Degree of parabolic quantum groups

Abstract

We study some elementary properties of the quantum enveloping algebra associated to a parabolic subalgebra $\mathfrak{p}$ of a semisimple Lie algebra $\mathfrak{g}$. In particular we prove an explicit formula for the degree of this algebra, that extends the well known formula for the quantum enveloping algebra associated to $\mathfrak{g}$ and $\mathfrak{b}$, where $\mathfrak{b}i$ is a Borel subalgebra of $\mathfrak{g}$.