Some aspects of the theory of quantum groups. (English. Russian original) Zbl 0839.17011

Russ. Math. Surv. 48, No. 6, 41-79 (1993); translation from Usp. Mat. Nauk 48, No. 6(294), 39-74 (1993).
This is a survey of some of the basic material on quantum groups. From the author’s introduction we quote: “We do not set ourselves the goal of surveying all the various theoretical and applied aspects of quantum groups, but prefer to take one example – multiparameter deformation of the group \(GL_n\) – in more or less detail, stating and partially proving some general results on the way. Thus we hope to reproduce a significant part of the theory of quantum groups which is useful for a first acquaintance with the subject.”
The main topics treated are: Basic definitions in Hopf algebra theory, \(R\)-matrix algebras, representations of quantum groups, quantum flag spaces, Schur algebras, the Frobenius morphism, noncommutative differential calculus, quantum Weyl algebras.


17B37 Quantum groups (quantized enveloping algebras) and related deformations
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
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