Summary: TWO­DIMENSIONAL TOPOLOGICAL QUANTUM FIELD
THEORIES AND FROBENIUS ALGEBRAS
LOWELL ABRAMS
Johns Hopkins University
Department of Mathematics
3400 Charles Street
Baltimore, MD 21218
ABSTRACT
We characterize Frobenius algebras A as algebras having a comultiplication which
is a map of A­modules. This characterization allows a simple demonstration of the
compatibility of Frobenius algebra structure with direct sums. We then classify
the indecomposable Frobenius algebras as being either ``annihilator algebras'' ---
algebras whose socle is a principal ideal --- or field extensions. The relationship
between two­dimensional topological quantum field theories and Frobenius algebras
is then formulated as an equivalence of categories. The proof hinges on our new
characterization of Frobenius algebras.
These results together provide a classification of the indecomposable two­dimensional
topological quantum field theories.
Keywords: topological quantum field theory, frobenius algebra, two­dimensional
cobordism, category theory

Source: Abrams, Lowell - Department of Mathematics, George Washington University

Collections: Mathematics