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Title: Quantum Groups (Graduate Texts in Mathematics, Vol 155) by Christian Kassel, Ollman, Park ISBN: 0-387-94370-6 Publisher: Springer Verlag Pub. Date: October, 1995 Format: Hardcover Volumes: 1 List Price(USD): $69.95 |
Average Customer Rating: 5 (2 reviews)
Rating: 5
Summary: For quantum mathematicians
Comment: This book is further evidence of the tremendous influence that quantum physics, especially quantum field theory and superstring theory, has had on modern mathematics. Very rich mathematical structures and simplified methods of proof have resulted from looking at mathematics from a quantum point of view. Because of the enormous success of viewpoint, examples being proofs of the Atiyah-Singer index theorem, the Jones polynomial, and the Seiberg-Witten equations, one could justify a rephrasing of the remark by Eugene Wigner and now speak of "the reasonable effectiveness of physics in mathematics".
The book gives a fine overview of a field that has only been around for a few decades, and is manifested by brilliant developments. Those who have worked with the Yang-Baxter equations from the theory of exactly solved models in statistical mechanics will see these equations come alive here in a much more general form. In addition, knot theorists and geometric topologists will appreciate the discussion of how their constructions can be cast in the tensor and tangle categories that are explained in detail in this book. The title of the book is a little strange, given that the structures treated are more specific than groups, but the author has explained well the theory of quantum groups, as is it is now referrred to in journal classification schemes.
An in-depth reading of the book is time-consuming, and no doubt the average reader will not read it from cover to cover but instead will peruse only the areas of immediate interest. Part One of the book is an overview of what the author calls quantum SL(2), which is an example of a Hopf algebra. The first two chapters are purely a review of algebra, with the third being an introduction to coalgebras, which the author, in a categorical sense, identifies as being dual to an algebra. The notion of a bialgebra is also discussed, which is essentially a vector space equipped with both an algebra structure and a coalgebra structure. Taking a tensor product of this vector space with itself and examining certain morphisms between these structures gives a set of compatibility conditions that define the bialgebra structure. A Hopf algebra is then a bialgebra that has a special endomorphism of the underlying vector space. The algebraic topologist reader will be familiar with Hopf algebras via studies of product manifolds such as Lie groups. Quantum groups have given many examples of non-commutative non-cocommutative bialgebras than were known before this research area had taken off. The author also discusses the quantum plane as an object that generalizes the affine plane, namely the two variables x, y generating the plane no longer commute but instead satisfy yx = q xy. The author investigates in detail the quantum group SLq(n), which is based on the classical Lie group. References are given for quantum groups based on the other Lie groups, such as the orthogonal and symplectic groups. The Lie algebra Uq(sl(2)) is given a detailed treatment by the author when q is not a root of unity. This Hopf algebra is a 1-parameter deformation of the enveloping algebra of the Lie algebra sl(2) considered in earlier chapters. The reader interested in the renormalization is strongly urged to read this first part, as recently it has been shown that for any quantum field theory, the combinatorics of Feynman diagrams gives rise to a Hopf algebra which is commutative as an algebra, and is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by one particle irreducible Feynman diagrams. The Lie bracket of two diagrams is computed from insertions of one graph in the other and vice versa, and the Lie group G is the group of characters of the Hopf algebra. This structure is used to go on and formulate the renormalization problem rigorously.
Part two is an overview of the famous Yang-Baxter equation whose exact solutions in terms of R-matrices have generated a vast amount of research. The author introduces the concept of a braided bialgebra, which contain a "universal" R-matrix which induces a solution of the Yang-Baxter equation on all of their modules, and thus giving a systematic method for constructing solutions of the Yang-Baxter equation. The duals of these bialgebras give a cobraided bialgebra, and the author shows how to construct a cobraided bialgebra out of any solution of the Yang-Baxter equation. It is also shown how the quantum groups GLq(2) and SLq(2) can be obtained by this method, and it is proven that they are cobraided. The famous Drinfeld quantum double, yielding a braided Hopf algebra out of any finite-dimensional Hopf algebra with invertible antipode, is discussed in great detail.
The next part is basically low-dimensional topology in the form of knots, links, and braids, wherein the author discusses the relationship between the Jones polynomial and R-matrices. The connection between knot theory and quantum groups is given by the representation theory of Hopf algebras, this connection taking place in the tensor category. A certain strict tensor category is built out of tangles, and shown to give isotopy invariants of links. Braiding in the tensor category is used to formalize the notion of crossing in link and tangle diagrams. Tensor categories modeled on framed tangles or "ribbons" are introduced to illustrate duality. The concept of a quasi-bialgebra is introduced and braid group representations of these are constructed. When quasi-bialgebras are equivalent under a "gauge transformation" introduced here, they have the same braid group representation.
The last part considers the role of monodromy in the theory of quantum groups. The quantum enveloping algebras due to Drinfeld and Jimbo are discussed and shown to provide isotopy invariants of links. The monodromy of the Knizhnik-Zamolodchikov system is shown to be equivalent to the braid group representation of this system. Knot invariants of finite type are shown to be universal invariants for quantum groups.
Rating: 5
Summary: Kassel's Quantum Groups
Comment: This is a very well-written book covering not only quantum groups but also the connections with low-dimensional topology. When comparing the presentation in the book with other presentations in the literature I realized how much work the author put in trying to make the material accessible. Part I covers Hopf algebras and quantum SL(2). Part II covers the Yang-Baxter equation and its solutions (R-matrices) and Drinfeld's quantum double construction. Part III deals with the applications to low-dimensional topology, i.e., the construction of knot invariants via braided tensor categories, in particular the construction of the Jones-Conway polynomial. Part IV is an account of Drinfeld's treatment of the monodromy of the Knizhnik-Zamolodchikov equations. The last chapter discusses finite type knot invariants, and discusses Cartier's combinatorial version of Kontsevich's universal knot invariant.
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