Abstract
Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and comultiplication. In this paper, we explore algebraic operations that branch lines derive under TQFT. In particular, we investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically.
Highlights
Frobenius algebras have been used extensively in the study of categorification of the Jones polynomial [9], via 2dimensional Topological Quantum Field Theory (2D-TQFT, [10])
2D-TQFT has been characterized [10] in terms of commutative Frobenius algebras, foams have not been algebraically characterized in terms of TQFT
In the case of a foam, we examine the associated algebraic operations that might be associated to branching circles in relation to the Frobenius algebra structure that occurs on the unbranched surfaces
Summary
Frobenius algebras have been used extensively in the study of categorification of the Jones polynomial [9], via 2dimensional Topological Quantum Field Theory (2D-TQFT, [10]). We study the types of algebraic operations that appear along the branch curves of foams in relation to 2D-TQFT. In the case of a foam, we examine the associated algebraic operations that might be associated to branching circles in relation to the Frobenius algebra structure that occurs on the unbranched surfaces. We identify and study Lie algebra and bialgebra structures in relation to branch curves and study their relations to the Frobenius algebra structure.
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