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