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D. Lebedev

Publications and source records attributed to D. Lebedev.

At least 19 recordsLinked to original sources
arXiv2008

On Baxter Q-operators And Their Arithmetic Implications

We consider Baxter Q-operators for various versions of quantum affine Toda chain. The interpretation of eigenvalues of the finite Toda chain Baxter operators as local Archimedean L-functions proposed recently is generalized to the case of affine Lie algebras. We also introduce a simple generalization of Baxter operators and local L-functions compatible with this identification. This gives a connection of the Toda chain Baxter Q-operators with an Archimedean version of the Polya-Hilbert operator proposed by Berry-Kitting. We also elucidate the Dorey-Tateo spectral interpretation of eigenvalues of Q-operators. Using explicit expressions for eigenfunctions of affine/relativistic Toda chain we obtain an Archimedean analog of Casselman-Shalika-Shintani formula for Whittaker function in terms of characters.

math.RT↗
arXiv2007

Baxter operator and Archimedean Hecke algebra

In this paper we introduce Baxter integral Q-operators for finite-dimensional Lie algebras gl(n+1) and so(2n+1). Whittaker functions corresponding to these algebras are eigenfunctions of the Q-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G=GL(n+1) proved earlier by Stade. We also identify eigenvalues of the Baxter Q-operator acting on Whittaker functions with local Archimedean L-factors. The Baxter Q-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra H(G(R),K), K being a maximal compact subgroup of G. Finally we stress an analogy between Q-operators and certain elements of the non-Archimedean Hecke algebra H(G(Q_p),G(Z_p)).

math.RT↗
arXiv2007

New Integral Representations of Whittaker Functions for Classical Lie Groups

We propose integral representations of the Whittaker functions for the classical Lie algebras sp(2l), so(2l) and so(2l+1). These integral representations generalize the integral representation of gl(l+1)-Whittaker functions first introduced by Givental. One of the salient features of the Givental representation is its recursive structure with respect to the rank of the Lie algebra gl(l+1). The proposed generalization of the Givental representation to the classical Lie algebras retains this property. It was shown elsewhere that the integral recursion operator for gl(l+1)-Whittaker function in the Givental representation coincides with a degeneration of the Baxter Q-operator for $\hat{gl(l+1)}$-Toda chains. We construct Q-operator for affine Lie algebras $\hat{so(2l)}$, $\hat{so(2l+1)}$ and a twisted form of $\hat{gl(2l)}$. We demonstrate that the relation between recursion integral operators of the generalized Givental representation and degenerate Q-operators remains valid for all classical Lie algebras.

math.RT↗
arXiv2006

Baxter Q-operator and Givental integral representation for C_n and D_n

Recently integral representations for the eigenfunctions of quadratic open Toda chain Hamiltonians for classical groups was proposed. This representation generalizes Givental representation for A_n. In this note we verify that the wave functions defined by these integral representations are common eigenfunctions for the complete set of open Toda chain Hamiltonians. We consider the zero eigenvalue wave functions for classical groups C_n and D_n thus completing the generalization of the Givental construction in these cases. The construction is based on a recursive procedure and uses the formalism of Baxter Q-operators. We also verify that the integral Q-operators for C_n, D_n and twisted affine algebra A_{2n-1}^{(2)} proposed previously intertwine complete sets of Hamiltonian operators. Finally we provide integral representations of the eigenfunctions of the quadratic $D_n$ Toda chain Hamiltonians for generic nonzero eigenvalues.

math.RT↗
arXiv2006

Givental Integral Representation for Classical Groups

We propose integral representations for wave functions of B_n, C_n, and D_n open Toda chains at zero eigenvalues of the Hamiltonian operators thus generalizing Givental representation for A_n. We also construct Baxter Q-operators for closed Toda chains corresponding to Lie algebras B_{\infty}, C_{\infty}, D_{\infty}, affine Lie algebras B^{(1)}_n, C^{(1)}_n, D^{(1)}_n and twisted affine Lie algebras A^{(2)}_{2n-1} and A^{(2)}_{2n}. Our approach is based on a generalization of the connection between Baxter Q-operator for A_n^{(1)} closed Toda chain and Givental representation for the wave function of A_n open Toda chain uncovered previously.

