A quantum version of the algebra of distributions of operatorname{SL}₂

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 $\operatorname{SL}_2$ as a filtration by finite-dimensional subalgebras.