Frobenius algebra

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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 Brauer and Nesbitt and were named after Frobenius. Nakayama discovered the beginnings of a rich duality theory in his (Nakayama 1939) and especially in his (Nakayama 1941). Dieudonné used this to characterize Frobenius algebras in his (Dieudonné 1958) where he called this property of Frobenius algebras a perfect duality. 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.

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[edit] Definition

A finite dimensional, unital, associative algebra A defined over a field k is a Frobenius algebra if and only if A can be equipped with a special kind of nondegenerate bilinear form, σ:A × A → k which is "associative" insofar as σ(a·b,c)=σ(a,b·c). Such a bilinear form is called a Frobenius form of the algebra.

Equivalently, one may require a linear functional λ:A→k such that the kernel of λ contains no nonzero left ideal of A.

A Frobenius algebra is called a symmetric algebra if σ can be chosen to be symmetric, or equivalently that λ can be chosen so that λ(a·b) = λ(b·a).

There is also a different, mostly unrelated notion of the symmetric algebra of a vector space.

[edit] Examples

  • Any matrix algebra defined over a field k is a Frobenius algebra with Frobenius form σ(a,b)=tr(a·b) where tr denotes the trace.
  • Every group ring of a finite group over a field is a Frobenius algebra, with Frobenius form σ(a,b) the coefficient of the identity element of a·b.
  • For a field k, the four-dimensional k-algebra k[x,y]/(x2, y2) is a Frobenius algebra.
  • For a field k, the three-dimensional k-algebra k[x,y]/(x, y)2 is not a Frobenius algebra.

[edit] Properties

  • The direct product and tensor product of Frobenius algebras are Frobenius algebras.
  • A finite-dimensional commutative local algebra over a field is Frobenius if and only if the right regular module is injective, if and only if the algebra has a unique minimal ideal.
  • Commutative, local Frobenius algebras are precisely the zero-dimensional local Gorenstein rings containing their residue field and finite dimensional over it.
  • The right regular representation of a Frobenius algebra is always injective.
  • For a field k, a finite-dimensional, unital, associative algebra is Frobenius if and only if the injective right A-module Homk(A,k) is isomorphic to the right regular representation of A.
  • For an infinite field k, a finite dimensional, unitial, associative k-algebra is a Frobenius algebra if it has only finitely many minimal right ideals.
  • If F is a finite dimensional extension field of k, then a finite dimensional F-algebra is naturally a finite dimensional k-algebra via restriction of scalars, and is a Frobenius F-algebra if and only if it is a Frobenius k-algebra. In other words, the Frobenius property does not depend on the field, as long as the algebra remains a finite dimensional algebra.
  • Similarly, if F is a finite dimensional extension field of k, then every k-algebra A gives rise naturally to a F algebra, F ⊗k A, and A is a Frobenius k-algebra if and only if F ⊗k A is a Frobenius F-algebra.
  • Amongst those finite-dimensional, unital, associative algebras whose right regular representation is injective, the Frobenius algebras A are precisely those whose simple modules M have the same dimension as their A-duals, HomA(M,A). Amongst these algebras, the A-duals of simple modules are always simple.

[edit] 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 (A,\mu,\eta,\delta,\varepsilon) in a monoidal category (C,\otimes,I) consists of an object A of C together with four morphisms

\mu:A\otimes A\to A,\qquad \eta:I\to A,\qquad\delta:A\to A\otimes A\qquad\mathrm{and}\qquad\varepsilon:A\to I

such that

  • the diagrams
Image:Frobenius_obj_coh_1.png

and

Image:Frobenius_obj_coh_2.png

commute (for simplicity the diagrams are given here in the case where the monoidal category C\, is strict).

[edit] 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. 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 namely determines uniquely (up to isomorphism) a 2-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) to VectK (the category of vector spaces over K).

[edit] References

[edit] See also