Degree cones and monomial bases of Lie algebras and quantum groups

TL;DR

This paper introduces degree filtrations on quantum groups and Lie algebras, leading to monomial bases and skew-polynomial algebra structures, with proven cases and conjectures for all types.

Contribution

It constructs specific filtrations on quantum groups that produce monomial bases and skew-polynomial structures, extending classical filtrations and proposing a general conjecture.

Findings

Filtrations induce skew-polynomial algebras on quantum groups.

Existence of degrees leading to monomial ideals in certain Lie types.

Conjecture that such degrees exist for all simple Lie algebras.

Abstract

We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}^-$. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an $\mathbb{N}$-filtration on any finite dimensional simple $\mathfrak{g}$-module. We prove for type $\tt{A}_n$, $\tt{C}_n$, $\tt{B}_3$, $\tt{D}_4$ and $\tt{G}_2$ that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$.