Abstract. Let Bq be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix q. We consider the graded dual Lq of the distinguished pre-Nichols algebra Bq from [A3] and the quantum divided power algebra Uq, a suitable Drinfeld double of Lq#kZ θ . We provide basis and presentations by generators and relations of Lq and Uq, and prove that they are noetherian and have finite Gelfand-Kirillov dimension.

Let Bq be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix q ∈ k θ×θ . Let Lq be the Lusztig algebra associated to Bq [AAR]. We present Lq as an extension (as braided Hopf algebras) of Bq by Zq where Zq is isomorphic to the universal enveloping algebra of a Lie algebra nq. We compute the Lie algebra nq when θ = 2.

A new algorithm for computing branching rules and Clebsch-Gordan coefficients of unitary representations of compact groups