Non-commutative Gröbner bases in algebras of solvable type

Abstract

We introduce a class of non-commutative polynomial rings over fields intermediate betweencommutative polynomial rings and general non-commutative polynomial rings. This class of solvable polynomial rings includes many rings arising naturally in mathematics and physics, such as iterated Ore extensions of fields and enveloping algebras of finite dimensional Lie algebras. We present algorithms that compute Gröbner bases of one- and two-sided ideals in solvable polynomial rings. They extend Buchberger's algorithm (see Buchberger, 1985 ) in the commutative case and Apel and Lassner's algorithms (see Apel & Lassner, 1988 ) for one-sided ideals in enveloping algebras of Lie algebras, as well as the results on one-sided standard bases in Weyl algebras, sketched in Galligo (1985) . We show that reduced one- and two-sided Gröbner bases in solvable polynomial rings are unique, and we solve the word problem and the ideal membership problem for algebras of solvable type, in particular in Clifford algebras. Further applications include the computation of elimination ideals, computing in residue modules and the computation of generators for modules of syzygies.