Uq(sl(2)) Satisfies a Bernstein Duality

Let R be a K-algebra over a field K. In the particular case of an enveloping algebra of a finite-dimensional solvable Lie algebra a local duality was studied by G. Barou and M. P. Malliavin in Barou and Malliavin (1985). Later this duality was extended to a wide class of algebras by the authors (see, Gómez et al., 1996; Jara and Jódar, 2000). Algebras in this class satisfy certain properties: Auslander–Gorenstein condition, they are Cohen–Macaulay, idim(R) = GKdim(R) and in addition they have the strong second layer condition. We refer to Gómez et al. (1996) and Jara and Jódar (2000) as basic papers on this theory where appear many examples of algebras satisfying all these properties. Local dualities defined by R, the underlying R-bimodule of the dual coalgebra R, are characterized as those such that, for any cofinite prime ideal P of R, the K-dimension of R/P and its image by the duality are equal. See Proposition (1.17) in Jara and Jódar (2000). Later, applying this study to the Bernstein duality, i.e. the duality defined by ExtμR(−, R), being μ = idim(R) = GKdim(R), we show that in many examples R defines the Bernstein duality. See Section 2 in Jara and Jódar (2000). The aim of this work is to extend the number of examples in which the Bernstein duality may be defined by R. In this case we study the algebra Uq(sl(2)), being q a root of unity (in the case in which q is not a root of unity the algebra Uq(sl(2)) does not satisfy the strong second layer condition). Therefore first we classify the finite dimensional irreducible representations M of Uq(sl(2)), after that we study what homological properties satisfies Uq(sl(2)) to finally study the dimension of ExtμUq(sl(2))(M,Uq(sl(2))). The paper is divided into three sections. The first two sections include the necessary background. In Section 1 we show some properties of Uq(sl(2)) and deal with the finite dimensional irreducible representations of Uq(sl(2)); our main references are Kassel’s and Jantzen’s books, where we address the readers to get complete proofs of the results