437 research outputs found

### Introduction to co-split Lie algebras

In this work, we introduce a new concept which is obtained by defining a new
compatibility condition between Lie algebras and Lie coalgebras. With this
terminology, we describe the interrelation between the Killing form and the
adjoint representation in a new perspective

### Integral formulas for wave functions of quantum many-body problems and representations of gl(n)

We derive explicit integral formulas for eigenfunctions of quantum integrals
of the Calogero-Sutherland-Moser operator with trigonometric interaction
potential. In particular, we derive explicit formulas for Jack's symmetric
functions. To obtain such formulas, we use the representation of these
eigenfunctions by means of traces of intertwining operators between certain
modules over the Lie algebra $\frak gl_n$, and the realization of these modules
on functions of many variables.Comment: 6 pages. One reference ([FF]) has been corrected. New references and
an introduction have been adde

### On pointed Hopf algebras associated to unmixed conjugacy classes in S_n

Let s in S_n be a product of disjoint cycles of the same length, C the
conjugacy class of s and rho an irreducible representation of the isotropy
group of s. We prove that either the Nichols algebra B(C, rho) is
infinite-dimensional, or the braiding of the Yetter-Drinfeld module is
negative

### Twisted traces of quantum intertwiners and quantum dynamical R-matrices corresponding to generalized Belavin-Drinfeld triples

This paper is a continuation of math.QA/9907181 and math.QA/9908115. We
consider traces of intertwiners between certain representations of the
quantized enveloping algebra associated to a semisimple complex Lie algebra g,
which are twisted by a ``generalized Belavin-Drinfeld triple'', i.e a triple
consisting of two subdiagrams of the Dynkin diagram of g together with an
isomorphism between them. The generating functions F(lambda,mu) for such traces
depend on two weights lambda and mu. We show that F(lambda,mu) satisfy two sets
of difference equations in the variable lambda: the Macdonald-Ruijsenaars (MR)
equations and the quantum Knizhnik-Zamolodchikov
(qKZB) equations. These equations involve as a main ingredient the quantum
dynamical R-matrices constructed in math.QA/9912009. When the generalized
Belavin-Drinfeld triple is an automorphism, we show that F(lambda,mu) satisfy
another two sets of difference equations with respect to the weight mu. These
dual MR and dual qKZB equations involve the usual Felder's dynamical R-matrix.
These results were first obtained by the first author and A. Varchenko in the
special case of the trivial Belavin-Drinfeld triple. However, the symmetry
between lambda and mu which exists in that case is destroyed in the twisted
setting. At the end, we brielfly treat the (simialr) case of Kac-Moody algebras
g and derive the classical limits of all the previous results.Comment: 30 pages, late

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