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In the previous paper, we showed that the wave functions of the quantum Ruijsenaars hyperbolic system diagonalize Baxter Q-operators. Using this property and duality relation, we prove orthogonality and completeness relations for the wave functions or, equivalently, unitarity of the corresponding integral transform. [ABSTRACT FROM AUTHOR]

In the previous paper, we introduced a commuting family of Baxter Q-operators for the quantum Ruijsenaars hyperbolic system. In the present work, we show that the wave functions of the quantum system found by M. Hallnäs and S. Ruijsenaars also diagonalize Baxter operators. Using this property, we prove the conjectured duality relation for the wave function. As a corollary, we show that the wave function solves bispectral problems for pairs of dual Macdonald and Baxter operators. Besides, we prove the conjectured symmetry of the wave function with respect to spectral variables and obtain new integral representation for it. [ABSTRACT FROM AUTHOR]

We introduce Baxter Q-operators for the quantum Ruijsenaars hyperbolic system. We prove that they represent a commuting family of integral operators and also commute with Macdonald difference operators, which are gauge equivalent to the Ruijsenaars Hamiltonians of the quantum system. The proof of commutativity of the Baxter operators uses a hypergeometric identity on rational functions that generalize Ruijsenaars kernel identities. [ABSTRACT FROM AUTHOR]

We present a proof of hypergeometric identities which play a crucial role in the theory of Baxter operators in the Ruijsenaars model. [ABSTRACT FROM AUTHOR]

We develop an operator approach to the evaluation of multiple integrals for multiloop Feynman massless diagrams. A commutative family of graph building operators Hα for ladder diagrams is constructed and investigated. The complete set of eigenfunctions and the corresponding eigenvalues for the operators Hα are found. This enables us to explicitly express a wide class of four-point ladder diagrams and a general two-loop propagator-type master diagram (with arbitrary indices on the lines) as Mellin-Barnes-type integrals. Special cases of these integrals are explicitly evaluated. A certain class of zig-zag four-point and two-point planar Feynman diagrams (relevant to the bi-scalar D-dimensional "fishnet" field theory and to the calculation of the β-function in ϕ4-theory) is considered. The graph building operators and convenient integral representations for these Feynman diagrams are obtained. The explicit form of the eigenfunctions for the graph building operators of the zig-zag diagrams is fixed by conformal symmetry and these eigenfunctions coincide with the 3-point correlation functions in D-dimensional conformal field theories. By means of this approach, we exactly evaluate the diagrams of the zig-zag series in special cases. In particular, we find a fairly simple derivation of the values for the zig-zag multi-loop two-point diagrams for D = 4. The role of conformal symmetry in this approach, especially a connection of the considered graph building operators with conformal invariant solutions of the Yang-Baxter equation is investigated in detail. [ABSTRACT FROM AUTHOR]

We show that the complex hypergeometric function describing -symbols for the group is a special degeneration of the -function—an elliptic analogue of the Euler–Gauss hypergeometric function. For this function, we derive mixed difference–recurrence relations as limit forms of the elliptic hypergeometric equation and some symmetry transformations. At the intermediate steps of computations, there emerge a function describing the -symbols for the Faddeev modular double and the corresponding difference equations and symmetry transformations. [ABSTRACT FROM AUTHOR]

The note is devoted to the factorization formula for the matrix with the generators of the group GL(N, ℝ) in the regular representation. The factorization formula helps to evaluate these generators and Casimir operators in the case of arbitrary N and clarifies the connection between GL(N, ℝ) and the quantum Toda chain. [ABSTRACT FROM AUTHOR]

The basic formulas for the unitary principal series representations of the group SL(2, ℂ) are given, and the decomposition of a tensor product of two representations into irreducible representations is considered. A simple proof of the completeness of the 3j-symbols is given. [ABSTRACT FROM AUTHOR]

The quantum SL(3,ℂ)-invariant spin magnet with infinite-dimensional principal series representation in local spaces is considered. We construct eigenfunctions of the Sklyanin B-operator which define the representation of separated variables of the model. [ABSTRACT FROM AUTHOR]

