Groebner Bases in Non-Commutative Algebras

INTRODUCTION Recently, the use of Groebner bases and Buchberger algorithm [BUC1,2,4] has been generalised from the case of commutative polynomials to finitely generated algebras R over a field k, R = k, s.t. for each i < j, for some cij ∈ k, for some commutative polynomial pij ∈ k[X1,...,Xn], one has xj xi cij xi xj = pij(x1,...,xn). The first results in this direction were due to Galligo [GAL], which studied Groebner bases for left ideals in Weyl algebras (R = k with di xi xi di = 1 for each i, di xj = xj di, xi xj = xj xi, di dj = dj di if i ≠ j); and to Apel Lassner [A-L], which studied Groebner bases for left ideals in tensor algebras over Lie algebras (cij = 1, pij linear). Kandri-Rody Weispfenning [KRW] were the first to study Groebner bases for two-sided ideals, in "solvable polynomial rings". Solvable polynomial rings can be described as follows: impose on the non-commutative polynomial ring k a graduation on Nn, by assigning to each term s the vector of exponents of the abelianization of s, and impose a semigroup well-ordering < on Nn. Clearly R = k/H, where H is the ideal generated by {Xj Xi cij Xi Xj pij}. Solvable polynomial rings are then those R s.t. for each i,j, cij ≠ 0 and deg(pij) < deg(Xi Xj); they include Weyl algebras, tensor algebras of Lie algebras, iterated Ore extensions. Kandri Rody and Weispfenning studied Groebner bases for two sided ideals in solvable polynomial rings and gave an algorithm for their computation, under the further restriction that H doesn't contain commutative polynomials.