Frobenius algebra

The product and coproduct on a Frobenius algebra can be interpreted as the functor of a (1+1)-dimensional topological quantum field theory, applied to a pair of pants.

Recently, it has been seen that they play an important role in the algebraic treatment and axiomatic foundation of topological quantum field theory. A commutative Frobenius algebra determines uniquely (up to isomorphism) a (1+1)-dimensional TQFT. More precisely, the category of commutative Frobenius ${\displaystyle K}$-algebras is equivalent to the category of symmetric strong monoidal functors from ${\displaystyle 2}$-${\displaystyle {\textbf {Cob}}}$ (the category of 2-dimensional cobordisms between 1-dimensional manifolds) to ${\displaystyle {\textbf {Vect}}_{K}}$ (the category of vector spaces over ${\displaystyle K}$).

The correspondence between TQFTs and Frobenius algebras is given as follows:

• 1-dimensional manifolds are disjoint unions of circles: a TQFT associates a vector space with a circle, and the tensor product of vector spaces with a disjoint union of circles,
• a TQFT associates (functorially) to each cobordism between manifolds a map between vector spaces,
• the map associated with a pair of pants (a cobordism between 1 circle and 2 circles) gives a product map ${\displaystyle V\otimes V\to V}$ or a coproduct map ${\displaystyle V\to V\otimes V}$, depending on how the boundary components are grouped – which is commutative or cocommutative, and
• the map associated with a disk gives a counit (trace) or unit (scalars), depending on grouping of boundary.

This relation between Frobenius algebras and (1+1)-dimensional TQFTs can be used to explain Khovanov's categorification of the Jones polynomial.[6][7]

Let B be a subring sharing the identity element of a unital associative ring A. This is also known as ring extension A | B. Such a ring extension is called Frobenius if

• There is a linear mapping E: A → B satisfying the bimodule condition E(bac) = bE(a)c for all b,c ∈ B and a ∈ A.
• There are elements in A denoted ${\displaystyle \{x_{i}\}_{i=1}^{n}}$ and ${\displaystyle \{y_{i}\}_{i=1}^{n}}$ such that for all a ∈ A we have:

${\displaystyle \sum _{i=1}^{n}E(ax_{i})y_{i}=a=\sum _{i=1}^{n}x_{i}E(y_{i}a)}$

The map E is sometimes referred to as a Frobenius homomorphism and the elements ${\displaystyle x_{i},y_{i}}$ as dual bases. (As an exercise it is possible to give an equivalent definition of Frobenius extension as a Frobenius algebra-coalgebra object in the category of B-B-bimodules, where the equations just given become the counit equations for the counit E.)

For example, a Frobenius algebra A over a commutative ring K, with associative nondegenerate bilinear form (-,-) and projective K-bases ${\displaystyle x_{i},y_{i}}$ is a Frobenius extension A | K with E(a) = (a,1). Other examples of Frobenius extensions are pairs of group algebras associated to a subgroup of finite index, Hopf subalgebras of a semisimple Hopf algebra, Galois extensions and certain von Neumann algebra subfactors of finite index. Another source of examples of Frobenius extensions (and twisted versions) are certain subalgebra pairs of Frobenius algebras, where the subalgebra is stabilized by the symmetrizing automorphism of the overalgebra.

The details of the group ring example are the following application of elementary notions in group theory. Let G be a group and H a subgroup of finite index n in G; let g₁, ..., g_n. be left coset representatives, so that G is a disjoint union of the cosets g₁H, ..., g_nH. Over any commutative base ring k define the group algebras A = k[G] and B = k[H], so B is a subalgebra of A. Define a Frobenius homomorphism E: A → B by letting E(h) = h for all h in H, and E(g) = 0 for g not in H : extend this linearly from the basis group elements to all of A, so one obtains the B-B-bimodule projection

${\displaystyle E\left(\sum _{g\in G}n_{g}g\right)=\sum _{h\in H}n_{h}h\ \ \ {\text{ for }}n_{g}\in k}$

