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.