Regular representations of finite-dimensional separable semisimple algebras and Groebner bases

摘要

Let A be a semisimple n-dimensional commutative algebra over a field F. It is easy to see that, given a basis 25 of A, the transposes of the matrices over F that represent a ε A regularly with respect to B can be simultaneously diagonalized over many fields. Using the multiplication table of the algebra we construct an ideal I of F[x_1, . . . , x_n] given in terms of a Groebner basis of the ideal I with respect to a total degree lexicographic monomial ordering and show that A is isomorphic to F[x_1, . . . , x_n]/I. We will then use Groebner basis properties to prove the properties of the algebra.