An Elimination Lemma for Algebras with PBW Bases

TL;DR

This paper proves an elimination lemma for algebras with PBW bases, showing how nonzero left ideals with certain GK-dimension properties intersect specific subspaces, and explores implications for binomial skew polynomial rings and solvable polynomial algebras.

Contribution

It establishes an elimination property for algebras with PBW bases and characterizes when left ideals have GK-dimension less than the algebra, especially in binomial skew polynomial rings and solvable polynomial algebras.

Findings

Nonzero left ideals with GK.dim < n intersect specific subspaces.

Elimination property holds for binomial skew polynomial rings.

Elimination property holds for solvable polynomial algebras.

Abstract

Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a finitely generated $K$-algebra with the PBW $K$-basis ${\cal B}=\{a_{1}^{\alpha_1}\cdots a_{n}^{\alpha_n}~|~(\alpha_1,\ldots ,\alpha_n)\in\mathbb{N}^n\}$. It is shown that if $L$ is a nonzero left ideal of $A$ with GK.dim$(A/L)=d<n$ ($=$ the number of generators of $A$), then $L$ has the {\it elimination property} in the sense that ${\bf V}(U)\cap L\ne \{0\}$ for every subset $U=\{ a_{i_1},\ldots ,a_{i_{d+1}}\}\subset\{a_1,\ldots ,a_n\}$ with $i_1<i_2<\cdots <i_{d+1}$, where ${\bf V}(U)=K$-span$\{a_{i_1}^{\alpha_1}\cdots a_{i_{d+1}}^{\alpha_{d+1}}~|~(\alpha_1,\ldots ,\alpha_{d+1})\in\mathbb{N}^{d+1}\}$. In terms of the structural properties of $A$, it is also explored when the condition GK.dim$(A/L)<n$ may hold for a left ideal $L$ of $A$. Moreover, from the viewpoint of realizing the elimination property by means of Gr\"obner bases, it is demonstrated that if $A$ is in the class of binomial skew polynomial rings [G-I2, Serdica Math. J., 30(2004)] or in the class of solvable polynomial algebras [K-RW, J. Symbolic Comput., 9(1990)], then every nonzero left ideal $L$ of $A$ satisfies GK.dim$(A/L)<$ GK.dim$A=n$ ($=$ the number of generators of $A$), thereby $L$ has the elimination property.