A Quantum Version of the Algebra of Distributions of SL$_2$

Abstract

Let $\lambda$ be a primitive root of unity of order $\ell$. We introduce a family of finite-dimensional algebras $\{\mathcal{D}_{\lambda, N}{\mathfrak{sl}_2}\}_{N\in\mathbb N_0}$ over the complex numbers, such that $\mathcal{D}_{\lambda, N}{\mathfrak{sl}_2}$ is a subalgebra of $\mathcal{D}_{\lambda,M}{\mathfrak{sl}_2}$ if $N < M$, and $\mathcal{D}_{\lambda, N-1}{\mathfrak{sl}_2}\subset \mathcal{D}_{\lambda, N}{\mathfrak{sl}_2}$ is a $\mathfrak u_{\lambda}(\mathfrak{sl}_2)$-cleft extension.

The simple $\mathcal{D}_{\lambda,N}{\mathfrak{sl}_2}$-modules $(\mathcal{L}_{N}{p})_{0\le p<\ell^{N+1}}$ are highest weight modules, which admit a tensor product decomposition: the first factor is a simple $\mathfrak u_{\lambda}(\mathfrak{sl}_2)$-module and the second factor is a simple $\mathcal{D}_{\lambda,N-1}{\mathfrak{sl}_2}$-module. This factorization resembles the corresponding Steinberg decomposition, and the family of algebras resembles the presentation of the algebra of distributions of SL$_2$ as a filtration by finite-dimensional subalgebras.