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Frobenius algebra

Frobenius algebra

Idea

A Frobenius algebra is a vector space that is both an algebra and a coalgebra in a compatible way. This sort of compatibility is different (and more “topological”) from that involved in a bialgebra/Hopf algebra. More generally, Frobenius algebras can be defined in any monoidal category, and even in any polycategory, in which case they are sometimes called Frobenius monoids.

Frobenius algebras have connections to TQFTs; for some more historical motivation see this MO question.

Definition

There are a number of equivalent definitions of the concept of Frobenius algebra.

The origial definition is an associative algebra with a suitable linear form? on it.

In the context of 2d TQFT what crucially matters is that this is equivalent to an associative algebra structure with a compatible coalgebra structure

There are

As associative algebra with coalgebra structure

Definition

A Frobenius algebra in a monoidal category is a quintuple (A,δ,ϵ,μ,η)(A, \delta, \epsilon, \mu, \eta) such that

  1. (A,μ,η)(A, \mu, \eta) is a monoid with multiplication μ:AAA\mu:A\otimes A\to A and unit η:IA\eta:I\to A,
  2. (A,δ,ϵ)(A, \delta, \epsilon) is a comonoid with comultiplication δ:AAA\delta:A\to A\otimes A and counit ϵ:AI\epsilon:A\to I, and
  3. the Frobenius laws hold: (1μ)(δ1)=δμ=(μ1)(1δ)(1 \otimes \mu) \circ (\delta \otimes 1) = \delta \circ \mu = (\mu \otimes 1) \circ (1 \otimes \delta).

In terms of string diagrams, this definition says:

string diagrams for the Frobenius algebra axioms

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.

As associative algebra with linear form

Frobenius algebras were originally formulated in the category Vect of vector spaces with the following equivalent definition:

Definition

A Frobenius algebra is a unital, associative algebra (A,μ,η)(A, \mu, \eta) equipped with a linear form ϵ:Ak\epsilon : A \rightarrow k such that ϵμ\epsilon\mu is a non-degenerate pairing. I.e. the induced map

(1)uϵμ(1u) u \mapsto \epsilon\mu(1 \otimes u)

is an isomorphism of VV with its dual space V *V^*. In such a case, ϵ\epsilon is called a Frobenius form.

From this definition it is easy to see that every Frobenius algebra in Vect is necessarily finite-dimensional.

Further definition

There are about a dozen equivalent definitions of a Frobenius algebra. Ross Street (2004) lists most of them.

Types of Frobenius algebras

Commutative Frobenius algebras

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.

Symmetric Frobenius algebras

We can define ‘commutative’ or ‘symmetric’ Frobenius algebras in any symmetric monoidal category. A Frobenius algebra AA is symmetric if

ϵμS A,A=ϵμ\epsilon \mu \circ S_{A,A} = \epsilon \mu \,

where S A,A:AAAAS_{A,A} : A \otimes A \to A \otimes A is the symmetry, and ϵμ\epsilon\mu is the nondegenerate pairing induced as above from the multiplication and the counit. Any commutative Frobenius algebra is symmetric, but not conversely: for example the algebra of n×nn \times n matrices with entries in a field, with its usual trace as ϵ\epsilon, is symmetric but not commutative when n>1n \gt 1.

A theorem of Eilenberg and Nakayama says that in the category of vector spaces over a field kk, an algebra AA can be equipped with the structure of a symmetric Frobenius algebra if (but not only if) it is separable, meaning that for any field KK extending kk, A kKA \otimes_k K is a semisimple algebra over KK.

Special Frobenius algebras

If μδ=1\mu \circ \delta = 1, a Frobenius algebra is said to be special. In the category of vector spaces, any element aa of an associative unital algebra gives a left multiplication map

L a: A A b ab \array{ L_a : &A &\to& A \\ &b &\mapsto& a b }

which in turn gives a bilinear pairing g:A×Akg: A \times A \to k defined by

g(a,b)=tr(L aL b) g(a,b) = tr(L_a L_b)

One can show that the algebra AA can be equipped with the structure of a special Frobenius algebra if and only if gg is nondegenerate, i.e., if there is an isomorphism AA *A \to A^* given by

ag(a,) a \mapsto g(a, -)

In this case, there is just one way to make AA into a special Frobenius algebra, namely by taking the counit to be

ϵ(a)=tr(L a) \epsilon(a) = tr(L_a)

(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 gg 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 k[G]k[G] of a finite group GG is always separable but strongly separable if and only if the order of GG is invertible in the field kk. By the results mentioned, this means that k[G]k[G] can always be made into a symmetric Frobenius algebra, but only into a special Frobenius algebra when |G||G| is invertible in kk.

To see this, we can check that the group algebra k[G]k[G] becomes a symmetric Frobenius algebra if we define the counit ϵ:k[G]k\epsilon: k[G] \to k to pick out the coefficient of 1G1 \in G:

ϵ: gGa gga 1. \epsilon : \sum_{g \in G} a_g \, g \mapsto a_1 \,.

