30 research outputs found
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 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 -operator formalism. Finally the generalization of the
integral representation to the infinite and periodic quantum Toda wave
functions is discussed.Comment: Corrections in Sections (3.2) and (4.1
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.Comment: AmsLatex, 41 pages, 3 figure
Eigenfunctions of GL(N,\RR) Toda chain: The Mellin-Barnes representation
The recurrent relations between the eigenfunctions for GL(N,\RR) and
GL(N-1,\RR) quantum Toda chains is derived. As a corollary, the Mellin-Barnes
integral representation for the eigenfunctions of a quantum open Toda chain is
constructed for the -particle case.Comment: Latex+amssymb.sty, 7 pages; corrected some typos published in Pis'ma
v ZhETF (2000), vol. 71, 338-34
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 to the
Lie algebra . The construction is based on the Gelfand-Zetlin
maximal commuting subalgebra in . We discuss the connection
with the other known integrable systems based on . 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.Comment: LaTeX, 13 page
On a class of representations of the Yangian and moduli space of monopoles
A new class of infinite dimensional representations of the Yangians
and corresponding to a complex semisimple algebra
and its Borel subalgebra is constructed.
It is based on the generalization of the Drinfeld realization of ,
in terms of quantum minors to the case of an arbitrary
semisimple Lie algebra . 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
-monopoles defined as the components of the space of based maps of
into the generalized flag manifold . Thus the constructed
representations of the Yangian may be considered as a quantization of the
moduli space of the monopoles.Comment: 16 pages, LaTex2e, some misprints are fixe
Intertwining operators and Hirota bilinear equations
An interpretation of Hirota bilinear relations for classical functions
is given in terms of intertwining operators. Noncommutative example of
is presented.Comment: Latex, 13 pages, no figures. Contribution to the Proceedings of
Alushta Conference, June 199
On q-deformed gl(l+1)-Whittaker function II
A representation of a specialization of a q-deformed class one lattice
gl(\ell+1}-Whittaker function in terms of cohomology groups of line bundles on
the space QM_d(P^{\ell}) of quasi-maps P^1 to P^{\ell} of degree d is proposed.
For \ell=1, this provides an interpretation of non-specialized q-deformed
gl(2)-Whittaker function in terms of QM_d(\IP^1). In particular the (q-version
of) Mellin-Barnes representation of gl(2)-Whittaker function is realized as a
semi-infinite period map. The explicit form of the period map manifests an
important role of q-version of Gamma-function as a substitute of topological
genus in semi-infinite geometry. A relation with Givental-Lee universal
solution (J-function) of q-deformed gl(2)-Toda chain is also discussed.Comment: Extended version submitted in Comm. Math. Phys., 24 page