Infinite Dimensional Algebras and (2+1)-Dimensional Field Theories: 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] is by now well understood. Secondly, deformations of Poisson structures which 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 _A T ² — the Lie algebra of infinitesimal area preserving diffeomorphisms of the torus — see [6 – 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 are two parts to my talk: 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.