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linear topology, hopf algebras and *-autonomous categories
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.