INFINITE DIMENSIONAL ALGEBRAS AND 2+1 DIMENSIONAL FIELD THEORY: YET ANOTHER VIEW OF gl(∞) SOME NEW ALGEBRAS

Abstract

Over the past years, associative algebras have come to play a major role in several areas of theoretical physics. Firstly, it has been realized that Yang Baxter algebras [1] constitute the relevant structure underlying 1+1 dimensional integrable models; in addition, their relation to braid groups, the theory of knots and links, and the exchange algebras of 1+1 dimensional conformal field theories [2] has been quite well understood by now. Secondly, deformations of Poisson structures that appeared in 2+1 dimensional field theories as infinite dimensional symmetry algebras possess underlying associative structures, which have also been studied in some detail (concerning higher spin theories see, e.g., [3, 4] and references therein, concerning the enveloping algebra of sl(2, C) see, e.g., [5], concerning deformations of diff AT2 — the Lie algebra of infinitesimal area preserving diffeomorphisms of the Torus — see [6, 7, 8, 9]). Ideas on how both investigations could eventually converge (i.e., a relation between 2+1 and 1+1 dimensions) have, e.g., been expressed in [10]. As indicated by the two subtitles there will be two parts to my paper: the first one presents a view on something I met long ago [11], and recently got interested in again [5, 7, 9, 12], while the second part introduces some algebraic structures that seem to be interesting, and possibly new.