Quantum upper triangular matrix algebras

TL;DR

This paper constructs quantum analogs of upper triangular matrix algebras and groups, exploring their algebraic structures, derivations, automorphisms, and cohomology, providing new insights into quantum groups related to poset incidence algebras.

Contribution

It introduces the bialgebra $T_q(n)$ and the Hopf algebra $UT_q(n)$ that quantize classical upper triangular matrix structures, extending quantum group theory.

Findings

The structure on $UT_q(n)$ is neither commutative nor cocommutative.

Comparison of derivations and automorphisms for $n=2$.

Analysis of low-degree Hochschild cohomology for $n=2$.

Abstract

Following the ideas in~\cite{yM88} and some inspiration from~\cite{KO24}, we construct a bialgebra $T_q(n)$ and a pointed Hopf algebra $UT_q(n)$ which quantize the coordinate rings of the algebra of upper triangular matrices and of the group of invertible upper triangular matrices of size $n\geq 2$, respectively, where $q$ is a nonzero parameter. The resulting structure on $UT_q(n)$ is neither commutative nor cocommutative, so we obtain a quantum group. The motivation comes from the idea of quantizing the incidence algebra of a finite poset, as the latter can be embedded as a subalgebra of the algebra of upper triangular matrices. After defining the bialgebra $T_q(n)$ and the Hopf algebra $UT_q(n)$, we study and compare their Lie algebras of derivations, their automorphism groups and their low degree Hochschild cohomology, in case $n=2$.