KASHIWARA-MIWA MODEL AND OPERATORS ACTING ON SPIN CHAINS

1994 ◽  
Vol 09 (27) ◽  
pp. 4701-4716
Author(s):  
ZHAN-NING HU
Keyword(s):  
Hopf Algebra ◽  
Quantum Algebra ◽  
Spin Chains ◽  
Baxter Equation ◽  
Elliptic Case ◽  
Special Case

In this paper, the Kashiwara-Miwa broken ZN model is discussed and a proof of the Yang-Baxter equation (YBE) is given, which shows that the YBE obtained by Hasegawa and Yamada is a special case. The Q operators acting on spin chains have a structure which is similar to that of the ordinary Hopf algebra, and the quantum algebra sl q(2) with qN=1 can be obtained from these operators as a limit of the elliptic case.

scholarly journals On the Structure of Monodromy Algebras and Drinfeld Doubles

1997 ◽  
Vol 09 (03) ◽  
pp. 371-395
Author(s):  
Florian Nill
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Tensor Products ◽  
Spin Chains ◽  
Coadjoint Action ◽  
Braided Group ◽  
Gauge Transformations ◽  
Loop Algebras ◽  
Algebraic Tensor Product ◽  
Quasitriangular Hopf Algebra

We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double [Formula: see text]. Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov–Tian–Shansky, both algebras are isomorphic to the algebraic tensor product H ⊗ H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlachányi and the author. In the Appendix the multi-loop algebras ℒm of Alekseev and Schomerus [3] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the "bosonization formula" of [3] representing ℒm as H ⊗…⊗ H.

Analytical Bethe Ansatz for quantum-algebra-invariant spin chains

1992 ◽  
Vol 372 (3) ◽  
pp. 597-621 ◽  
Author(s):  
Luca Mezincescu ◽  
Rafael I. Nepomechie
Keyword(s):  
Bethe Ansatz ◽  
Quantum Algebra ◽  
Spin Chains

slq(n) Quantum Algebra As a Limit of the Elliptic Case

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 391-403 ◽  
Author(s):  
Bo-yu Hou ◽  
Kang-jie Shi ◽  
Zhong-xia Yang
Keyword(s):  
Quantum Algebra ◽  
Elliptic Case ◽  
Specific Relation

We study the limit of the algebra associated with the solution of Z n elliptic YBE. And we give its specific relation with the quantum algebra sl q (n) .

APPLICATIONS OF COMPUTER ALGEBRA IN QUANTUM GROUPS

1994 ◽  
Vol 05 (04) ◽  
pp. 701-706
Author(s):  
W.-H. STEEB
Keyword(s):  
Quantum Groups ◽  
Computer Algebra ◽  
Kronecker Product ◽  
Quantum Algebra ◽  
Theoretical Physics ◽  
Baxter Equation ◽  
Quantum Algebras ◽  
Helpful Tool

Quantum groups and quantum algebras play a central role in theoretical physics. We show that computer algebra is a helpful tool in the investigations of quantum groups. We give an implementation of the Kronecker product together with the Yang-Baxter equation. Furthermore the quantum algebra obtained from the Yang-Baxter equation is implemented. We apply the computer algebra package REDUCE.

scholarly journals Multiparametric and Colored Extensions of the Quantum Group GLq(N) and the Yangian Algebra Y(glN) Through a Symmetry Transformation of the Yang–Baxter Equation

1997 ◽  
Vol 12 (05) ◽  
pp. 945-962 ◽  
Author(s):  
B. Basu-Mallick ◽  
P. Ramadevi ◽  
R. Jagannathan
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Spectral Parameter ◽  
General Relation ◽  
Symmetry Transformation ◽  
Baxter Equation ◽  
Color Parameter ◽  
R Matrix ◽  
General Symmetry ◽  
Yangian Algebra

Inspired by Reshetikhin's twisting procedure to obtain multiparametric extensions of a Hopf algebra, a general "symmetry transformation" of the "particle conserving" R-matrix is found such that the resulting multiparametric R-matrix, with a spectral parameter as well as a color parameter, is also a solution of the Yang–Baxter equation (YBE). The corresponding transformation of the quantum YBE reveals a new relation between the associated quantized algebra and its multiparametric deformation. As applications of this general relation to some particular cases, multiparametric and colored extensions of the quantum group GL q(N) and the Yangian algebra Y(glN) are investigated and their explicit realizations are also discussed. Possible interesting physical applications of such extended Yangian algebras are indicated.