In this paper, we show that an operator introduced by A. A. Tsvetkov enjoys all the necessary properties of a Q-operator. It is shown that the Q-operator of the XXX spin chain with spin ℓ turns into Tsvetkov's operator in the continuous limit as ℓ→∞. Bibliography: 18 titles. [ABSTRACT FROM AUTHOR]

We study 6j-symbols or Racah coefficients for the tensor products of infinite-dimensional unitary principal series representations of the group SL(2, ℂ). Using the Feynman diagram technique, we reproduce the results of Ismagilov in constructing these symbols (up to a slight difference associated with equivalent representations). The resulting 6j-symbols are expressed either as a triple integral over complex plane or as an infinite bilateral sum of integrals of the Mellin–Barnes type. [ABSTRACT FROM AUTHOR]

We consider the problem of seeking the eigenvectors for a commuting family of quantum minors of the monodromy matrix for an SL(n,ℂ)-invariant inhomogeneous spin chain. The algebra generators and elements of the L-operator at each site of the chain are implemented as linear differential operators in the space of functions of n(n−1)/2 variables. In the general case, the representation of the sl(ℂ) algebra at each site is infinite-dimensional and belongs to the principal unitary series. We solve this problem using a recursive procedure with respect to the rank n of the algebra. We obtain explicit expressions for the eigenvalues and eigenvectors of the commuting family. We consider the particular cases n = 2 and n = 3 and also the limit case of the one-site chain in detail. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study solutions of the Yang-Baxter equation on the tensor product of an arbitrary finite-dimensional and an arbitrary infinite-dimensional representations of rank 1 symmetry algebra. We consider the cases of the Lie algebra sℓ, 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, finite-difference operators with trigonometric coefficients in the case of the modular double, or finite-difference operators with coefficients constructed of the 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 formulas. Bibliography: 44 titles. [ABSTRACT FROM AUTHOR]

A convenient operator representation for zig-zag four-point and two-point planar Feynman diagrams relevant to the bi-scalar D -dimensional "fishnet" field theory is obtained. This representation gives a possibility to evaluate exactly diagrams of the zig-zag series in special cases. In particular, we give a fairly simple proof of the Broadhurst-Kreimer conjecture about the values of zig-zag multi-loop two-point diagrams which make a significant contribution to the renormalization group β -function in the 4-dimensional ϕ 4 theory. [ABSTRACT FROM AUTHOR]

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 η and τ, Im τ > 0, Im η > 0. For two-dimensional lattices g = nη + mτ/2 and g = 1/2 + nη + mτ/2 with incommensurate 1, 2 η, τ and integers n, m > 0, 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. [ABSTRACT FROM AUTHOR]

We study the properties of conformal operators in the SL(2) sector of planar N=4 SYM and its supersymmetric SL(2|2) extension. The correlation functions of these operators and their form factors with respect to asymptotic on-shell states are determined by two different polynomials which can be identified as eigenstates of the dilatation operator in the coordinate and momentum representations, respectively. We argue that, in virtue of integrability of the dilatation operator, the two polynomials satisfy a duality relation - they are proportional to each other upon an appropriate identification of momenta and coordinates. Combined with the conventional N=4 superconformal symmetry, this leads to the dual superconformal symmetry of the dilatation operator. We demonstrate that this symmetry is powerful enough to fix the eigenspectrum of the dilatation operator to the lowest order in the coupling. We use the relation between the one-loop dilatation operator and Heisenberg spin chain to show that, to lowest order in the coupling, the dual symmetry is generated by the Baxter Q-operator in the limit of large spectral parameter. [ABSTRACT FROM AUTHOR]

Abstract: Yangian symmetry of amplitudes in super-Yang–Mills theory is formulated in terms of eigenvalue relations for monodromy matrix operators. The Quantum Inverse Scattering Method provides the appropriate tools to treat the extended symmetry and to recover as its consequences many known features like cyclic and inversion symmetry, BCFW recursion, Inverse Soft Limit construction, Grassmannian integral representation, R-invariants and on-shell diagram approach. [Copyright &y& Elsevier]

