 
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
