Noncommutative Grobner Bases for Almost Commutative Algebras

Abstract

Let $K$ be an infinite field and $K<X>=K<X_1,...,X_n>$ the free associative algebra generated by $X=\{X_1,...,X_n\}$ 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 $\mathbb{N}$-filtration $FA$, the associated $\mathbb{N}$-graded algebra $G(A)$ is commutative, then $I$ is generated by a finite Gröbner basis. Therefor, every quotient algebra of the enveloping algebra $U(\mathbf{g})$ of a finite dimensional $K$-Lie algebra $\mathbf{g}$ is, as a noncommutative algebra of the form $A=K<X>/I$, defined by a finite Gröbner basis in $K<X>$.

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