Graded Monomial Ordering for $\mathbb{N}$-graded and $\mathbb{N}$-filtered Solvable Polynomial Algebras of $(\mathcal{B},d())$-type

Abstract

Let $K$ be a field, and $A=K[a_1,\dots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. It is shown that if $A$ is an $\mathbb{N}$-graded algebra of $(\mathcal{B},d())$-type, then $A$ has a graded monomial ordering ${\prec }_{gr}$. It is also shown that $A$ is an $\mathbb{N}$-filtered algebra of $(\mathcal{B},d())$-type if and only if $A$ has a graded momomial ordering, where $\mathcal{B}$ is the PBW basis of $A$.

15 pages