# Introduction to quantum groups and crystal bases

This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum Yang-Baxter equation and quantum inverse scattering method developed by the Leningrad School Ludwig Faddeev , Leon Takhtajan , Evgeny Sklyanin , Nicolai Reshetikhin and Vladimir Korepin and related work by the Japanese School.

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One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfeld and Michio Jimbo as a deformation of the universal enveloping algebra of a semisimple Lie algebra or, more generally, a Kac—Moody algebra , in the category of Hopf algebras. The resulting algebra has additional structure, making it into a quasitriangular Hopf algebra.

There are various coassociative coproducts under which these algebras are Hopf algebras, for example,. Alternatively, the quantum group U q G can be regarded as an algebra over the field C q , the field of all rational functions of an indeterminate q over C. The center of quantum group can be described by quantum determinant. Just as there are many different types of representations for Kac—Moody algebras and their universal enveloping algebras, so there are many different types of representation for quantum groups.

As is the case for all Hopf algebras, U q G has an adjoint representation on itself as a module, with the action being given by. One important type of representation is a weight representation, and the corresponding module is called a weight module.

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A weight module is a module with a basis of weight vectors. A weight module is called integrable if the actions of e i and f i are locally nilpotent i. Of special interest are highest weight representations , and the corresponding highest weight modules.

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Similarly, a quantum group can have a lowest weight representation and lowest weight module, i. As is the case for all Hopf algebras, the tensor product of two modules is another module. In the specific case where G is a finite-dimensional Lie algebra as a special case of a Kac—Moody algebra , then the irreducible representations with dominant integral highest weights are also finite-dimensional.

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In the case of a tensor product of highest weight modules, its decomposition into submodules is the same as for the tensor product of the corresponding modules of the Kac—Moody algebra the highest weights are the same, as are their multiplicities. Strictly, the quantum group U q G is not quasitriangular, but it can be thought of as being "nearly quasitriangular" in that there exists an infinite formal sum which plays the role of an R -matrix.

The infinite formal sum is the product of two factors, [ citation needed ]. The formal infinite sum which plays the part of the R -matrix has a well-defined action on the tensor product of two irreducible highest weight modules, and also on the tensor product of two lowest weight modules. See also compact quantum group. Woronowicz introduced compact matrix quantum groups.

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The geometry of a compact matrix quantum group is a special case of a noncommutative geometry. Strictly, this does not make C G a Hopf algebra, unless G is finite. Note that u is a representation, but not a unitary representation. Note that w is a unitary representation.

## Introduction to Quantum Groups and Crystal Bases by Jin Hong, Seok-Jin Kang - pardusamederg.ga

Whereas compact matrix pseudogroups are typically versions of Drinfeld-Jimbo quantum groups in a dual function algebra formulation, with additional structure, the bicrossproduct ones are a distinct second family of quantum groups of increasing importance as deformations of solvable rather than semisimple Lie groups. This quantum group was linked to a toy model of Planck scale physics implementing Born reciprocity when viewed as a deformation of the Heisenberg algebra of quantum mechanics.

Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra the Iwasawa decomposition , and this provides a canonical bicrossproduct quantum group associated to g. For su 2 one obtains a quantum group deformation of the Euclidean group E 3 of motions in 3 dimensions.

From Wikipedia, the free encyclopedia. Algebraic construct of interest in theoretical physics.

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## Introduction Quantum Groups Crystal Bases by Jin Hong Seok Jin Kang

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