Noncommutative Gröbner Bases for Almost Commutative Algebras

Let K be an infinite field and K〈X〉 = K〈X1, ..., Xn〉 the free associative algebra generated by X = {X1, ..., Xn} over K. It is proved that if I is a two-sided ideal of K〈X〉 such that the K-algebra A = K〈X〉/I is almost commutative in the sense of [3], namely, with respect to its standard N-filtration FA, the associated N-graded algebra G(A) is commutative, then I is generated by a finite Gröbner basis. Therefor, every quotient algebra of the enveloping algebra U(g) of a finite dimensional K-Lie algebra g is, as a noncommutative algebra of the form A = K〈X〉/I, defined by a finite Gröbner basis in K〈X〉. 2000 Mathematics Classification Primary 16W70; Secondary 16Z05.