26 research outputs found
Unified description of quantum affine (super)algebras U_q(A_{1}^{(1)}) and U_q(C(2)^{(2)})
We show that the quantum affine algebra U_{q}(A_{1}^{(1)}) and the quantum
affine superalgebra U_{q}(C(2)^{(2)}) admit unified description. The difference
between them consists in the phase factor which is equal to 1 for
U_{q}(A_{1}^{(1)}) and is equal to -1 for U_{q}(C(2)^{(2)}). We present such a
description for the construction of Cartan-Weyl generators and their
commutation relations, as well for the universal R-matrices.Comment: 16 pages, LaTeX. Talk by V.N. Tolstoy at XIV-th Max Born Symposium
"New Symmetries and Integrable Models", Karpacz, September 1999; in press in
Proceedings, Ed. World Scientific, 200
Quantum Affine (Super)Algebras and
We show that the quantum affine algebra and the quantum
affine superalgebra admit a unified description. The
difference between them consists in the phase factor which is equal to 1 for
and it is equal to -1 for . We present
such a description for the actions of the braid group, for the construction of
Cartan-Weyl generators and their commutation relations, as well for the
extremal projector and the universal R-matrix. We give also a unified
description for the 'new realizations' of these algebras together with explicit
calculations of corresponding R-matrices.Comment: 22 pages, LaTe
Extremal projectors for contragredient Lie (super)symmetries (short review)
A brief review of the extremal projectors for contragredient Lie
(super)symmetries (finite-dimensional simple Lie algebras, basic classical Lie
superalgebras, infinite-dimensional affine Kac-Moody algebras and
superalgebras, as well as their quantum -analogs) is given. Some
bibliographic comments on the applications of extremal projectors are
presented.Comment: 21 pages, LaTeX; typos corrected, references adde
Q-power function over Q-commuting variables and deformed XXX, XXZ chains
We find certain functional identities for the Gauss q-power function of a sum
of q-commuting variables. Then we use these identities to obtain two-parameter
twists of the quantum affine algebra U_q (\hat{sl}_2) and of the Yangian
Y(sl_2). We determine the corresponding deformed trigonometric and rational
quantum R-matrices, which then are used in the computation of deformed XXX and
XXZ Hamiltonians.Comment: LaTeX, 12 page
On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization
We study classical twists of Lie bialgebra structures on the polynomial
current algebra , where is a simple complex
finite-dimensional Lie algebra. We focus on the structures induced by the
so-called quasi-trigonometric solutions of the classical Yang-Baxter equation.
It turns out that quasi-trigonometric -matrices fall into classes labelled
by the vertices of the extended Dynkin diagram of . We give
complete classification of quasi-trigonometric -matrices belonging to
multiplicity free simple roots (which have coefficient 1 in the decomposition
of the maximal root). We quantize solutions corresponding to the first root of
.Comment: 41 pages, LATE
Twisted Classical Poincar\'{e} Algebras
We consider the twisting of Hopf structure for classical enveloping algebra
, where is the inhomogenous rotations algebra, with
explicite formulae given for Poincar\'{e} algebra
The comultiplications of twisted are obtained by conjugating
primitive classical coproducts by where
denotes any Abelian subalgebra of , and the universal
matrices for are triangular. As an example we show that
the quantum deformation of Poincar\'{e} algebra recently proposed by Chaichian
and Demiczev is a twisted classical Poincar\'{e} algebra. The interpretation of
twisted Poincar\'{e} algebra as describing relativistic symmetries with
clustered 2-particle states is proposed.Comment: \Large \bf 19 pages, Bonn University preprint, November 199
Universal integrability objects
We discuss the main points of the quantum group approach in the theory of
quantum integrable systems and illustrate them for the case of the quantum
group . We give a complete set of the
functional relations correcting inexactitudes of the previous considerations. A
special attention is given to the connection of the representations used to
construct the universal transfer operators and -operators.Comment: 21 pages, submitted to the Proceedings of the International Workshop
"CQIS-2012" (Dubna, January 23-27, 2012
Finite Dimensional Representations of the Quantum Superalgebra in a Reduced Basis
For generic we give expressions for the transformations of all
essentially typical finite-dimensional modules of the Hopf superalgebra
. The latter is a deformation of the universal enveloping algebra
of the Lie superalgebra . The basis within each module is similar to
the Gel'fand-Zetlin basis for . We write down expressions for the
transformations of the basis under the action of the Chevalley generators.Comment: 7 pages, emTeX, University of Ghent Int. Report TWI-93-2