Yang-Baxter equation and representation theory: I

Kulish, P. P.; Reshetikhin, Nicolai; Sklyanin, E. K.

doi:10.1007/bf02285311

Cited by 903 publications

(970 citation statements)

References 6 publications

Supporting

Mentioning

950

Contrasting

Unclassified

Order By: Relevance

“…Provided that all coefficients in (24)(25)(26)(27)(28)(29)(30) are well defined (which is indeed the case under the conditions required here), it is sufficient to show that the explicit actions (24)(25)(26)(27)(28)(29)(30) satisfy the relation (11)(12)(13)(14)(15)(16)(17)(18)(19) (plus, of course, also the f -Serre relations). The irreducibility then follows from the results of Zhang [35] or from the observation that for generic q a deformed matrix element in the GZ basis is zero only if the corresponding non-deformed matrix element vanishes.…”

Section: ] Be An Essentially Typical Highest Weight Then the Set Of mentioning

confidence: 86%

“…. , n + m − 2}; (18) e n e n−1 e n e n+1 +e n−1 e n e n+1 e n +e n e n+1 e n e n−1 +e n+1 e n e n−1 e n −(q +q −1 )e n e n−1 e n+1 e n = 0; (19) • The relations obtained from (16)(17)(18)(19) by replacing every e i by f i (f -Serre relations).…”

Section: ] Be An Essentially Typical Highest Weight Then the Set Of mentioning

confidence: 99%

“…A quantum (super)algebra U q [G] associated with a (simple) Lie (super)algebra G is a deformation of the universal enveloping algebra of G endowed with a Hopf algebra structure. The first example was given in [19,30], and soon followed the generalization to any Kac-Moody Lie algebra with symmtrizable Cartan matrix [4,12]. For the deformation of the enveloping algebra of a Lie superalgebra we mention the case of osp(1/2) [20,21], later to be extended to Lie superalgebras with a symmetrizable Cartan matrix [32] including the basic [16] Lie superalgebras [1,2].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

1994

View full text Add to dashboard Cite

Explicit expressions for the generators of the quantum superalgebra U q [gl(n/m)] acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations : for these a Gel'fand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set of q-number identities.MSC numbers: 16W30, 17B40, 81R50.--

show abstract

Section: ] Be An Essentially Typical Highest Weight Then the Set Of mentioning

confidence: 86%

Section: ] Be An Essentially Typical Highest Weight Then the Set Of mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

1994

View full text Add to dashboard Cite

show abstract

“…(62) This is also true for the case with Λ = Λ, for which (6.5) need be amended with trivial Λ/Λ powers which break the left-right symmetry. In (6.5), the last line corresponds to the usual Bethe Ansatz, (12) whereas the summations over the ε's correspond to all waves reflected at the two boundaries.…”

Section: Low T Bethe Ansatz For Diagonal Interfacesmentioning

confidence: 92%

The chiral Potts models revisited

Au-Yang

Perk

1995

J Stat Phys

View full text Add to dashboard Cite

Abstract:In honor of Onsager's ninetieth birthday, we like to review some exact results obtained so far in the chiral Potts models and to translate these results into language more transparent to physicists, so that experts in Monte Carlo calculations, high and low temperature expansions, and various other methods, can use them.We shall pay special attention to the interfacial tension ǫ r between the k state and the k − r state. By examining the ground states, it is seen that the integrable line ends at a superwetting point, on which the relation ǫ r = r ǫ 1 is satisfied, so that it is energetically neutral to have one interface or more. We present also some partial results on the meaning of the integrable line for low temperatures where it lives in the non-wet regime. We make Baxter's exact results more explicit for the symmetric case. By performing a Bethe Ansatz calculation with open boundary conditions we confirm a dilogarithm identity for the low-temperature expansion which may be new.We propose a new model for numerical studies. This model has only two variables and exhibits commensurate and incommensurate phase transitions and wetting transitions near zero temperature. It appears to be not integrable, except at one point, and at each temperature there is a point, where it is almost identical with the integrable chiral Potts model. KEY WORDS:Chiral Potts model; chiral clock model; star-triangle equations; Yang-Baxter equations; interfacial tension; wetting; superwetting; scaling; corrections to scaling; low-temperature expansions; dilogarithms; Bethe Ansatz. INTRODUCTIONWhen Onsager published his solution of the two-dimensional Ising model in 1944, (1) this was almost instantly recognized as a milestone in the development of statistical mechanics. Many new developments were inspired by his results. On the other hand, Onsager's techniques were far ahead of his time and, when he announced his incredibly simple result for the spontaneous magnetization as a comment to a conference talk, (2,3) his paper Onsager's (only partly published) work on two-point functions in the Ising model got extended in 1966, (18−20) but it was only in 1973, when Wu, McCoy, Tracy, announced the Painlevé equation for the scaled two-point correlation, that the theory of the two-dimensional Ising model went beyond Onsager's level. Also, the first two-dimensional models solved that were more complicated than the Ising model were Lieb's ice model (24,25) of 1967 and Baxter's eight-vertex model (9) of 1973.Onsager's loop-algebra solution method was generalized only in 1985 when Von Gehlen and Rittenberg (26) solved the Dolan-Grady (27) criterium within a one-dimensional generalization of the quantum Potts model. The connection with Onsager's 1944 paper was noticed by Perk, (28,29) showing that the chiral Potts model (30,31,10) is the first genuine generalization of the Ising model, with its "superintegrable" case (32) solvable for two reasons: startriangle integrability (30,31,10) and loop-algebra integrability. (28,26) The chiral Potts...

show abstract

“…Generally the monodromy matrix for this type of representation can be constructed from one for the vector representation by the fusion method [24], and in the sℓ 3 case the Lax matrix is simply defined from…”

Section: Lattice Vertex Operator and Transfer Matrixmentioning

confidence: 99%