Lie group representations that appeared mysteriously useful in the theory of p-adic Whit-taker functions, particularly on the metaplectic group. Pursuing this phenomenon leads to surprising connections with quantum groups and the Yang-Baxter equation. Also in the p-adic theory, Iwahori Whittaker functions are connected with representations of Hecke.

Template:Group theory sidebar. In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra.There is no single, all-encompassing definition, but instead a family of broadly similar objects.

In this paper we propose versions of the associative Yang-Baxter equation and higher order R-matrix identities which can be applied to quantum dynamical R-matrices. As is known quantum non-dynamical R-matrices of Baxter-Belavin type satisfy this equation. Together with unitarity condition and skew-symmetry it provides the quantum Yang-Baxter equation and a set of identities useful for.A novel approach to the construction of the universal R-matrix in the quantum group is presented.A general from of the universal R-matrix for arbitrary matrix quantum groups is derived.Withthe use of the generalization of the induced representation technique particular factorized scattering matrices associated with the considered quantum groups are determined.The relationship between classical and quantum theory is of central importance to the philos-ophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of.

The universal R-matrix allows applications of quantum groups in the construction of invariants of knots and links. The main component of the universal R-matrix is a quasi R-matrix, which has applications in other areas of representation theory, for instance in Lusztig’s and Kashiwara’s theory of canonical bases. Also essential to the theory of quantised enveloping algebras is the existence.

Read MoreLusztig's quantum group. Tilting modules. The Fusion Category in Action. Lecture 17. Affine Lie algebras. The fusion ring. Lecture 18. Kashiwara Crystals. Crystal bases and quantum groups. Crystals of tableaux. A peek at tableau combinatorics. Exercises. There are exercises at the last page in Lectures 2,3,4,5 and 9. Recommended texts.

Read MoreWe propose a general theory to study semidirect products of C -quantum groups in the framework of multiplicative unitaries. Starting from a quantum group with a projection we decompose its multiplicative unitary as a product of two unitary operators. One of them is again a multiplicative unitary in the standard sense; it describes the quotient.

Read MoreAbstract. Using the language of -Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group,, from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra .We apply the generalized FRST construction and obtain an -bialgebroid .Natural analogs of the exterior algebra and their matrix.

Read MoreThe central object of the quantum group approach is the universal R-matrix being an element of the tensor product of two copies of the quantum loop algebra. The integrability objects are constructed by choosing representations for the factors of that tensor product 7.

Read MoreIn the Lorentz group case, this perturbative approach is proved to coincide with the algebraic and combinatorial approach for knot invariants defined out of the formal R-matrix and formal ribbon elements in the Quantum Lorentz Group, and its infinite dimensional unitary representations.

Read MoreWe propose an algebraic scheme for quantizing the rational Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special parametrization of the cotangent bundle over. In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical -matrix. Quantizing both of them we find the quantum L-operator algebra and construct its.

Read MoreBerkeley Lectures on Lie Groups and Quantum Groups Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin, and Vera Serganova Last updated January 31, 2020.

Read MoreStructure of QUE algebras: the universal R-matrix; 9. Specializations of QUE algebras; 10. Representations of QUE algebras: the generic case; 11. Representations of QUE algebras: the root of unity case; 12. Infinite-dimensional quantum groups; 13. Quantum harmonic analysis; 14. Canonical bases; 15. Quantum group invariants of knots and 3-manifolds; 16. Quasi-Hopf algebras and the Knizhnik.

Read MoreThe 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.

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