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

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