Related papers: Sklyanin invariant integration
A new formula connecting the elliptic $6j$-symbols and the fusion of the vertex-face intertwining vectors is given. This is based on the identification of the $k$ fusion intertwining vectors with the change of base matrix elements from…
In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels which realize the given action by bounded operators on a Krein space. Applications to the GNS…
Intertwining operators for infinite-dimensional representations of the Sklyanin algebra with spins l and -l-1 are constructed using the technique of intertwining vectors for elliptic L-operator. They are expressed in terms of elliptic…
Reflection algebras is a class of algebras associated with integrable models with boundaries. The coefficients of Sklyanin determinant generate the center of the reflection algebra. We give a combinatorial description of Sklyanin…
The geometric crystal operators and geometric $R$-matrices (or geometric Weyl group actions) give commuting actions on the field of rational functions in $mn$ variables. We study the invariants of various combinations of these actions,…
In this paper we discuss some properties of Baxter's TQ equation for the eight-vertex elliptic Sklyanin algebra it its compact representation based on the elliptic Gamma-functions. As the main result, we establish the structure of the…
These notes contain essentially a rewriting of several properties of two well-known quantities, the so-called theta symbol (or triangular symbol), which is rational, and the 6j symbol, which is usually irrational, in terms of two related…
The Sklyanin algebra $S_{\eta}$ has a well-known family of infinite-dimensional representations $D(\mu)$, $\mu \in C^*$, in terms of difference operators with shift $\eta$ acting on even meromorphic functions. We show that for generic…
We suppose that $G$ is a locally compact abelian group, $Y$ is a measure space, and $H$ is a reproducing kernel Hilbert space on $G\times Y$ such that $H$ is naturally embedded into $L^2(G\times Y)$ and it is invariant under the…
The representation theory of a 3-dimensional Sklyanin algebra $S$ depends on its (noncommutative projective algebro-) geometric data: an elliptic curve $E$ in $\mathbb{P}^2$, and an automorphism $\sigma$ of $E$ given by translation by a…
We will attach a scalar invariant to a tetrahedron whose edges are labelled by irreducible representations of a ternary orthogonal group $\mathrm{SO}_3$ over a local field. This generalizes the $6j$ symbol whose theory was developed by…
Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson's biorthogonal 10-W-9 functions. We…
The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second order Heun operators on quadratic grids with no diagonal terms are determined. These special or…
S-Heun operators on linear and $q$-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The Continuous Hahn and Big $q$-Jacobi polynomials are functions on which these…
We present other examples illustrating the operator-theoretic approach to invariant integrals on quantum homogeneous spaces developed by Kuersten and the second author. The quantum spaces are chosen such that their coordinate algebras do…
In this article a complete description is given of the simple representations of a 3-dimensional Sklyanin algebra associated to a torsion point. In order to determine these irreducible representations, a review is given of classical results…
The spectral eta-invariant of a self-adjoint elliptic differential operator on a closed manifold is rigid, provided that the parity of the order is opposite to the parity of dimension of the manifold. The paper deals with the calculation of…
Action-angle type variables for spin generalizations of the elliptic Ruijsenaars-Schneider system are constructed. The equations of motion of these systems are solved in terms of Riemann theta-functions. It is proved that these systems are…
We propose a conjectural formula for correlation functions of the Z-invariant (inhomogeneous) eight-vertex model. We refer to this conjecture as Ansatz. It states that correlation functions are linear combinations of products of three…
When solving data analysis problems it is important to integrate prior knowledge and/or structural invariances. This paper contributes by a novel framework for incorporating algebraic invariance structure into kernels. In particular, we…