Quantum upper triangular matrix algebras

Following the ideas in~\cite{yM88} and some 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, 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 $U{T}_q(n)$, we study and compare their Lie algebras of derivations, their automorphism groups and their low degree Hochschild cohomology, in case $n=2$.