Frobenius algebra

In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Ferdinand Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory , . Jean Dieudonné used this to characterize Frobenius algebras . Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory.
Definition.
A finite-dimensional, unital, associative algebra "A" defined over a field "k" is said to be a Frobenius algebra if "A" is equipped with a nondegenerate bilinear form that satisfies the following equation: . This bilinear form is called the Frobenius form of the algebra.
Equivalently, one may equip "A" with a linear functional such that the kernel of "λ" contains no nonzero left ideal of "A".
A Frobenius algebra is called symmetric if "σ" is symmetric, or equivalently "λ" satisfies .
There is also a different, mostly unrelated notion of the symmetric algebra of a vector space.
Category-theoretical definition.
In category theory, the notion of Frobenius object is an abstract definition of a Frobenius algebra in a category. A Frobenius object formula_1 in a monoidal category formula_2 consists of an object "A" of "C" together with four morphisms
such that
and
commute (for simplicity the diagrams are given here in the case where the monoidal category "C" is strict) and are known as Frobenius conditions.
More compactly, a Frobenius algebra in C is a so-called Frobenius monoidal functor A:1 → C, where 1 is the category consisting of one object and one arrow.
A Frobenius algebra is called isometric or special if formula_6.
Applications.
Frobenius algebras originally were studied as part of an investigation into the representation theory of finite groups, and have contributed to the study of number theory, algebraic geometry, and combinatorics. They have been used to study Hopf algebras, coding theory, and cohomology rings of compact oriented manifolds.
Topological quantum field theories.
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 "K"-algebras is equivalent to the category of symmetric strong monoidal functors from 2-Cob (the category of 2-dimensional cobordisms between 1-dimensional manifolds) to Vect"K" (the category of vector spaces over "K").
The correspondence between TQFTs and Frobenius algebras is given as follows:
This relation between Frobenius-algebras and (1+1)-dimensional TQFTs can be used to explain the Khovanov's categorification of the Jones polynomial.
Generalization: Frobenius extension.
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
The map "E" is sometimes referred to as a Frobenius homomorphism and the elements formula_10 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 formula_11 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 "g"1, ..., "gn". be left coset representatives, so that "G" is a disjoint union of the cosets "g"1"H", ..., "gnH". 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
(The orthonormality condition formula_13 follows.) The dual base is given by formula_14, since
The other dual base equation may be derived from the observation that G is also a disjoint union of the right cosets formula_16.
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:
By DeMeyer's criterion "A" is "G"-Galois over "B" if there are elements formula_18 in "A" satisfying:
whence also
Then "A" is a Frobenius extension of "B" with "E": "A" → "B" defined by
which satisfies
(Furthermore, an example of a separable algebra extension since formula_23 is a separability element satisfying "ea = ae" for all "a" in "A" as well as formula_24. Also an example of a depth two subring ("B" in "A") since
where
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("AB") where the mapping is given by "a" ↦ "λa"("x") and "λa"("x") = "ax" for each "a,x" ∈ "A". Endomorphism ring theorems and converses were investigated later by Mueller, Morita, Onodera and others.