Decision of Algebra Isomorphisms Using Gröbner Bases
Kiyoshi Shirayanagi · Birkhäuser Boston eBooks · 1993
Given two finite-dimensional non-commutative finitely presented algebras A and B over a field k , this paper presents two algorithms for deciding whether or not they are isomorphic as k -algebras. We reduce this problem to a radical ideal membership problem in a commutative ring, by Hilbert’s zero point theorem. That is, A and B are isomorphic if and only if the determinant f of a k -linear mapping: A → B does not lie in the radical of an ideal I derived from the relations of A and B . Moreover an efficient technique for computing f is provided. We propose two Gröbner basis methods for solving our problem. One is to compute the radical of I directly and then to rewrite the above determinant f modulo this radical. The other method is to judge solvability of a certain system of algebraic equations: the union of I and a new polynomial. As a result, it is shown that the isomorphism problem for finitely presented algebras is decidable. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.