A quantum version of the algebra of distributions of ${\mathrm{SL}}_2$

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 algebra of distributions of ${\mathrm{SL}}_2$ as a filtration by finite-dimensional subalgebras.

16 pages, comments are welcome. Second version: 18 pages, some preliminaries were added

References in corpus (1)

• Algebraic representations and constructible sheaves

Cited by in corpus (2)

• Billiards and Tilting Characters for ${\mathrm{SL}}_3$
• Algebraic representations and constructible sheaves