math.RT↗
arXiv2005

On a Gauss-Givental Representation of Quantum Toda Chain Wave Function

We propose group theory interpretation of the integral representation of the quantum open Toda chain wave function due to Givental. In particular we construct the representation of $U((\mathfrak{gl}(N))$ in terms of first order differential operators in Givental variables. The construction of this representation turns out to be closely connected with the integral representation based on the factorized Gauss decomposition. We also reveal the recursive structure of the Givental representation and provide the connection with the Baxter $Q$-operator formalism. Finally the generalization of the integral representation to the infinite and periodic quantum Toda wave functions is discussed.

math.RT↗
arXiv2005

On a Class of Representations of Quantum Groups

This paper is a short account of the construction of a new class of the infinite-dimensional representations of the quantum groups. The examples include finite-dimensional quantum groups $U_q(\mathfrak{g})$, Yangian $Y(\mathfrak{g})$ and affine quantum groups at zero level $U_q(\hat{\mathfrak{g}})_{c=0}$ corresponding to an arbitrary finite-dimensional semisimple Lie algebra $\mathfrak{g}$. At the intermediate step we construct the embedding of the quantum groups into the algebra of the rational functions on the quantum multi-dimensional torus. The explicit parameterization of the quantum groups used in this paper turns out to be closely related to the parameterization of the moduli spaces of the monopoles. As a result the proposed constructions of the representations provide a quantization of the moduli spaces of the monopoles on $\RR^3$ and $\RR^2\times S^1$.

math.QA↗
arXiv2004

On a class of representations of the Yangian and moduli space of monopoles

A new class of infinite dimensional representations of the Yangians $Y(\frak{g})$ and $Y(\frak{b})$ corresponding to a complex semisimple algebra $\frak{g}$ and its Borel subalgebra $\frak{b}\subset\frak{g}$ is constructed. It is based on the generalization of the Drinfeld realization of $Y(\frak{g})$, $\frak{g}=\frak{gl}(N)$ in terms of quantum minors to the case of an arbitrary semisimple Lie algebra $\frak{g}$. The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of $G$-monopoles defined as the components of the space of based maps of $\mathbb{P}^1$ into the generalized flag manifold $X=G/B$. Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles.

math.AG↗
arXiv2004

Representation theory and quantum integrability

We describe new constructions of the infinite-dimensional representations of $U(\mathfrak{g})$ and $U_q(\mathfrak{g})$ for $\mathfrak{g}$ being $\mathfrak{gl}(N)$ and $\mathfrak{sl}(N)$. The application of these constructions to the quantum integrable theories of Toda type is discussed. With the help of these infinite-dimensional representations we manage to establish direct connection between group theoretical approach to the quantum integrability and Quantum Inverse Scattering Method based on the representation theory of Yangian and its generalizations. In the case of $U_q(\mathfrak{g})$ the considered representation is naturally supplied with the structure of $U_q(\mathfrak{g})\otimes U_{\tilde q}(\check{\mathfrak{g}})$-bimodule where $\check {\mathfrak{g}}$ is Langlands dual to $\mathfrak{g}$ and $\log q/2πi=- (\log{\tilde q}/2πi)^{-1}$. This bimodule structure is a manifestation of the Morita equivalence of the algebra and its dual.

math.QA↗
arXiv2003

Representation Theory and the Quantum Inverse Scattering Method: The Open Toda Chain and the Hyperbolic Sutherland Model

Using the representation theory of $\frak{gl}(N,\RR)$, we express the wave function of the $GL(N,\RR)$ Toda chain, which two of us recently obtained by the Quantum Inverse Scattering Method, in terms of multiple integrals. The main tool is our generalization of the Gelfand-Zetlin method to the case of infinite-dimensional representations of $\frak{gl}(N,\RR)$. The interpretation of this generalized construction in terms of the coadjoint orbits is given and the connection with the Yangian $Y(\frak{gl}(N))$ is discussed. We also give the hyperbolic Sutherland model eigenfunctions expressed in terms of integrals in the Gelfand-Zetlin representation. Using the example of the open Toda chain, we discuss the connection between the Quantum Inverse Scattering Method and Representation Theory.

