Computation of Gelfand-Kirillov dimension for $B$-type structures

Abstract

Let $\mathcal{O}(\mbox{Spin}_{q^{1/2}}(2n+1))$ and $\mathcal{O}(SO_q(2n+1))$ be the quantized algebras of regular functions on the Lie groups $\mbox{Spin}(2n+1)$ and $SO(2n+1)$, respectively. In this article, we prove that the Gelfand-Kirillov dimension of a simple unitarizable $\mathcal{O}(\mbox{Spin}_{q^{1/2}}(2n+1))$-module $V_{t,w}^{\mbox{Spin}}$ is the same as the length of the Weyl word $w$. We show that the same result holds for the $\mathcal{O}(SO_q(2n+1))$-module $V_{t,w}$, which is obtained from $V_{t,w}^{\mbox{Spin}}$ by restricting the algebra action to the subalgebra $\mathcal{O}(SO_q(2n+1))$ of $\mathcal{O}(\mbox{Spin}_{q^{1/2}}(2n+1))$. Moreover, we consider the quantized algebras of regular functions on certain homogeneous spaces of $SO(2n+1)$ and $\mbox{Spin}(2n+1)$ and show that its Gelfand-Kirillov dimension is equal to the dimension of the homogeneous space as a real differentiable manifold.