# A Quantum Version of the Algebra of Distributions of SL$_{2}$

### Iván Ezequiel Angiono

Universidad Nacional de Córdoba, Argentina

## Abstract

Let $λ$ be a primitive root of unity of order $ℓ$. We introduce a family of finite-dimensional algebras ${D_{λ,N}sl_{2}}_{N∈N_{0}}$ over the complex numbers, such that $D_{λ,N}sl_{2}$ is a subalgebra of $D_{λ,M}sl_{2}$ if $N<M$, and $D_{λ,N−1}sl_{2}⊂D_{λ,N}sl_{2}$ is a $u_{λ}(sl_{2})$-cleft extension.

The simple $D_{λ,N}sl_{2}$-modules $(L_{N}p)_{0≤p<ℓ_{N+1}}$ are highest weight modules, which admit a tensor product decomposition: the first factor is a simple $u_{λ}(sl_{2})$-module and the second factor is a simple $D_{λ,N−1}sl_{2}$-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.

## Cite this article

Iván Ezequiel Angiono, A Quantum Version of the Algebra of Distributions of SL$_{2}$. Publ. Res. Inst. Math. Sci. 54 (2018), no. 1, pp. 141–161

DOI 10.4171/PRIMS/54-1-5