Фробениус алгебралары: дуалдық қасиеттері мен қолданыстары

Algebraic structure with "nice" duality properties

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 Georg 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.

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 σ : A × A → k that satisfies the following equation: 1=σ(a·b, c) = σ(a, b·c). This bilinear form is called the Frobenius form of the algebra. Equivalently, one may equip A with a linear functional λ : A → k 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 1=λ(a·b) = λ(b·a). There is also a different, mostly unrelated notion of the symmetric algebra of a vector space.

For a Frobenius algebra A with σ as above, the automorphism ν of A such that 1=σ(a, b) = σ(ν(b), a) is Nakayama automorphism associated to A and σ.

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. Any finite dimensional unital associative algebra A has a natural homomorphism to its own endomorphism ring End(A). A bilinear form can be defined on A in the sense of the previous example. If this bilinear form is nondegenerate, then it equips A with the structure of a Frobenius algebra. Every group ring k[G] of a finite group G over a field k is a symmetric Frobenius algebra, with Frobenius form σ(a,b) given by the coefficient of the identity element in a·b. For a field k, the four dimensional k algebra k[x,y]/ (x2, y2) is a Frobenius algebra. This follows from the characterization of commutative local Frobenius rings below, since this ring is a local ring with its maximal ideal generated by x and y, and unique minimal ideal generated by xy. For a field k, the three dimensional k algebra A=k[x,y]/ (x, y)2 is not a Frobenius algebra. The A homomorphism from xA into A induced by x ↦ y cannot be extended to an A homomorphism from A into A, showing that the ring is not self injective, thus not Frobenius. Any finite dimensional Hopf algebra, by a 1969 theorem of Larson Sweedler on Hopf modules and integrals.

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. Frobenius algebras are quasi Frobenius rings, and in particular, they are left and right Artinian and left and right self 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, unital, 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. A finite dimensional bi Frobenius algebra or strict double Frobenius algebra is a k vector space A with two multiplication structures as unital Frobenius algebras (A, • , 1) and (A, , ): there must be multiplicative homomorphisms and of A into k with and non degenerate, and a k isomorphism S of A onto itself which is an anti automorphism for both structures, such that This is the case precisely when A is a finite dimensional Hopf algebra over k and S is its antipode. The group algebra of a finite group gives an example.

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.