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Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension

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 Added by Edward S. Letzter
 Publication date 2008
  fields
and research's language is English




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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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