But when |G||G| is invertible in kk, we can check that k[G]k[G] becomes a special symmetric Frobenius algebra if we normalize the counit as follows:

ϵ: gGa gga 1|G|. \epsilon : \sum_{g \in G} a_g \, g \mapsto \frac{a_1}{|G|} \, .

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.

†-Frobenius algebras

If a Frobenius algebra lives in a monoidal †-category, (δ) =μ(\delta)^\dagger = \mu and (ϵ) =η(\epsilon)^\dagger = \eta, then it is said to be a †-Frobenius algebra. These crop up in the theory of 2d TQFTs, and also in the foundations of quantum theory.

Properties

General

Proposition

A Frobenius algebra AA in a monoidal category is an object dual to itself.

Proof

Let II be the monoidal unit. To say AA is dual to itself means there are maps e:IAAe: I \to A \otimes A and p:AAIp: A \otimes A \to I such that the usual triangular equations hold. The maps are defined by

e=(IηAδAA),p=(AAμAϵI)e = (I \stackrel{\eta}{\to} A \stackrel{\delta}{\to} A \otimes A), \qquad p = (A \otimes A \stackrel{\mu}{\to} A \stackrel{\epsilon}{\to} I)

and one of the triangular equations uses one of the Frobenius laws and unit and counit axioms to derive the following commutative diagram:

A 1η AA 1δ AAA 1 μ μ1 A δ AA 1 ϵ1 A\array{ A & \stackrel{1 \otimes \eta}{\to} & A \otimes A & \stackrel{1 \otimes \delta}{\to} & A \otimes A \otimes A \\ & {}_1\searrow & \downarrow \mathrlap{\mu} & & \downarrow \mathrlap{\mu \otimes 1} \\ & & A & \overset{\delta}{\to} & A \otimes A \\ & & & {}_{1}\searrow & \downarrow \mathrlap{\epsilon \otimes 1} \\ & & & & A }

The other triangular equation uses the other Frobenius law and unit and counit axioms.

As a result, we see that in the monoidal category Mod kMod_k of modules over a commutative ring kk, Frobenius algebras AA considered as modules over kk are finitely generated and projective. This is because A kA \otimes_k -, being adjoint to itself, is a left adjoint and therefore preserves all colimits. That A kA \otimes_k - preserves arbitrary small coproducts means AA is finitely generated over kk, and that A kA \otimes_k- preserves coequalizers means AA is projective over kk.

  • Every Frobenius algebra AA is a quasi-Frobenius algebra?: projective and injective left (right) modules over AA coincide.

  • Every Frobenius algebra AA is a pseudo-Frobenius algebra?: AA is an injective cogenerator in the category of left (right) AA-modules.

PROPs for Frobenius algebras

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:

string diagrams for the Frobenius algebra axioms

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:

string diagram for the "symmetric" law in a Frobenius algebra

The PROP for commutative Frobenius algebras is 2Cob?, as noted by many people and formally proved in (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:

diagrams for the monoid laws in 2Cob

The comonoid laws look like this:

diagrams for the comonoid laws in 2Cob

The Frobenius laws look like this:

diagrams for the Frobenius laws in 2Cob

and the commutative law looks like this:

diagrams for the commutative law in 2Cob

The PROP for special commutative Frobenius algebras is Cospan(FinSet), as proved by Rosebrugh, Sabadini and Walters. This is worth comparing to the PROP for commutative bialgebras, which is Span(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 mδ=1m \circ \delta = 1 says that

string diagram for the "special" law in a Frobenius algebra

Classification of 2d TQFT

2d TQFT (“TCFT”)coefficientsalgebra structure on space of quantum states
open topological stringVect k{}_kFrobenius algebra AAfolklore+(Abrams 96)
open topological string with closed string bulk theoryVect k{}_kFrobenius algebra AA with trace map BZ(A)B \to Z(A) and Cardy condition(Lazaroiu 00, Moore-Segal 02)
non-compact open topological stringCh(Vect)Calabi-Yau A-∞ algebra(Kontsevich 95, Costello 04)
non-compact open topological string with various D-branesCh(Vect)Calabi-Yau A-∞ category
non-compact open topological string with various D-branes and with closed string bulk sectorCh(Vect)Calabi-Yau A-∞ category with Hochschild cohomology
local closed topological string2Mod(Vect k{}_k) over field kkseparable symmetric Frobenius algebras(SchommerPries 11)
non-compact local closed topological string2Mod(Ch(Vect))Calabi-Yau A-∞ algebra(Lurie 09, section 4.2)
non-compact local closed topological string2Mod(S)(\mathbf{S}) for a symmetric monoidal (∞,1)-category S\mathbf{S}Calabi-Yau object in S\mathbf{S}(Lurie 09, section 4.2)

Frobenius algebras in polycategories

In fact, Frobenius algebras can be defined in any polycategory, and hence in any linearly distributive category. The essential point is that the monoidal structure used for the monoid structure could be different from the monoidal structure used for the comonoid structure, i.e. we could have μ:AAA\mu:A\otimes A \to A but δ:AAA\delta :A \to A \parr A. The compatibility between \otimes and \parr in a linearly distributive category (or between their “multicategorical” analogues in a polycategory) is precisely what is required to write down the composites involved in the Frobenius laws. For instance, we can have

AA1 AδA(AA)lindistrib(AA)Aμ1 AAA A\otimes A \xrightarrow{1_A \otimes \delta} A \otimes (A \parr A) \xrightarrow{lin-distrib} (A \otimes A) \parr A \xrightarrow{\mu \parr 1_A} A\parr A

and one of the Frobenius laws says that this composite is equal to

AAμAδAA. A \otimes A \xrightarrow{\mu} A \xrightarrow{\delta} A\parr A.

