Abstract
We study the Hopf algebra structure of Lusztig's quantum groups. First we show that the zero part is the tensor product of the group algebra of a finite abelian group with the enveloping algebra of an abelian Lie algebra. Second we build them from the plus, minus and zero parts by means of suitable actions and coactions within the formalism presented by Sommerhauser to describe triangular decompositions.
Highlights
There are two versions of quantum groups at roots of 1: the one introduced and studied by De Concini, Kac and Procesi [9, 10] and the quantum divided power algebra of Lusztig [15, 16, 17, 18]
The small quantum groups appear as quotients of the first and Hopf subalgebras of the second; in both cases they fit into suitable exact sequences of Hopf algebras
We show in Theorem 3.10 that it splits as the tensor product of the group algebra of a finite group and the enveloping algebra of the Cartan subalgebra of the corresponding Lie algebra
Summary
There are two versions of quantum groups at roots of 1: the one introduced and studied by De Concini, Kac and Procesi [9, 10] and the quantum divided power algebra of Lusztig [15, 16, 17, 18]. The graded duals of those distinguished pre-Nichols algebras were studied in [4] under the name of Lusztig algebras; when the braiding is of Cartan type one recovers in this way the positive (and the negative) parts of Lusztig’s quantum groups. In order to construct the analogues of Lusztig’s quantum groups at roots of one for Nichols algebras of diagonal type, we still need to define the 0-part and the interactions with the positive and negative parts. This leads us to understand the Hopf algebra structure of a Lusztig’s quantum group which is the objective of this Note. Other versions of triangular decompositions similar to [23] appear in [12, 20]
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