Quantum upper triangular matrix algebras

Abstract

Following the ideas in~\cite{yM88}, \cite{T90} and inspiration from~\cite{KO24}, we construct a bialgebra $T_q(n)$ and a pointed Hopf algebra $U{T}_q(n)$ which quantize the coordinate rings of the algebra of upper triangular matrices and of the group of invertible upper triangular matrices of size $n\ge 2$, respectively, where $q$ is a nonzero parameter. The resulting structure on $U{T}_q(n)$ is neither commutative nor cocommutative and it can be seen as a Hopf quotient of the Takeuchi's two-parameter quantization~\cite{T90} of $\mathrm{GL}(n)$ corresponding to a specific choice of parameters. 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. We further study and compare the Lie algebras of derivations, the automorphism groups and the low degree Hochschild cohomology of these algebras in case $n=2$.