Universal R-Matrix for Esoteric Quantum Groups

Journal of Pure and Applied Algebra, 2001

Letĝ be an untwisted affine Kac-Moody algebra. The quantum group U q (ĝ) is known to be a quasitriangular Hopf algebra (to be precise, a braided Hopf algebra). Here we prove that its unrestricted specializations at odd roots of 1 are braided too: in particular, specializing q at 1 we have that the function algebra F H of the Poisson proalgebraic group H dual of G (a Kac-Moody group with Lie algebraĝ) is braided. This in turn implies also that the action of the universal R-matrix on the tensor products of pairs of Verma modules can be specialized at odd roots of 1.

Journal of Geometry and Physics, 1988

A relation between quantum R-matrices and certain factorization problem in Hopf algebras is established. A definition of dressing transformation in the quantum case is also given. The term "Quantum groups" has been recently proposed by V. Drinfeld [3] to cover a specific class of Hopf algebras that are intrinsically connected with the Quantum Inverse Scattering Method [1, 2]. As a matter of fact, the invention of Q.l.S.M. has provided a vast source of new examples for the theory of Hopf algebras. In this respect we should first of all mention of definition of quantized enveloping algebras of simple Lie algebras given by Drinfeld [4] and Jimbo [5] following the work of Kulish and Reshetikhin [6 ] that deals with the sl(2) case. An alternative approach which keeps much closer to the original ideas of Q.I.S.M. has been developed by Faddeev, Reshetikhin and Takhtajan in [7]. The theory of Quantum groups has as its semi-classical counterpart the theory of Poisson Lie group, that is Lie groups equipped with Poisson bracket such that group multiplication is a Poisson mapping. The theory of Poisson Lie groups is also relatively new [8] and was motivated both by its relation to Quantum groups and by applications to an important class of classical integrable systems on 1-dimensional lattices described by difference I..ax equations [9].

Arxiv preprint math-ph/0105002, 2001

R. JAGANATHAN The Institute of Mathematical Sciences 4th Cross Road, Central Institutes of Technology Campus, Tharamani Chennai - 600 113, India E-Mail:jagan@imsc.ernet.in ... Some very elementary ideas about quantum groups and quantum algebras are introduced and ...

1997

Journal of Mathematical Physics, 2015

Let g be a complex orthogonal or symplectic Lie algebra and g ′ ⊂ g the Lie subalgebra of rank rk g ′ = rk g − 1 of the same type. We give an explicit construction of generators of the Mickelsson algebra Z q (g, g ′ ) in terms of Chevalley generators via the R-matrix of U q (g).

Communications in Mathematical Physics, 1992

We construct complexified versions of the quantum groups associated with the Lie algebras of type A n-\, B n , C n , and D n. Following the ideas of Faddeev, Reshetikhin and Takhtajan we obtain the Hopf algebras of regular functionals Ujg on these complexified quantum groups. In the special example A\ we derive the g-deformed enveloping algebra U q (sl(2,C)). In the limit q-> 1 the undeformed U(sl(2, C)) is recovered.

Lie Theory and Its Applications in Physics V, 2004

In the classification of solutions of the Yang-Baxter equation, there are solutions that are not deformations of the trivial solution (essentially the identity). We consider the algebras defined by these solutions, and the corresponding dual algebras. We then study the representations of the latter. We are also interested in the Baxterisation of these R-matrices and in the corresponding quantum planes.

This is an introduction to the quantum groups, or rather to the simplest quantum groups. The idea is that the unitary group $U_N$ has a free analogue $U_N^+$, whose standard coordinates $u_{ij}\in C(U_N^+)$ are allowed to be free, and the closed subgroups $G\subset U_N^+$ can be thought of as being the compact quantum Lie groups. There are many interesting examples of such quantum groups, for the most designed in order to help with questions in quantum mechanics and statistical mechanics, and some general theory available as well, including Peter-Weyl theory, Tannakian duality, Brauer theorems and Weingarten integration. We discuss here the basic aspects of all this.

This work is dedicated to my loving wife Emi for all her support and understanding; to my wise and patient mentor Metin Arık for teaching me a lot of what I know and to my parents who have shown me how to think scientifically about nature. iv ABSTRACT QUANTUM GROUP STRUCTURES ASSOCIATED WITH INVARIANCES OF SOME PHYSICAL ALGEBRAS In this study, the anticommuting spin algebra is introduced and it is shown to be invariant under the action of the quantum group SO q=−1 (3). Furthermore, its representations and Hopf algebra structure are studied and found to be closely resemble the similar results for the angular momentum algebra. The invariance properties of the bosonic and fermionic oscillator algebras under inhomogeneous transformations are also studied. The bosonic inhomogeneous symplectic group, BISp(2d, R) , and the fermionic inhomogeneous orthogonal group, F IO(2d, R) , are defined as the inhomogeneous invariance quantum groups of these algebras. The sub(quantum)groups and contractions of these quantum groups are studied as a source for new quantum groups. Finally, the fermionic inhomogeneous orthogonal quantum group is defined for odd number of dimensions and its sub(quantum)groups and contractions are studied. inho v OZET BAZI FİZİKSEL CEBİRLERİN DEGİŞMEZLİGİİLEİLGİLİ KUANTUM GRUP YAPILARI Buçalışmada, ters-degişmeli spin cebri tanımlanmış ve bu cebrin SO q=−1 (3)

2002

In this paper inhomogeneous group G is a semidirect product of a finite-dimensional vector space V by a group G0 of linear transformations of V . Then V is a normal subgroup of G and G0 = G/V is the factor group. The Hopf algebra A = Poly(G) of polynomial functions on G is generated by the sub-algebra A0 = Poly(G0) and N distinguished linearly independent elements elements x1, x2, . . . xN (N = dim V ) being the linear forms on V . The comultiplication ∆ restricted to A0 coincides with the one describing the group structure of G0 and for any k = 1, 2, . . . , N