From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation

Chicherin, Dmitry; Derkachov, S. E.; Spiridonov, V. P.

doi:10.3842/sigma.2016.028

Cited by 13 publications

(35 citation statements)

References 44 publications

(158 reference statements)

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“…The factorization formula (10) is equivalent to the generating formula for matrix elements of the restricted R-operator, Eq. (6), hence it confirms the efficiency of the result (6) obtained initially in [8]. Let us mention that it would be difficult to guess the factorization formula (10) taking as a starting point the classical result (4).…”

Section: Introductionsupporting

confidence: 68%

“…we will prove the factorization formula (10). Besides the reduction formula (6) for sℓ 2 the analogous restrictions (to a finite-dimensional representation) of the general R-operator has been obtained in the paper [8] for the Lie group SL(2, C) and for the modular double of U q (sℓ 2 ) [17].…”

Section: Introductionmentioning

confidence: 99%

“…In other words it is a (n + 1) × (n + 1) matrix, whose entries are differential operators acting on the space of polynomials C[z]. In [8] the restriction of the R-operator has been calculated and the generating formula for its matrix matrix elements has been found. More exactly, the R-operator being applied to the generating function (z 1 − x) n of the finite-dimensional representation in the first space and to a polynomial Φ(z) from the second space 3 gives the following 4…”

Section: Introductionmentioning

confidence: 99%

“…(5) has to be calculated carefully, since the divergences arise in both factors. In [8] it has been shown that at (half)-integer spin 2s = n ∈ Z ≥0 the operator (5) can be restricted to a finite-dimensional invariant subspace in the first space of the tensor product. The restricted operator acts on a tensor product of the (n + 1)-dimensional space (where spin s = n 2 representation is realized) and an infinite-dimensional space (where spin ℓ representation is realized).…”

Section: Introductionmentioning

confidence: 99%

“…does there exist a factorized matrix form of the R-operator at (half)-integer spin s = n 2 which would be analogous to (8) ?…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Matrix Factorization for Solutions of the Yang–Baxter Equation

Derkachov

Chicherin

2016

J Math Sci

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We study solutions of the Yang-Baxter equation on a tensor product of an arbitrary finitedimensional and an arbitrary infinite-dimensional representations of the rank one symmetry algebra. We consider the cases of the Lie algebra sℓ 2 , the modular double (trigonometric deformation) and the Sklyanin algebra (elliptic deformation). The solutions are matrices with operator entries. The matrix elements are differential operators in the case of sℓ 2 , finite-difference operators with trigonometric coefficients in the case of the modular double, or finite-difference operators with coefficients constructed out of Jacobi theta functions in the case of the Sklyanin algebra. We find a new factorized form of the rational, trigonometric, and elliptic solutions, which drastically simplifies them. We show that they are products of several simply organized matrices and obtain for them explicit formulae.

show abstract

Section: Introductionsupporting

confidence: 68%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…does there exist a factorized matrix form of the R-operator at (half)-integer spin s = n 2 which would be analogous to (8) ?…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Matrix Factorization for Solutions of the Yang–Baxter Equation

Derkachov

Chicherin

2016

J Math Sci

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New Elliptic Solutions of the Yang–Baxter Equation

Chicherin¹,

Derkachov

Spiridonov

2016

Commun. Math. Phys.

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We consider finite-dimensional reductions of an integral operator with the elliptic hypergeometric kernel describing the most general known solution of the Yang-Baxter equation with a rank 1 symmetry algebra. The reduced R-operators reproduce at their bottom the standard Baxter's R-matrix for the 8-vertex model and Sklyanin's L-operator. The general formula has a remarkably compact form and yields new elliptic solutions of the Yang-Baxter equation based on the finite-dimensional representations of the elliptic modular double. The same result is also derived using the fusion formalism.

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On tensor product decomposition of positive representations of $${\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))$$

2021

Lett Math Phys

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We study the tensor product decomposition of the split real quantum group U q q (sl(2, R)) from the perspective of finite dimensional representation theory of compact quantum groups. It is known that the class of positive representations of U q q (sl(2, R)) is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch-Gordan coefficients of the tensor product decomposition of finite dimensional representations of the compact quantum group Uq(sl2) by solving certain functional equations arising from analytic continuation and using normalization arising from tensor products of canonical basis.

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From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation

Cited by 13 publications

References 44 publications

Matrix Factorization for Solutions of the Yang–Baxter Equation

Matrix Factorization for Solutions of the Yang–Baxter Equation

New Elliptic Solutions of the Yang–Baxter Equation

On tensor product decomposition of positive representations of $${\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))$$

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