This is analogous to how a bimonoid can be defined in any duoidal category. In fact, it is a sort of microcosm principle; it is shown in (Egger2010) that Frobenius monoids in the linearly distributive category Sup are precisely *-autonomous cocomplete posets (and hence, in particular, linearly distributive).

In polycategorical language we can give another unbiased definition of a commutative Frobenius monoid: it is equipped with exactly one morphism (A,A,,A)n(A,A,,A)m\overset{n}{\overbrace{(A,A,\dots,A)}} \to \overset{m}{\overbrace{(A,A,\dots,A)}} of each possible (two-sided) arity, such that any (symmetric) polycategorical composite of two such morphisms is equal to another such. The monoid structure consists of the morphisms of arity (2,1)(2,1) and (0,1)(0,1), while the comonoid structure is the morphisms of arity (1,2)(1,2) and (1,0)(1,0), and the Frobenius relations say that three ways to compose these to produce a morphism of arity (2,2)(2,2) are equal. (The morphism of arity (0,0)(0,0) is the composite ϵη\epsilon\eta; no axiom is required on it, because in a polycategory there is no other morphism ()()()\to () to compare it to.) In other words, the free symmetric polycategory containing a commutative Frobenius monoid is the terminal symmetric polycategory. In this way Frobenius algebras are to polycategories in the same way that monoids are to multicategories.

Mike Shulman: I have not carefully checked the above statement, but it seems that the Frobenius laws should suffice to manipulate any such composite into any other. Personal communications from other people who should know are in agreement.

References

Frobenius algebras were introduced by Brauer and Nesbitt and were named after Ferdinand Frobenius.

See for instance

  • Samuel Eilenberg and Tadasi Nakayama (1955), On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings, Nagoya Math. J. 9, 1–16. (web)
  • Marcelo Aguiar (2000), A note on strongly separable algebras, Boletín de la Academia Nacional de Ciencias (Córdoba, Argentina), special issue in honor of Orlando Villamayor, 65, 51–60. (pdf)

Their role in 2d TQFT is discussed for instance in

  • Lowell Abrams , Two-dimensional topological quantum field theories and Frobenius algebra, Jour. Knot. Theory and its Ramifications 5, 569–587 (1996) (ps)

  • John Baez, This Week’s Finds in Mathematical Physics, week268 and week299.

  • Joachim Kock (2004), Frobenius Algebras and 2d Topological Quantum Field Theories, Cambridge U. Press, Cambridge.
  • Joachim Kock (2006), Remarks on the history of the Frobenius equation. (web)
  • Aaron Lauda (2006), Frobenius algebras and ambidextrous adjunctions, Theory and Applications of Categories 16, 84-122. (web (arXiv)

For applications in proof theory of classical and linear logic or linguistics:

  • Martin Hyland, Abstract Interpretation of Proofs: Classical Propositional Calculus , pp.6-21 in Marcinkowski, Tarlecki (eds.), Computer Science Logic (CSL 2004) , LNCS 3210 Springer Heidelberg 2004. (preprint)

  • Richard Garner, Three investigations into linear logic , PhD report Cambridge 2006. (pdf)

  • D. Kartsaklis, M. Sadrzadeh, S. Pulman, Bob Coecke, Reasoning about Meaning in Natural Language with Compact Closed Categories and Frobenius Algebras , arXiv:1401.5980 (2014). (pdf)

Frobenius algebras in linearly distributive categories are discussed in

  • Jeff Egger, The Frobenius relations meet linear distributivity, TAC

See also

  • Bertfried Fauser, Some Graphical Aspects of Frobenius Structures , preprint (2012). arXiv:1202.6380

  • Stephen Lack (2004), Composing PROPs, Theory and Applications of Categories 13, 147–163. (web)

  • R. Rosebrugh, N. Sabadini and R.F.C. Walters (2005), Generic commutative separable algebras and cospans of graphs, Theory and Applications of Categories 15 (Proceedings of CT2004), 164–177. (web)

  • Ross Street (2004), Frobenius monads and pseudomonoids, J. Math. Phys. 45. (web)

  • R. F. C. Walters, R. J. Wood, Frobenius Objects in Cartesian Bicategories , TAC 20 no. 3 (2008) pp.25-47. (pdf)

  • F. W. Lawvere, Ordinal Sums and Equational Doctrines , pp.141-155 in Eckmann (ed.), Seminar on Triples and Categorical Homology Theory , LNM 80 Springer Heidelberg 1969. (TAC Reprint of vol. 80)

Revised on August 1, 2016 12:53:11 by Mike Shulman (24.43.127.99)