Flow equivalence, bialgebras and combinatorial spacetimes
by
David Hillman

Let A and B be square matrices of nonnegative integers. Let \sigma_A and \sigma_B be the subshifts of finite type associated with A and B. Let F_A and F_A be the topological suspensions of \sigma_A and \sigma_B. Let f: F_A --> F_A be a flow equivalence map. Parry and Sullivan showed that such an f exists if and only if A and B are related by a finite sequence of simple matrix transformations. The Bowen-Franks group is a well-known invariant of these transformations.

The matrices A and B are associated with directed graphs, which in turn can be associated with scalars in an abstract tensor algebra. A vertex with i incoming edges and j outgoing edges is represented by a symmetric tensor with i downstairs indices and j upstairs indices. Then it turns out that the matrix transformations of Parry and Sullivan correspond precisely to the axioms of a commutative cocommutative bialgebra, where the tensor is thought of as an i-fold multiplication followed by a j-fold comultiplication. From this point of view the Bowen-Franks group is most naturally seen as a group bialgebra which may be easily generalized to obtain an invariant semigroup bialgebra.

The set of topological conjugacy maps of the form h: \sigma_A --> \sigma_A is precisely the set of reversible 1+1-dimensional cellular automata. It turns out that flow equivalence maps of the form f: F_A --> F_A are closely related to a simple generalization of reversible 1+1-dimensional cellular automata in which spacetime curvature is permitted. (I call such maps "combinatorial spacetimes.") Here the components of F_A are interpreted as the one-dimensional combinatorial spaces; the map f induces a map between components, so it describes how these spaces evolve.

Date received: February 1, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caab-60.