186 research outputs found
Yang-Baxter R operators and parameter permutations
We present an uniform construction of the solution to the Yang- Baxter
equation with the symmetry algebra and its deformations: the
q-deformation and the elliptic deformation or Sklyanin algebra. The R-operator
acting in the tensor product of two representations of the symmetry algebra
with arbitrary spins and is built in terms of products of
three basic operators which are
constructed explicitly. They have the simple meaning of representing elementary
permutations of the symmetric group , the permutation group of
the four parameters entering the RLL-relation.Comment: 22 pages LaTex, comments added, version to be published in Nucl.
Phys.
Matrix factorization for solutions of the Yang-Baxter equation
We study solutions of the Yang-Baxter equation on a tensor product of an
arbitrary finite-dimensional 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
Baxter operators for arbitrary spin
We construct Baxter operators for the homogeneous closed spin
chain with the quantum space carrying infinite or finite dimensional
representations. All algebraic relations of Baxter operators and transfer
matrices are deduced uniformly from Yang-Baxter relations of the local building
blocks of these operators. This results in a systematic and very transparent
approach where the cases of finite and infinite dimensional representations are
treated in analogy. Simple relations between the Baxter operators of both cases
are obtained. We represent the quantum spaces by polynomials and build the
operators from elementary differentiation and multiplication operators. We
present compact explicit formulae for the action of Baxter operators on
polynomials.Comment: 37 pages LaTex, 7 figures, version for publicatio
Finite-dimensional representations of the elliptic modular double
We investigate the kernel space of an integral operator M(g) depending on the
"spin" g and describing an elliptic Fourier transformation. The operator M(g)
is an intertwiner for the elliptic modular double formed from a pair of
Sklyanin algebras with the parameters and , Im,
Im. For two-dimensional lattices and with incommensurate and integers , the operator
M(g) has a finite-dimensional kernel that consists of the products of theta
functions with two different modular parameters and is invariant under the
action of generators of the elliptic modular double.Comment: 25 pp., published versio
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