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Let $\lambda$ be a primitive root of unity of order $\ell$. We introduce afamily of finite-dimensional algebras$\{\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)\}_{N\in\mathbb{N}_0}$ over thecomplex numbers, such that $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$ is asubalgebra 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, whichadmit a tensor product decomposition: the first factor is a simple$\mathfrak{u}_{\lambda}(\mathfrak{sl}_2)$-module and the second factor is asimple $\mathcal{D}_{\lambda,N-1}(\mathfrak{sl}_2)$-module. This factorizationresembles the corresponding Steinberg decomposition, and the family of algebrasresembles the presentation of algebra of distributions of $\operatorname{SL}_2$as a filtration by finite-dimensional subalgebras.

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