The following paper is available by anonymous ftp from the site triples.math.mcgill.ca. It is in the file pub/blute in dvi and ps format(compressed). Anyone who would like the paper and cannot access it this way can of course contact me. Regards, Rick Blute LINEAR TOPOLOGY, HOPF ALGEBRAS AND *-AUTONOMOUS CATEGORIES Richard F. Blute McGill University Abstract The goal of this paper is to construct models of variants of linear logic in categories of vector spaces. We begin by recalling work of Barr, in which vector spaces are augmented with linear topologies to obtain new examples of symmetric monoidal closed (autonomous) categories. The advantage of these categories is that they have large subcategories of reflexive objects which are also autonomous, in fact *-autonomous. Thus, we obtain models of (a fragment of) classical linear logic. These models have an advantage over other models arising from linear algebra in that they do not identify the two multiplicative connectives, tensor and par. Furthermore, the models will be complete and cocomplete. We then extend Barr's work to vector spaces with additional structure, in particular to representations of Hopf algebras. As special cases, we examine cocommutative Hopf algebras, and quasitriangular Hopf algebras, also known as quantum groups. In the quasitriangular case, the representations actually form a braided *-autonomous category, first defined by the author. They thus give a model of a braided variant of linear logic, also defined by the author. This is the first example of such a model which does not identify the two multiplicative connectives. Hopf algebras, and more generally bialgebras, have been of interest recently as a possible framework for the study of concurrency, in that they encode the paradigm of interleaving processes, also known as fair merge, as well as other structural operations on concurrent processes. This was first observed by D. Benson. As a result of the work in this paper, such bialgebras will yield models of a fragment of linear logic. While these models do not equate the multiplicative connectives they do validate the MIX rule, thus giving models of a slightly larger theory, first studied by Fleury and Retore. Also, if additional restrictions are placed on the Hopf algebra, we obtain models of cyclic linear logic, defined by Yetter. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++