math.QA↗
arXiv2003

On a class of integrable systems connected with GL(N,\RR)

In this paper we define a new class of the quantum integrable systems associated with the quantization of the cotangent bundle $T^*(GL(N))$ to the Lie algebra $\frak{gl}_N$. The construction is based on the Gelfand-Zetlin maximal commuting subalgebra in $U(\frak{gl}_N)$. We discuss the connection with the other known integrable systems based on $T^*GL(N)$. The construction of the spectral tower associated with the proposed integrable theory is given. This spectral tower appears as a generalization of the standard spectral curve for integrable system.

math.QA↗
arXiv2001

Unitary representations of $U_{q}(\mathfrak{sl}(2,\RR))$, the modular double, and the multiparticle q-deformed Toda chains

The paper deals with the analytic theory of the quantum q-deformed Toda chain; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L.Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N-particle q-deformed open Toda chain are given as a multiple integral of the Mellin-Barnes type. For the periodic chain the two dual Baxter equations are derived.

hep-th↗
arXiv2000

Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism

The integral representations for the eigenfunctions of $N$ particle quantum open and periodic Toda chains are constructed in the framework of Quantum Inverse Scattering Method (QISM). Both periodic and open $N$-particle solutions have essentially the same structure being written as a generalized Fourier transform over the eigenfunctions of the $N-1$ particle open Toda chain with the kernels satisfying to the Baxter equations of the second and first order respectively. In the latter case this leads to recurrent relations which result to representation of the Mellin-Barnes type for solutions of an open chain. As byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra function in the case of $GL(N,\RR)$ group.

hep-th↗
arXiv1999

Integral representation for the eigenfunctions of quantum periodic Toda chain

Integral representation for the eigenfunctions of quantum periodic Toda chain is constructed for N-particle case. The multiple integral is calculated using the Cauchy residue formula. This gives the representation which reproduces the particular results obtained by Gutzwiller for N=2,3 and 4-particle chain. Our method to solve the problem combines the ideas of Gutzwiller and R-matrix approach of Sklyanin with the classical results in the theory of the Whittaker functions. In particular, we calculate Sklyanin's invariant scalar product from the Plancherel formula for the Whittaker functions derived by Semenov-Tian-Shansky thus obtaining the natural interpretation of the Sklyanin measure in terms of the Harish-Chandra function.

hep-th↗
arXiv1998

Intertwining Operators and Soliton Equations

In this paper we generalize the fermionic approach to the KP hierarchy sudgested in the papers of Kyoto school 1981-1984 (Sato,Jimbo, Miwa...). The main idea is that the components of the intertwiningoperators are in some sense a generalization of free fermions for $gl_{\infty}$. We formulate in terms of intertwining operators the integrable hierarchies related to Kac-Moody Lie algebra symmetries. We write down explicitly the bosonization of these operators for different choices of Heisenberg subalgebras. These different realizations lead to different hierarchies of soliton equations. For example, for $sl_N$-symmetries we get hierarchies obtained as $(n_1,..., n_s)$-reduction from $s$-component KP $(n_1+...+n_s = N)$ introduced by V.Kac and J.Van de Leur.

hep-th↗
arXiv1998

Classical Limit of the Scaled Elliptic Algebra $A_{\hbar,η}(sl_2)$

The classical limit of the scaled elliptic algebra $A_{\hbar,η}(sl_2)$ is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic $sl_2$ valued functions on a strip and as an extended algebra of decreasing automorphic $sl_2$ valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra $A_{q,p}(sl_2)$ is also briefly presented.

q-alg↗
arXiv1998

Yangian Algebras and Classical Riemann Problems

We investigate different Hopf algebras associated to Yang's solution of quantum Yang-Baxter equation. It is shown that for the precise definition of the algebra one needs the commutation relations for the deformed algebra of formal currents and the specialization of the Riemann problem for the currents. Two different Riemann problems are considered. They lead to the central extended Yangian double associated with ${sl}_2$ and to the degeneration of scaling limit of elliptic affine algebra. Unless the defining relations for the generating functions of the both algebras coincide their properties and the theory of infinite-dimensional representations are quite different. We discuss also the Riemann problem for twisted algebras and for scaled elliptic algebra.

q-alg↗