(The orthonormality condition ${\displaystyle E(g_{i}^{-1}g_{j})=\delta _{ij}1}$ follows.) The dual base is given by ${\displaystyle x_{i}=g_{i},y_{i}=g_{i}^{-1}}$, since

${\displaystyle \sum _{i=1}^{n}g_{i}E\left(g_{i}^{-1}\sum _{g\in G}n_{g}g\right)=\sum _{i}\sum _{h\in H}n_{g_{i}h}g_{i}h=\sum _{g\in G}n_{g}g}$

The other dual base equation may be derived from the observation that G is also a disjoint union of the right cosets ${\displaystyle Hg_{1}^{-1},\ldots ,Hg_{n}^{-1}}$.

Also Hopf-Galois extensions are Frobenius extensions by a theorem of Kreimer and Takeuchi from 1989. A simple example of this is a finite group G acting by automorphisms on an algebra A with subalgebra of invariants:

${\displaystyle B=\{x\in A\mid \forall g\in G,g(x)=x\}.}$

By DeMeyer's criterion A is G-Galois over B if there are elements ${\displaystyle \{a_{i}\}_{i=1}^{n},\{b_{i}\}_{i=1}^{n}}$ in A satisfying:

${\displaystyle \forall g\in G:\ \ \sum _{i=1}^{n}a_{i}g(b_{i})=\delta _{g,1_{G}}1_{A}}$

whence also

${\displaystyle \forall g\in G:\ \ \sum _{i=1}^{n}g(a_{i})b_{i}=\delta _{g,1_{G}}1_{A}.}$

Then A is a Frobenius extension of B with E: A → B defined by

${\displaystyle E(a)=\sum _{g\in G}g(a)}$

which satisfies

${\displaystyle \forall x\in A:\ \ \sum _{i=1}^{n}E(xa_{i})b_{i}=x=\sum _{i=1}^{n}a_{i}E(b_{i}x).}$

(Furthermore, an example of a separable algebra extension since ${\textstyle e=\sum _{i=1}^{n}a_{i}\otimes _{B}b_{i}}$ is a separability element satisfying ea = ae for all a in A as well as ${\textstyle \sum _{i=1}^{n}a_{i}b_{i}=1}$. Also an example of a depth two subring (B in A) since

${\displaystyle a\otimes _{B}1=\sum _{g\in G}t_{g}g(a)}$

where

${\displaystyle t_{g}=\sum _{i=1}^{n}a_{i}\otimes _{B}g(b_{i})}$

for each g in G and a in A.)

Frobenius extensions have a well-developed theory of induced representations investigated in papers by Kasch and Pareigis, Nakayama and Tzuzuku in the 1950s and 1960s. For example, for each B-module M, the induced module A ⊗_B M (if M is a left module) and co-induced module Hom_B(A, M) are naturally isomorphic as A-modules (as an exercise one defines the isomorphism given E and dual bases). The endomorphism ring theorem of Kasch from 1960 states that if A | B is a Frobenius extension, then so is A → End(A_B) where the mapping is given by a ↦ λ_a(x) and λ_a(x) = ax for each a,x ∈ A. Endomorphism ring theorems and converses were investigated later by Mueller, Morita, Onodera and others.

As already hinted at in the previous paragraph, Frobenius extensions have an equivalent categorical formulation. Namely, given a ring extension ${\displaystyle S\subset R}$, the induced induction functor ${\displaystyle R\otimes _{S}-\colon {\text{Mod}}(S)\to {\text{Mod}}(R)}$ from the category of, say, left S-modules to the category of left R-modules has both a left and a right adjoint, called co-restriction and restriction, respecitvely. The ring extension is then called Frobenius if and only if the left and the right adjoint are naturally isomorphic.

This leads to the obvious abstraction to ordinary category theory: An adjunction ${\displaystyle F\dashv G}$ is called a Frobenius adjunction iff also ${\displaystyle G\dashv F}$. A functor F is a Frobenius functor if it is part of a Frobenius adjunction, i.e. if it has isomorphic left and right adjoints.