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Quantum groups. (English) Zbl 0667.16003
Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798-820 (1987).
[For the entire collection see Zbl 0657.00005.]
In the last decade rather rich algebraic structures as bialgebras, Hopf algebras etc. have become important in the context of quantum theory. The paper under review is a report on Hopf algebras and connected structures in that field of mathematical physics. The motivation for definitions and examples comes from the study of integrable quantum systems. The question “What is a quantum group?” heading point 1. is answered by the following phrases: “It is important, that a quantum group is not a group, nor even a group object in the category of quantum spaces” and “So the notions of Hopf algebra and quantum group are in fact equivalent, but the second one has some geometric flavor.” Starting with the function algebra on a group as a commutative Hopf algebra the general notion is discussed and the interest in Hopf algebras which are neither commutative nor cocommutative is sketched. In spite of constructing examples, the notion of Poisson- and coPoisson-Hopf algebra and its quantization is introduced. Quantization of an associative $$k$$-algebra $$A_ 0$$ is a deformation into a $$k[[h]]$$-algebra $$A$$ such that the quotient A/hA with its natural Poisson structure equals $$A_ 0$$. A Poisson-Lie group is a Lie group with a Poisson bracket on its function space making this a Poisson-Hopf algebra. Duality theory implies a close connection with Lie bialgebras as formulated in
Theorem 1. The category of connected and simply connected Poisson-Lie groups is equivalent to the category of finite dimensional Lie bialgebras.
Theorem 2. Let $$\delta: U\mathfrak g\to U\mathfrak g\times U\mathfrak g$$ be a Poisson cobracket which makes $$U\mathfrak g$$ a coPoisson-Hopf algebra. Then $$\delta (\mathfrak g)\subseteq \mathfrak g\otimes \mathfrak g$$ and $$(\mathfrak g,\delta | \mathfrak g)$$ is a Lie bialgebra.” $$U{\mathfrak g}$$ denotes the enveloping algebra of the Lie algebra $$\mathfrak g$$. There are examples of Lie bialgebras.
It is pointed out, how a Kac-Moody algebra with a fixed invariant scalar product admits the structure of a Lie bialgebra. The notion of a coboundary triangular Lie bialgebra is defined with respect to solutions of the classical Yang-Baxter equation. The quantizations of the examples of Lie bialgebras turn out to be closely related to trigonometric solutions of the quantum Yang-Baxter equation. The quantized enveloping algebras and the quantized formal series Hopf algebras are discussed and presented as dual objects to each other. The notion of coboundary triangular Hopf algebra is introduced and considering representations it is noted how triangular quantized enveloping algebras can be constructed from cocommutative Hopf algebras and are in fact isomorphic (as algebras) to enveloping algebras. The quantum inverse scattering method is sketched from the Hopf algebra point of view. There are a lot of hints and references to the 83 papers cited at the end of the report.

##### MSC:
 16T05 Hopf algebras and their applications 16T10 Bialgebras 16T15 Coalgebras and comodules; corings 16T20 Ring-theoretic aspects of quantum groups 16T25 Yang-Baxter equations 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 81U40 Inverse scattering problems in quantum theory 53D17 Poisson manifolds; Poisson groupoids and algebroids 53D55 Deformation quantization, star products