11,354 research outputs found
q-Analogs of symmetric function operators
For any homomorphism V on the space of symmetric functions, we introduce an
operation which creates a q-analog of V. By giving several examples we
demonstrate that this quantization occurs naturally within the theory of
symmetric functions. In particular, we show that the Hall-Littlewood symmetric
functions are formed by taking this q-analog of the Schur symmetric functions
and the Macdonald symmetric functions appear by taking the q-analog of the
Hall-Littlewood symmetric functions in the parameter t. This relation is then
used to derive recurrences on the Macdonald q,t-Kostka coefficients.Comment: 17 pages - minor revisions to appear in Discrete Mathematics issue
for LaCIM'200
The c-function expansion of a basic hypergeometric function associated to root systems
We derive an explicit c-function expansion of a basic hypergeometric function
associated to root systems. The basic hypergeometric function in question was
constructed as explicit series expansion in symmetric Macdonald polynomials by
Cherednik in case the associated twisted affine root system is reduced. Its
construction was extended to the nonreduced case by the author. It is a
meromorphic Weyl group invariant solution of the spectral problem of the
Macdonald q-difference operators. The c-function expansion is its explicit
expansion in terms of the basis of the space of meromorphic solutions of the
spectral problem consisting of q-analogs of the Harish-Chandra series. We
express the expansion coefficients in terms of a q-analog of the Harish-Chandra
c-function, which is explicitly given as product of q-Gamma functions. The
c-function expansion shows that the basic hypergeometric function formally is a
q-analog of the Heckman-Opdam hypergeometric function, which in turn
specializes to elementary spherical functions on noncompact Riemannian
symmetric spaces for special values of the parameters.Comment: 42 pages; In version 2 we removed a technical condition on the
lattice and corrected some small misprints; in version 3 we have made some
minor revisions, and we have revised Subsections 5.2 and 5.3 establishing the
link between the rank one case of the theory and the theory on basic
hypergeometric series; in version 4 small typos corrected, references update
q and q,t-Analogs of Non-commutative Symmetric Functions
We introduce two families of non-commutative symmetric functions that have
analogous properties to the Hall-Littlewood and Macdonald symmetric functions.Comment: Different from analogues in math.CO/0106191 - v2: 26 pages - added a
definition in terms of triangularity/scalar product relations - to be
submitted FPSAC'0
Braided algebras and their applications to Noncommutative Geometry
We introduce the notion of a braided algebra and study some examples of
these. In particular, R-symmetric and R-skew-symmetric algebras of a linear
space V equipped with a skew-invertible Hecke symmetry R are braided algebras.
We prove the "mountain property" for the numerators and denominators of their
Poincare-Hilbert series (which are always rational functions).
Also, we further develop a differential calculus on modified Reflection
Equation algebras. Thus, we exhibit a new form of the Leibniz rule for partial
derivatives on such algebras related to involutive symmetries R. In particular,
we present this rule for the algebra U(gl(m)). The case of the algebra U(gl(2))
and its compact form U(u(2)) (which can be treated as a deformation of the
Minkowski space algebra) is considered in detail. On the algebra U(u(2)) we
introduce the notion of the quantum radius, which is a deformation of the usual
radius, and compute the action of rotationally invariant operators and in
particular of the Laplace operator. This enables us to define analogs of the
Laplace-Beltrami operators corresponding to certain Schwarzschild-type metrics
and to compute their actions on the algebra U(u(2)) and its central extension.
Some "physical" consequences of our considerations are presented.Comment: LaTeX file, 24 page
Baxter operator formalism for Macdonald polynomials
We develop basic constructions of the Baxter operator formalism for the
Macdonald polynomials associated with root systems of type A. Precisely we
construct a dual pair of mutually commuting Baxter operators such that the
Macdonald polynomials are their common eigenfunctions. The dual pair of Baxter
operators is closely related to the dual pair of recursive operators for
Macdonald polynomials leading to various families of their integral
representations. We also construct the Baxter operator formalism for the
q-deformed gl(l+1)-Whittaker functions and the Jack polynomials obtained by
degenerations of the Macdonald polynomials associated with the type A_l root
system. This note provides a generalization of our previous results on the
Baxter operator formalism for the Whittaker functions. It was demonstrated
previously that Baxter operator formalism for the Whittaker functions has deep
connections with representation theory. In particular the Baxter operators
should be considered as elements of appropriate spherical Hecke algebras and
their eigenvalues are identified with local Archimedean L-factors associated
with admissible representations of reductive groups over R. We expect that the
Baxter operator formalism for the Macdonald polynomials has an interpretation
in representation theory of higher-dimensional arithmetic fields.Comment: 22 pages, typos are fixe
Covariant q-differential operators and unitary highest weight representations for U_q su(n,n)
We investigate a one-parameter family of quantum Harish-Chandra modules of
U_q sl(2n). This family is an analog of the holomorphic discrete series of
representations of the group SU(n,n) for the quantum group U_q su(n, n). We
introduce a q-analog of "the wave" operator (a determinant-type differential
operator) and prove certain covariance property of its powers. This result is
applied to the study of some quotients of the above-mentioned quantum
Harish-Chandra modules. We also prove an analog of a known result by J.Faraut
and A.Koranyi on the expansion of reproducing kernels which determines the
analytic continuation of the holomorphic discrete series.Comment: 26 page
From Reflection Equation Algebra to Braided Yangians
In general, quantum matrix algebras are associated with a couple of
compatible braidings. A particular example of such an algebra is the so-called
Reflection Equation algebra. In this paper we analyse its specific properties,
which distinguish it from other quantum matrix algebras (in first turn, from
the RTT one). Thus, we exhibit a specific form of the Cayley-Hamilton identity
for its generating matrix, which in a limit turns into the Cayley-Hamilton
identity for the generating matrix of the enveloping algebra U(gl(m)). Also, we
consider some specific properties of the braided Yangians, recently introduced
by the authors. In particular, we establish an analog of the Cayley-Hamilton
identity for the generating matrix of such a braided Yangian. Besides, by
passing to a limit of the braided Yangian, we get a Lie algebra similar to that
entering the construction of the rational Gaudin model. In its enveloping
algebra we construct a Bethe subalgebra by the method due to D.Talalaev
Braided affine geometry and q-analogs of wave operators
The main goal of this review is to compare different approaches to
constructing geometry associated with a Hecke type braiding (in particular,
with that related to the quantum group U_q(sl(n))). We make an emphasis on
affine braided geometry related to the so-called Reflection Equation Algebra
(REA). All objects of such type geometry are defined in the spirit of affine
algebraic geometry via polynomial relations on generators.
We begin with comparing the Poisson counterparts of "quantum varieties" and
describe different approaches to their quantization. Also, we exhibit two
approaches to introducing q-analogs of vector bundles and defining the
Chern-Connes index for them on quantum spheres. In accordance with the
Serre-Swan approach, the q-vector bundles are treated as finitely generated
projective modules over the corresponding quantum algebras.
Besides, we describe the basic properties of the REA used in this
construction and compare different ways of defining q-analogs of partial
derivatives and differentials on the REA and algebras close to them. In
particular, we present a way of introducing a q-differential calculus via
Koszul type complexes. The lements of the q-calculus are applied to defining
q-analogs of some relativistic wave operators.Comment: A review submitted to Journal of Physics A: Mathematical and
Theoretica
- …