1,236 research outputs found

### 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

### 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

### Basic quasi-Hopf algebras over cyclic groups

Let $m$ a positive integer, not divisible by 2,3,5,7. We generalize the
classification of basic quasi-Hopf algebras over cyclic groups of prime order
given in \cite{EG3} to the case of cyclic groups of order $m$. To this end, we
introduce a family of non-semisimple radically graded quasi-Hopf algebras
$A(H,s)$, constructed as subalgebras of Hopf algebras twisted by a quasi-Hopf
twist, which are not twist equivalent to Hopf algebras. Any basic quasi-Hopf
algebra over a cyclic group of order $m$ is either semisimple, or is twist
equivalent to a Hopf algebra or a quasi-Hopf algebra of type $A(H,s)$.Comment: 32page

### Generalizations of Felder's elliptic dynamical r-matrices associated with twisted loop algebras of self-dual Lie algebras

A dynamical $r$-matrix is associated with every self-dual Lie algebra \A
which is graded by finite-dimensional subspaces as \A=\oplus_{n \in \cZ}
\A_n, where \A_n is dual to \A_{-n} with respect to the invariant scalar
product on \A, and \A_0 admits a nonempty open subset \check \A_0 for
which \ad \kappa is invertible on \A_n if $n\neq 0$ and \kappa \in \check
\A_0. Examples are furnished by taking \A to be an affine Lie algebra
obtained from the central extension of a twisted loop algebra \ell(\G,\mu) of
a finite-dimensional self-dual Lie algebra \G. These $r$-matrices, R: \check
\A_0 \to \mathrm{End}(\A), yield generalizations of the basic trigonometric
dynamical $r$-matrices that, according to Etingof and Varchenko, are associated
with the Coxeter automorphisms of the simple Lie algebras, and are related to
Felder's elliptic $r$-matrices by evaluation homomorphisms of \ell(\G,\mu)
into \G. The spectral-parameter-dependent dynamical $r$-matrix that
corresponds analogously to an arbitrary scalar-product-preserving finite order
automorphism of a self-dual Lie algebra is here calculated explicitly.Comment: LaTeX2e, 22 pages. Added a reference and a remar

### Representation theory in complex rank, I

P. Deligne defined interpolations of the tensor category of representations
of the symmetric group S_n to complex values of n. Namely, he defined tensor
categories Rep(S_t) for any complex t. This construction was generalized by F.
Knop to the case of wreath products of S_n with a finite group. Generalizing
these results, we propose a method of interpolating representations categories
of various algebras containing S_n (such as degenerate affine Hecke algebras,
symplectic reflection algebras, rational Cherednik algebras, etc.) to complex
values of n. We also define the group algebra of S_n for complex n, study its
properties, and propose a Schur-Weyl duality for Rep(S_t). In version 2, same
more details have been added.Comment: 26 pages, late

### On Vafa's theorem for tensor categories

In this note we prove two main results. 1. In a rigid braided finite tensor
category over C (not necessarily semisimple), some power of the Casimir element
and some even power of the braiding is unipotent. 2. In a (semisimple) modular
category, the twists are roots of unity dividing the algebraic integer D^{5/2},
where D is the global dimension of the category (the sum of squares of
dimensions of simple objects). Both results generalize Vafa's theorem, saying
that in a modular category twists are roots of unity, and square of the
braiding has finite order. We also discuss the notion of the quasi-exponent of
a finite rigid tensor category, which is motivated by results 1 and 2 and the
paper math/0109196 of S.Gelaki and the author.Comment: 6 pages, late

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