Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension

Details Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension

2007-08-23

arxiv.org/abs/0708.3190v3

Mathematics

Rings and Algebras

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

Scientific paper

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 $R \to M_n(k')$, 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 $n = 3$.

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