arXiv subjects

Joseph Bernstein

Publications and source records attributed to Joseph Bernstein.

23 records · Page 2Linked to original sources

arXiv2000

A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors

We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with the n-th tensor power of the fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by projective functors and the commuting action of the Temperley-Lieb algebra by Zuckerman functors. Indecomposable projective functors correspond to Lusztig canonical basis in U(sl(2)). In the dual realization the n-th tensor power of the fundamental representation is identified with a direct sum of parabolic blocks of the highest weight category. Translation across the wall functors act as generators of the Temperley-Lieb algebra while Zuckerman functors act as generators of U(sl(2)).

math.QA↗

arXiv1999

Analytic continuation of representations and estimates of automorphic forms

Properties of analytic vectors in representations of SL(2,R) are used to give new bounds for the triple products recently considered by P. Sarnak. A conjecture of Sarnak about such products is proved. The results of this paper generalize results of A. Good and M. Jutila about special cases, but the techniques are entirely different. One consequence of these results is a new estimate of the magnitude of the Fourier coefficients of cusp forms for non-arithmetic sub-groups of SL(2,R).

math.RT↗

arXiv1997

Quantum groups and representations with highest weight

We consider a special category of Hopf algebras, depending on parameters $Σ$ which possess properties similar to the category of representations of simple Lie group with highest weight $λ$. We connect quantum groups to minimal objects in this categories---they correspond to irreducible representations in the category of representations with highest weight $λ$. Moreover, we want to correspond quantum groups only to finite dimensional irreducible representations. This gives us a condition for $λ$: $λ$--- is dominant means the minimal object in the category of representations with highest weight $λ$ is finite dimensional. We put similar condition for $Σ$. We call $Σ$ dominant if the minimal object in corresponding category has polynomial growth. Now we propose to define quantum groups starting from dominant parameters $Σ$.

q-alg↗

arXiv1995

Sacler lectures

The series of three lectures given at Tel-Aviv University in 1992: 1. Tensor categories. 2. Quantum groups. 3. Topological (quantum) field theories. Published as the preprint IAS 897-92 of Tel-Aviv University and The Mortimer and Raymond Sacler Institute of Advanced Studies.

q-alg↗

arXiv1994

On quantum group SL_q(2)

We start with the observation that the quantum group SL_q(2), described in terms of its algebra of functions has a quantum subgroup, which is just a usual Cartan group. Based on this observation we develop a general method of constructing quantum groups with similar property. We also describe this method in the language of quantized universal enveloping algebras, which is another common method of studying quantum groups. We carry our method in detail for root systems of type SL(2); as a byproduct we find a new series of quantum groups - metaplectic groups of SL(2)-type. Representations of these groups can provide interesting examples of bimodule categories over monoidal category of representations of SL_q(2).

hep-th↗