A finite-dimensional Lie algebra arising from a Nichols algebra of diagonal type (rank 2)

Abstract

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.