Duke Mathematical Journal 

About: Duke Mathematical Journal is an academic journal. The journal publishes majorly in the area(s): Bounded function & Conjecture. It has an ISSN identifier of 0012-7094. Over the lifetime, 5898 publication(s) have been published receiving 238855 citation(s).

On crystal bases of the $Q$-analogue of universal enveloping algebras

Abstract: 0. Introduction. The notion of the q-analogue of universal enveloping algebras is introduced independently by V. G. Drinfeld and M. Jimbo in 1985 in their study of exactly solvable models in the statistical mechanics. This algebra Uq(g) contains a parameter q, and, when q 1, this coincides with the universal enveloping algebra. In the context of exactly solvable models, the parameter q is that of temperature, and q 0 corresponds to the absolute temperature zero. For that reason, we can expect that the q-analogue has a simple structure at q 0. In [K1] we named crystallization the study at q 0, and we introduced the notion of crystal bases. Roughly speaking, crystal bases are bases of Uq(9)-modules at q 0 that satisfy certain axioms. There, we proved the existence and the uniqueness of crystal bases of finite-dimensional representations of U(g) when g is one of the classical Lie algebras A,, B,, C, and D,. K. Misra and T. Miwa ([M]) proved the existence of a crystal base of the basic representation of U,(A1)) and gave its combinatorial description. The aim of this article is to give the proof of the existence and uniqueness theorem of crystal bases for an arbitrary symmetrizable Kac-Moody Lie algebra I. Moreover, we globalize this notion. Namely, with the aid of a crystal base we construct a base named the global crystal base of any highest weight irreducible integrable

Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations

Abstract: A simple duality argument shows these two problems are completely equivalent ifp and q are dual indices, (]/) + (I/q) ]. ]nteresl in Problem A when S is a sphere stems from the work of C. Fefferman [3], and in this case the answer is known (see [l I]). Interest in Problem B was recently signalled by 1. Segal [6] who studied the special case S {(x, y) yZ xz I} and gave the interpretation of the answer as a space-time decay for solutions of the Klein-Gordon equation with finite relativistic-invariant norm. In this paper we give a complete solution when S is a quadratic surface given by

A categorification of the Jones polynomial

Abstract: Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.