Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension

2007/08/23 by Edward S. Letzter, Letzter, Edward S.
Mathematics · #13P10 (Secondary) #16R30 #16Z05 (Primary) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AC #math.RA #msc:13P10 #msc:16R30 #msc:16Z05

paper · pdf · doi:10.48550/arxiv.0708.3190

12 pages, no figures. Revised; to appear in Journal of Algebra (Computational Section)

arxiv created 2008/07/20 · arxiv updated 2009/12/01

Abstract

Let n be a positive integer, and let k be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented k-algebra R has infinitely many equivalence classes of semisimple representations R → M_n(k'), where k' is the algebraic closure of k. The test reduces the problem to computational commutative algebra over k, via famous results of Artin, Procesi, and Shirshov. The test is illustrated by explicit examples, with n = 3.

Related

• Counting equivalence classes of irreducible representations
• Effective Detection of Nonsplit Module Extensions
• Constructing irreducible representations of finitely presented algebras
• Infinite dimensional representations of finite dimensional algebras and amenability
• Counting finite-dimensional algebras over finite fields