Generalization of Drinfeld quantum affine algebras (original) (raw)
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Generalization and Deformation of Drinfeld quantum affine algebras
1996
Drinfeld gave a current realization of the quantum affine algebras as a Hopf algebra with a simple comultiplication for the quantum current operators. In this paper, we will present a generalization of such a realization of quantum Hopf algebras. As a special case, we will choose the structure functions for this algebra to be elliptic functions to derive certain elliptic
Encyclopedia of Mathematical Physics, 2006
Affine quantum groups are certain pseudo-quasitriangular Hopf algebras that arise in mathematical physics in the context of integrable quantum field theory, integrable quantum spin chains, and solvable lattice models. They provide the algebraic framework behind the spectral parameter dependent Yang-Baxter equation (0.1) R 12 (u)R 13 (u + v)R 23 (v) = R 23 (v)R 13 (u + v)R 12 (u). One can distinguish three classes of affine quantum groups, each leading to a different dependence of the R-matrices on the spectral parameter u: Yangians lead to rational R-matrices, quantum affine algebras lead to trigonometric R-matrices and elliptic quantum groups lead to elliptic R-matrices. We will mostly concentrate on the quantum affine algebras but many results hold similarly for the other classes. After giving mathematical details about quantum affine algebras and Yangians in the first two section, we describe how these algebras arise in different areas of mathematical physics in the three following sections. We end with a description of boundary quantum groups which extend the formalism to the boundary Yang-Baxter (reflection) equation.
1997
We consider the algebra isomorphism found by Frenkel and Ding between the RLL and the Drinfeld realizations of Uq(widehatgl(2))U_q(\widehat{gl(2)})Uq(widehatgl(2)). After we note that this is not a Hopf algebra isomorphism, we prove that there is a unique Hopf algebra structure for the Drinfeld realization so that this isomorphism becomes a Hopf algebra isomorphism. Though more complicated, this Hopf algebra structure is also closed, just as the one found previously by Drinfeld.
Infinite Hopf Families of Algebras and Yang-Baxter Relations
2001
A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure of an infinite Hopf family of (super-) algebras is proposed. The structure of the infinite Hopf family is given explicitly on the level of LLL matrices. The relation with the Drinfeld current realization is established in the case of 4times44\times44times4 RRR-matrices by studying the analogue of the Ding-Frenkel theorem. By use of the concept of algebra ``comorphisms'' (which generalize the notion of algebra comodules for standard Hopf algebras), a possible way of constructing infinitely many commuting operators out of the generalized RLLRLLRLL algebras is given. Finally some examples of the generalized RLLRLLRLL algebras are briefly discussed.
The Hopf algebra of vector fields on complex quantum groups
1992
We derive the equivalence of the complex quantum enveloping algebra and the algebra of complex quantum vector fields for the Lie algebra types An, B n , en and Dn by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals.
Quantum affine algebras and their representations
Journal of Pure and Applied Algebra, 1995
We prove a highest weight classification of the finite-dimensional irreducible representations of a quantum affine algebra, in the spirit of Cartan's classification of the finite-dimensional irreducible representations of complex simple Lie algebras in terms of dominant integral weights. We also survey what is currently known about the structure of these representations.
Quantum Affine Algebras at Roots of Unity
1996
We study the restricted form of the qaunatized enveloping algebra of an untwisted affine Lie algebra and prove a triangular decomposition for it. In proving the decomposition we prove several new identities in the quantized algebra, one of these show a connection between the quantized algebra and Young diagrams. These identities are all invisible in the non-quantum case of the problem which was considered by Garland in 1978. We then study the finite-dimensional irreducible representations and prove a factorization theorem for such representations.
The Euclidean Hopf algebra Uq(eN) and its fundamental Hilbert-space representations
Journal of Mathematical Physics, 1995
We construct the Euclidean Hopf algebra U q (e N ) dual of F un(R N q >⊳SO q −1 (N)) by realizing it as a subalgebra of the differential algebra Diff (R N q ) on the quantum Euclidean space R N q ; in fact, we extend our previous realization [1] of U q −1 (so(N)) within Diff (R N q ) through the introduction of q-derivatives as generators of q-translations. The fundamental Hilbert space representations of U q (e N ) turn out to be of highest weight type and rather simple " lattice-regularized " versions of the classical ones. The vectors of a basis of the singlet (i.e. zero-spin) irrep can be realized as normalizable functions on R N q , going to distributions in the limit q → 1.
Polyadic Hopf Algebras And Quantum Groups
This article continues the study of concrete algebra-like structures in our polyadic approach , when the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions. In this way, the associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, the dimension of the algebra can be not arbitrary, but " quantized " ; the polyadic convolution product and bialgebra can be defined, when algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to the quantum group theory, we introduce the polyadic version of the braidings, almost co-commutativity, quasitriangularity and the equations for R-matrix (that can be treated as polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.