Chari, Vyjayanthi & Andrew Pressley - Guide to quantum groups
Quantum groups first arose in the physics literature, particularly in the work
of L. D. Faddeev and the Leningrad school, from the 'inverse scattering
method', which had been developed to construct and solve 'integrable' quan-
tum systems. They have excited great interest in the past few years because
of their unexpected connections with such, at first sight, unrelated parts of
mathematics as the construction of knot invariants and the representation
theory of algebraic groups in characteristic p.
In their original form, quantum groups are associative algebras whose defin-
ing relations are expressed in terms of a matrix of constants (depending on the
integrable system under consideration) called a quantum R-matrix. It was
realized independently by V. (]. Drinfel'd and M. Jimbo around 1985 that
these algebras are Hopf algebras, which, in many cases, are deformations of
'universal enveloping algebras' of Lie algebras. A little later, Yu. I. Manin
and S. L. Woronowicz independently constructed non-commutative deforma-
tions of the algebra of functions on the groups $L2() and $U2, respectively,
and showed that many of the classical results about algebraic and topological
groups admit analogues in the non-commutative case.
Thus, although many of the fundamental papers on quantum groups are
written in the language of integrable systems, their properties are accessible
by more conventional mathematical techniques, such as the theory of topo-
logical and algebraic groups and Lie algebras. Our aim in this book is to
present the theory of quantum groups from this latter point of view. In fact,
we shall concentrate on the study of the 'Lie algebras' of quantum groups,
which seems to be the approach which has proved most powerful, particularly
in applications, but we shall also discuss, in rather less detail, their relation
with 'non-commutative algebraic geometry and topology'.
- Paperback: 668 pages
- Publisher: Cambridge University Press; 1st PB Edition edition (October 27, 1995)
- Language: English
- ISBN-10: 0521558840
- ISBN-13: 978-0521558846
- Product Dimensions: 0.9 x 0.6 x 0.1 inches
Quantum groups first arose in the physics literature, particularly in the work
of L. D. Faddeev and the Leningrad school, from the 'inverse scattering
method', which had been developed to construct and solve 'integrable' quan-
tum systems. They have excited great interest in the past few years because
of their unexpected connections with such, at first sight, unrelated parts of
mathematics as the construction of knot invariants and the representation
theory of algebraic groups in characteristic p.
In their original form, quantum groups are associative algebras whose defin-
ing relations are expressed in terms of a matrix of constants (depending on the
integrable system under consideration) called a quantum R-matrix. It was
realized independently by V. (]. Drinfel'd and M. Jimbo around 1985 that
these algebras are Hopf algebras, which, in many cases, are deformations of
'universal enveloping algebras' of Lie algebras. A little later, Yu. I. Manin
and S. L. Woronowicz independently constructed non-commutative deforma-
tions of the algebra of functions on the groups $L2() and $U2, respectively,
and showed that many of the classical results about algebraic and topological
groups admit analogues in the non-commutative case.
Thus, although many of the fundamental papers on quantum groups are
written in the language of integrable systems, their properties are accessible
by more conventional mathematical techniques, such as the theory of topo-
logical and algebraic groups and Lie algebras. Our aim in this book is to
present the theory of quantum groups from this latter point of view. In fact,
we shall concentrate on the study of the 'Lie algebras' of quantum groups,
which seems to be the approach which has proved most powerful, particularly
in applications, but we shall also discuss, in rather less detail, their relation
with 'non-commutative algebraic geometry and topology'.