ON COMMUTANTS, DUAL PAIRS AND NON-SEMISIMPLE ALGEBRAS FROM STATISTICAL MECHANICS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 675-705 ◽  
Author(s):  
PAUL P. MARTIN ◽  
DAVID S. MCANALLY
Keyword(s):  
Dual Pair ◽  
Quantum Spin ◽  
Spin Chains ◽  
Quantum Spin Chains ◽  
Complex Vector ◽  
Dual Pairs ◽  
Morita Equivalent ◽  
Finite Dimensional ◽  
Semisimple Algebras ◽  
Special Case

For M a finite dimensional complex vector space and A a certain type of (unital) subalgebra of End(M) (including some specific types of physical significance in the field of quantum spin chains) we give an algorithm for constructing the centraliser or commutant B of A on M. We give examples, and discuss the conditions for centralising to be an involution, i.e. A, B a dual pair, and for B and A to be Morita equivalent. A special case of one example shows that Hn(q), Uq(sl2) act as a dual pair on the tensored vector representation for all q.

scholarly journals INTEGRABILITY OF OPEN SPIN CHAINS WITH QUANTUM ALGEBRA SYMMETRY

1991 ◽  
Vol 06 (29) ◽  
pp. 5231-5248 ◽  
Author(s):  
LUCA MEZINCESCU ◽  
RAFAEL I. NEPOMECHIE
Keyword(s):  
Spin Chain ◽  
Quantum Algebra ◽  
Quantum Spin ◽  
Spin Chains ◽  
Invariant Chain ◽  
Quantum Spin Chain ◽  
Fundamental Representation ◽  
R Matrix ◽  
Boundary Terms ◽  
Xxz Chain

We construct an open quantum spin chain from the “twisted” [Formula: see text]R matrix in the fundamental representation which has the quantum algebra symmetry Uq[ su (2)]. This anisotropic spin-1 chain is different from the Uq[ su (2)]-invariant chain constructed from the “untwisted” [Formula: see text] spin-1 R matrix (namely, the spin-1 XXZ chain of Fateev-Zamolodchikov with boundary terms) but, nevertheless, is also completely integrable. We discuss the general case of an R matrix of the type g(k), where k∈{1, 2, 3}, and g is any simple Lie algebra.

scholarly journals Anti-flexible bialgebras

2021 ◽  
pp. 2250212
Author(s):  
Mafoya Landry Dassoundo ◽  
Chengming Bai ◽  
Mahouton Norbert Hounkonnou
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Manin Triple ◽  
Unexpected Consequence ◽  
Special Case

We establish a bialgebra theory for anti-flexible algebras in this paper. We introduce the notion of an anti-flexible bialgebra which is equivalent to a Manin triple of anti-flexible algebras. The study of a special case of anti-flexible bialgebras leads to the introduction of anti-flexible Yang–Baxter equation in an anti-flexible algebra which is an analogue of the classical Yang–Baxter equation in a Lie algebra or the associative Yang–Baxter equation in an associative algebra. It is unexpected consequence that both the anti-flexible Yang–Baxter equation and the associative Yang–Baxter equation have the same form. A skew-symmetric solution of anti-flexible Yang–Baxter equation gives an anti-flexible bialgebra. Finally the notions of an [Formula: see text]-operator of an anti-flexible algebra and a pre-anti-flexible algebra are introduced to construct skew-symmetric solutions of anti-flexible Yang–Baxter equation.

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