Homology, Homotopy and Applications

Support varieties and representation type of self-injective algebras

Jörg Feldvoss and Sarah Witherspoon

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Abstract

We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg: If a finite dimensional self-injective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed in "Support varieties and representation type of small quantum groups," Internat. Math. Res. Notices 2010, no. 7, 1346–1362. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half-quantum groups, and Nichols (quantum symmetric) algebras.

Article information

Source
Homology Homotopy Appl., Volume 13, Number 2 (2011), 197-215.

Dates
First available in Project Euclid: 30 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.hha/1335806750

Mathematical Reviews number (MathSciNet)
MR2854335

Zentralblatt MATH identifier
1242.16018

Subjects
Primary: 16D50: Injective modules, self-injective rings [See also 16L60] 16L60: Quasi-Frobenius rings [See also 16D50] 16G60: Representation type (finite, tame, wild, etc.) 16G10: Representations of Artinian rings 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 16T05: Hopf algebras and their applications [See also 16S40, 57T05] 17B35: Universal enveloping (super)algebras [See also 16S30] 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 20C08: Hecke algebras and their representations

Keywords
Support variety Hochschild cohomology complexity representation type wild tame block self-injective algebra Hecke algebra reduced universal enveloping algebra small half-quantum group Nichols algebra quantum symmetric algebra Hopf algebra

Citation

Feldvoss, Jörg; Witherspoon, Sarah. Support varieties and representation type of self-injective algebras. Homology Homotopy Appl. 13 (2011), no. 2, 197--215. https://projecteuclid.org/euclid.hha/1335806750


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