of prime ideals is liftable to U. Moreover, for any simple U-module M, all associated prime ideals of M in Spec A have the same coheight. Hence,thecoheight of the associated prime ideals of A is an invariant of a given simple U-module. This implies a stratification of the category of $U$-modules controlled by the coheight of associated prime ideals of A. Our approach can be viewed as a generalization of the classical paper by R.Block, it allows in particular to study representations of gl(n) beyond the classical category of weight or generalized weight modules

Automorphism groups of smooth quintic threefolds

We study automorphism groups of smooth quintic threefolds. Especially, we describe all the maximal ones with explicit examples of target quintic threefolds. There are exactly $22$ such groups

Endotrivial modules for the Schur covers of the symmetric and alternating groups

We investigate the endotrivial modules for the Schur covers of the symmetric and alternating groups and determine the structure of their group of endotrivial modules in all characteristics. We provide a full description of this group by generators and relations in all cases.Comment: changes from (v21): corrected minor typos, added reference