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  • Print publication year: 2003
  • Online publication date: January 2010

2 - Frobenius algebras

Summary

Summary

A first preliminary section reviews some basic notions of vector spaces, pairings, algebras and modules, and establishes notation and terminology.

Section 2.2 is devoted to ‘classical’ theory of Frobenius algebras. A Frobenius algebra can be characterised equivalently as: a finite-dimensional algebra A equipped with an associative nondegenerate pairing, or equipped with a linear functional whose nullspace contains no nontrivial ideals, or equipped with an A-linear isomorphism to the dual space A*. Then we give a long list of examples of Frobenius algebras. Some of these examples require more algebra than presumed elsewhere in the text, but dont panic! – these examples are not really needed elsewhere in the text.

The main result of this chapter is established in Section 2.3. It is yet another equivalent characterisation of Frobenius algebras: a Frobenius algebra is an algebra which is also a coalgebra, with a compatibility between multiplication and comultiplication. This compatibility condition is actually of topological nature, and a second important goal of this chapter is to develop a graphical language for the algebraic operations involved, which provides important insight in the structures.

In Section 2.4 we collect some results on the category of Frobenius algebras: we observe that Frobenius algebra homomorphisms are always invertible, and that the tensor product of two Frobenius algebras is again a Frobenius algebra in a canonical way. Finally we make a digression on Hopf algebras and compare their axioms with those for Frobenius algebras.