340 research outputs found

### Scattering matrices and affine Hecke algebras

We construct the scattering matrices for an arbitrary Weyl group in terms of
elementary operators which obey the generalised Yang-Baxter equation. We use
this construction to obtain the affine Hecke algebras. The center of the affine
Hecke algebras coincides with commuting Hamiltonians. These Hamiltonians have
q-deformed affine Lie algebras as symmetry algebra.Comment: 22 pages, harvmac, no figures, Lecture at Schladming, March 4,11 199

### Representations of affine Lie algebras, elliptic r-matrix systems, and special functions

There were some errors in paper hep-th/9303018 in formulas 6.1, 6.6, 6.8,
6.11. These errors have been corrected in the present version of this paper.
There are also some minor changes in the introduction.Comment: 33 pages, no figure

### Irreducibility of fusion modules over twisted Yangians at generic point

With any skew Young diagram one can associate a one parameter family of
"elementary" modules over the Yangian \Yg(\g\l_N). Consider the twisted
Yangian \Yg(\g_N)\subset \Yg(\g\l_N) associated with a classical matrix Lie
algebra \g_N\subset\g\l_N. Regard the tensor product of elementary Yangian
modules as a module over \Yg(\g_N) by restriction. We prove its
irreducibility for generic values of the parameters.Comment: Replaced with journal version, 18 page

### Completely splittable representations of affine Hecke-Clifford algebras

We classify and construct irreducible completely splittable representations
of affine and finite Hecke-Clifford algebras over an algebraically closed field
of characteristic not equal to 2.Comment: 39 pages, v2, added a new reference with comments in section 4.4,
added two examples (Example 5.4 and Example 5.11) in section 5, mild
corrections of some typos, to appear in J. Algebraic Combinatoric

### Isospectral flow in Loop Algebras and Quasiperiodic Solutions of the Sine-Gordon Equation

The sine-Gordon equation is considered in the hamiltonian framework provided
by the Adler-Kostant-Symes theorem. The phase space, a finite dimensional
coadjoint orbit in the dual space \grg^* of a loop algebra \grg, is
parametrized by a finite dimensional symplectic vector space $W$ embedded into
\grg^* by a moment map. Real quasiperiodic solutions are computed in terms of
theta functions using a Liouville generating function which generates a
canonical transformation to linear coordinates on the Jacobi variety of a
suitable hyperelliptic curve.Comment: 12 pg

### Whittaker Limits of Difference Spherical Functions

The main aim of this paper is to introduce global q–Whittaker functions as the limit t → 0 of the (renormalized) generalized symmetric spherical functions constructed in [C5] for arbitrary reduced root systems (see [Sto] in the C∨C–case). This work is inspired by [GLO1] and [GLO2], though our approach is different. For instance, we obtain a q–version of the classical Shintani-Casselman-Shalika formula [Shi, CS] via the q–Mehta -Macdonald integral in the Jackson setting. The Shintani-type formulas (in the case of GLn) play an important role in [GLO1, GLO2], but the q–Gauss integrals are not considered there as well as globally-defined q–Whittaker functions. We use these formulas to obtain a q, t–generalization of the Harish-Chandra asymptotic formula for the classical spherical function

### The decomposition of level-1 irreducible highest weight modules with respect to the level-0 actions of the quantum affine algebra $U'_q(\hat{sl}_n)$

We decompose the level-1 irreducible highest weight modules of the quantum
affine algebra $U_q(\hat{sl}_n)$ with respect to the level-0 $U'_q
(\hat{sl}_n)$--action defined in q-alg/9702024. The decomposition is
parameterized by the skew Young diagrams of the border strip type.Comment: 22 pages, AMSLaTe

### On the idempotents of Hecke algebras

We give a new construction of primitive idempotents of the Hecke algebras
associated with the symmetric groups. The idempotents are found as evaluated
products of certain rational functions thus providing a new version of the
fusion procedure for the Hecke algebras. We show that the normalization factors
which occur in the procedure are related to the Ocneanu--Markov trace of the
idempotents.Comment: 11 page

### A simple construction of elliptic $R$-matrices

We show that Belavin's solutions of the quantum Yang--Baxter equation can be
obtained by restricting an infinite $R$-matrix to suitable finite dimensional
subspaces. This infinite $R$-matrix is a modified version of the
Shibukawa--Ueno $R$-matrix acting on functions of two variables.Comment: 6 page

### Bethe Ansatz solutions for Temperley-Lieb Quantum Spin Chains

We solve the spectrum of quantum spin chains based on representations of the
Temperley-Lieb algebra associated with the quantum groups ${\cal U}%
_{q}(X_{n})$ for $X_{n}=A_{1},$ $B_{n},$ $C_{n}$ and $D_{n}$. The tool is a
modified version of the coordinate Bethe Ansatz through a suitable choice of
the Bethe states which give to all models the same status relative to their
diagonalization. All these models have equivalent spectra up to degeneracies
and the spectra of the lower dimensional representations are contained in the
higher-dimensional ones. Periodic boundary conditions, free boundary conditions
and closed non-local boundary conditions are considered. Periodic boundary
conditions, unlike free boundary conditions, break quantum group invariance.
For closed non-local cases the models are quantum group invariant as well as
periodic in a certain sense.Comment: 28 pages, plain LaTex, no figures, to appear in Int. J. Mod. Phys.

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