Quantum groups at roots of 1. (English) Zbl 0714.17013

Lusztig, George

Geom. Dedicata 35, No. 1-3, 89-114 (1990).

The quantized deformation \(U\) of an enveloping algebra corresponding to the Cartan matrix of a Lie algebra is considered as a \({\mathbb{Q}}(v)\)-algebra. \({\mathbb{Q}}\) denotes the field of rationals and \(v\) is the quantum parameter. A subalgebra is defined as a \(Z[v,v^{-1}]\)-module. The author obtains a braid group action on \(U\) and describes a \({\mathbb{Q}}(v)\)-basis of \(U\). The question of a \(Z[v,v^{-1}]\)-basis of the subalgebra is reduced to rank two and solved explicitly in that case. In the last part \(v\) is assumed to be an \(\ell\)-th root of unity, \(\ell\) odd and in some cases not divisible by 3. A surjective Hopf algebra homomorphism is constructed mapping the quantum algebra onto the ordinary enveloping algebra, the kernel is a finite-dimensional Hopf algebra.
For some details in proof and an enlarged understanding the interested reader is referred to the author’s paper [J. Am. Math. Soc. 3, No. 1, 257–296 (1990; Zbl 0695.16006)].

Reviewer: Helmut Boseck (Greifswald)

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
16T20	Ring-theoretic aspects of quantum groups
20F36	Braid groups; Artin groups

Keywords:

quantum groups; quantum Hopf algebras; Cartan matrix; braid group