symmetric monoidal (∞,1)-category of spectra
A Frobenius algebra in a monoidal category is a quadruple such that
In terms of string diagrams, this definition says:
The first line here shows the associative law and left/right unit laws for a monoid. The second line shows the coassociative law and left/right counit laws for a comonoid. The third line shows the Frobenius laws.
A Frobenius algebra is a unital, associative algebra equipped with a linear form such that is a non-degenerate pairing. I.e. the induced map
From this definition it is easy to see that every Frobenius algebra in Vect is necessarily finite-dimensional.
There are about a dozen equivalent definitions of a Frobenius algebra. Ross Street (2004) lists most of them.
We can define ‘commutative’ Frobenius algebras in any symmetric monoidal category. Namely, a Frobenius algebra is commutative if its associated monoid is commutative — or equivalently, if its associated comonoid is cocommutative.
We can define ‘commutative’ or ‘symmetric’ Frobenius algebras in any symmetric monoidal category. A Frobenius algebra is symmetric if
where is the symmetry. Any commutative Frobenius algebra is symmetric, but not conversely: for example the algebra of matrices with entries in a field, with its usual trace as , is symmetric but not commutative when .
A theorem of Eilenberg and Nakayama says that in the category of vector spaces over a field , an algebra can be equipped with the structure of a symmetric Frobenius algebra if (but not only if) it is separable, meaning that for any field extending , is a semisimple algebra over .
If , a Frobenius algebra is said to be special. In the category of vector spaces, any element of an associative unital algebra gives a left multiplication map
which in turn gives a bilinear pairing defined by
One can show that the algebra can be equipped with the structure of a special Frobenius algebra if and only if is nondegenerate, i.e., if there is an isomorphism given by
In this case, there is just one way to make into a special Frobenius algebra, namely by taking the counit to be
(In any Frobenius algebra, the unit, multiplication and counit determine the comultiplication.)
In fact, all the results of the previous paragraph generalize to Frobenius algebras in any symmetric monoidal category, since the proofs can be done using string diagrams.
An associative unital algebra for which the bilinear pairing is nondegenerate is called strongly separable. So, any strongly separable algebra becomes a special Frobenius algebra in a unique way. For more details, see separable algebra and Aguiar (2000).
To get a feeling for some of the concepts we are discussing, an example is helpful. The group algebra of a finite group is always separable but strongly separable if and only if the order of is invertible in the field . By the results mentioned, this means that can always be made into a symmetric Frobenius algebra, but only into a special Frobenius algebra when is invertible in .
To see this, we can check that the group algebra becomes a symmetric Frobenius algebra if we define the counit to pick out the coefficient of :
But when is invertible in , we can check that becomes a special symmetric Frobenius algebra if we normalize the counit as follows:
We should warn the reader that Rosebrugh et al (2005) call a special Frobenius algebra ‘separable’. This usage conflicts with the standard definition of a separable algebra in the category of vector spaces over a field, so we suggest avoiding it.
A Frobenius algebra in a monoidal category is an object dual to itself.
Let be the monoidal unit. To say is dual to itself means there are maps and such that the usual triangular equations hold. The maps are defined by
and one of the triangular equations uses one of the Frobenius laws and unit and counit axioms to derive the following commutative diagram:
The other triangular equation uses the other Frobenius law and unit and counit axioms.
As a result, we see that in the monoidal category of modules over a commutative ring , Frobenius algebras considered as modules over are finitely generated and projective. This is because , being adjoint to itself, is a left adjoint and therefore preserves all colimits. That preserves arbitrary small coproducts means is finitely generated over , and that preserves coequalizers means is projective over .
Every Frobenius algebra is a quasi-Frobenius algebra?: projective and injective left (right) modules over coincide.
Every Frobenius algebra is a pseudo-Frobenius algebra?: is an injective cogenerator in the category of left (right) -modules.
Certain kinds of Frobenius algebras have nice PROPs or PROs. The PRO for Frobenius algebras is the monoidal category of planar thick tangles, as noted by Aaron Lauda Lauda (2006) and illustrated here:
Lauda and Pfeiffer Lauda (2008) showed that the PROP for symmetric Frobenius algebras is the category of ‘topological open strings’, since it obeys this extra axiom:
The PROP for commutative Frobenius algebras is 2Cob?, as noted by many people and formally proved by Lowell Abrams Abrams (1996). This means that any commutative Frobenius algebra gives a 2d TQFT. See Kock (2006) for a history of this subject and Kock (2004) for a detailed introduction. In 2Cob, the circle is a Frobenius algebra. The monoid laws look like this:
The comonoid laws look like this:
The Frobenius laws look like this:
and the commutative law looks like this:
The PROP for special commutative Frobenius algebras is Span(FinSet), as proved by Rosebrugh, Sabadini and Walters. This is worth comparing to the PROP for commutative bialgebras, which is Cospan(FinSet). For details, see Rosebrugh et al (2005), and also Lack (2004).
A special commutative Frobenius algebra gives a 2d TQFT that is insensitive to the genus of a 2-manifold, since in terms of pictures, the ‘specialness’ axioms says that
See for instance
Their role in 2-dimensional TQFT is discussed for instance in
(TO ADD: stuff from this MO post) (TO DO: is used for the comultiplication in the beginning, then for the linear form, and in the definition of symmetry for the bilinear pairing on a Frobenius algebra. Use three different letters for these)