We construct the R-operator—the solution of the Yang–Baxter equation acting in the tensor product of two infinite-dimensional representations of Faddeev’s modular double. This R-operator intertwines the product of two L-operators associated with the modular double and it is built from three basic operators generating the permutation group of four parameters . [ABSTRACT FROM AUTHOR]

The R-matrix acting in the tensor product of two spinor representation spaces of Lie algebra so(d) is considered thoroughly. The corresponding Yang-Baxter relation is proved and the underlying localYang-Baxter equation is established. [ABSTRACT FROM AUTHOR]

Abstract: Baxter operators are constructed for quantum spin chains with deformed symmetry. The parallel treatment of Yang–Baxter operators for the cases of undeformed, trigonometrically and elliptically deformed symmetries presented earlier and relying on the factorization regarding parameter permutations is extended to the global chain operators following the scheme worked out recently in the undeformed case. [Copyright &y& Elsevier]

Abstract: This paper presents the second part of our study devoted to the construction of Baxter operators for the homogeneous closed XXX spin chain with the quantum space carrying infinite or finite-dimensional representations. We consider the Baxter operators used in Bazhanov et al. (1996, 1997, 1999, 2010) , formulate their construction uniformly with the construction of our previous paper. The building blocks of all global chain operators are derived from the general Yang–Baxter operators and all operator relations are derived from general Yang–Baxter relations. This leads naturally to the comparison of both constructions and allows to connect closely the treatment of the cases of infinite-dimensional representation of generic spin and finite-dimensional representations of integer or half-integer spin. We prove not only the relations between the operators but present also their explicit forms and expressions for their action on polynomials representing the quantum states. [Copyright &y& Elsevier]

Abstract: We construct Baxter operators for the homogeneous closed XXX 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. [Copyright &y& Elsevier]

In this paper, we give a self-contained description of three methods of perturbation series summation for the resolvent in the quantum Coulomb problem. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The iterative construction of a generic representation of gℓ( n + 1) or of the trigonomentric deformation of its enveloping algebra is conveniently formulated in terms of Lax matrices. The Lax matrix of the constructed representation factorises into parts determined by the Lax matrix of a generic representation of the algebra with reduced rank and others appearing in the factorised expression of the Lax matrix of the special Jordan–Schwinger representation. [ABSTRACT FROM AUTHOR]

The general rational solution of the Yang-Baxter equation with the symmetry algebra sℓ(2) can be represented as a product of simpler building blocks called $$ \mathcal{R} $$ -operators. $$ \mathcal{R} $$ -operators are constructed explicitly and have simple structure. Using $$ \mathcal{R} $$ -operators, we construct the two-parametric Baxter’s Q-operator for the generic inhomogeneous XXX-spin chain. In the case of a homogeneous XXX-spin chain, it is possible to reduce the general Q-operator to a much simpler one-parametric Q-operator. Bibliography: 22 titles. [ABSTRACT FROM AUTHOR]

The Yang-Baxter operator is obtained as a product of operators that permute representation parameters in the Lax operators. The construction relies on a factorization of the Lax operator into triangular matrices. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

Abstract: 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 -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. [Copyright &y& Elsevier]

Abstract: We construct Baxter operators as generalized transfer matrices being traces of products of generic R matrices. The latter are shown to factorize into simpler operators allowing for explicit expressions in terms of functions of a Weyl pair of basic operators. These explicit expressions are the basis for explicit expression for Baxter Q-operators and for investigating their properties. [Copyright &y& Elsevier]

The Jordanian deformation of sl(2) bialgebra structure is studied in view of physical applications to breaking of conformal symmetry in the high energy asymptotics of scattering. Representations are formulated in terms of polynomials, generators in terms of differential operators. The deformed R operator with generic representations is analyzed in spectral and integral forms. [Copyright &y& Elsevier]

Eigenfunctions of the matrix elements of the monodromy matrix provide a convenient basis for studies of spin chain models. We present an iterative method for constructing the eigenfunctions in the case of spin chains. We derived an explicit integral representation for the eigenfunctions and calculated the corresponding scalar products (Sklyanin's measure). [ABSTRACT FROM AUTHOR]