A Quantum Version of the Algebra of Distributions of SL2

Let A be a primitive root of unity of order. We introduce a family of finite-dimensional algebras {D-lambda,D-N(sl(2))}(N is an element of N0) over the complex numbers, such that D-lambda,D-N (sl(2)) is a subalgebra of D-lambda,D-M(sl(2)) if N < M, and D-lambda,D-N-1(sl(2)) subset of D-lambda,D-N(sl(2)) is a u(lambda)(sl(2))-cleft extension. The simple D-lambda,D-N(sl(2))-modules (L-N(p))(0 <= p<l)(N+perpendicular to), are highest weight modules, which admit a tensor product decomposition: the first factor is a simple u(lambda)(sl(2))-module and the second factor is a simple D-lambda,D-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 SL2 as a filtration by finite-dimensional subalgebras.