nLab 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 original definition is an associative algebra with a suitable linear form? on it.

• As an associative algebra with linear form.

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

• As an associative algebra with compatible coalgebra structure.

There are

• Further equivalent definitions

As associative algebra with coalgebra structure

Definition

A Frobenius algebra in a monoidal category is a quintuple $(A, \delta, \epsilon, \mu, \eta)$ such that

1. $(A, \mu, \eta)$ is a monoid with multiplication $\mu:A\otimes A\to A$ and unit $\eta:I\to A$,
2. $(A, \delta, \epsilon)$ is a comonoid with comultiplication $\delta:A\to A\otimes A$ and counit $\epsilon:A\to I$, and
3. the Frobenius laws hold: $(1 \otimes \mu) \circ (\delta \otimes 1) = \delta \circ \mu = (\mu \otimes 1) \circ (1 \otimes \delta)$.

In terms of string diagrams, this definition says:

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.

In fact, although this seems rarely to be remarked, the two Frobenius laws can be replaced by the single axiom $(1 \otimes \mu) \circ (\delta \otimes 1) = (\mu \otimes 1) \circ (1 \otimes \delta)$. Here is a proof in string diagram notation that this axiom implies $(1 \otimes \mu) \circ (\delta \otimes 1) = \delta \circ \mu$ (taken from Pastro-Street 2008):

Here the first and fifth steps use the counitality property of the comonoid structure, the second and fourth steps use the assumed axiom (once in each direction), and the third step uses coassociativity.

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 $(A, \mu, \eta)$ equipped with a linear form $\epsilon : A \rightarrow k$ such that $\epsilon\circ\mu$ is a non-degenerate pairing. I.e. the induced map

$$u \mapsto (v\mapsto\epsilon\circ\mu(v \otimes u))$$

is an isomorphism of $V$ with its dual space $V^*$. In such a case, $\epsilon$ 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 $A$ is symmetric if

$$\epsilon \mu \circ S_{A,A} = \epsilon \mu \,$$

where $S_{A,A} : A \otimes A \to A \otimes A$ is the symmetry, and $\epsilon\mu$ 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 $n \times n$ matrices with entries in a field, with its usual trace as $\epsilon$, is symmetric but not commutative when $n \gt 1$.

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

Special Frobenius algebras

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

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

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

$$g(a,b) = tr(L_a L_b)$$

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

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

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

$$\epsilon(a) = tr(L_a)$$

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

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

$$\epsilon : \sum_{g \in G} a_g \, g \mapsto a_1 \,.$$

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

$$\epsilon : \sum_{g \in G} a_g \, g \mapsto \frac{a_1}{|G|} \, .$$

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, $(\delta)^\dagger = \mu$ and $(\epsilon)^\dagger = \eta$, 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 $A$ in a monoidal category is an object dual to itself.

Proof

Let $I$ be the monoidal unit. To say $A$ is dual to itself means there are maps $e: I \to A \otimes A$ and $p: A \otimes A \to I$ such that the triangle identities hold. The maps are defined by

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

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

$$\array{ A & \stackrel{1 \otimes \eta}{\to} & A \otimes A & \stackrel{1 \otimes \delta}{\to} & A \otimes A \otimes A \\ & {}_1\searrow & \downarrow \mathrlap{\mu} & & \downarrow \mathrlap{\mu \otimes 1} \\ & & A & \overset{\delta}{\to} & A \otimes A \\ & & & {}_{1}\searrow & \downarrow \mathrlap{\epsilon \otimes 1} \\ & & & & A }$$

The other triangle identity uses the other Frobenius law and unit and counit axioms.

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

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

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

Frobenius algebras are closely connected with ambidextrous adjunctions. For example, a Frobenius monad on a category $C$ is by definition a Frobenius monoid in the monoidal category of endofunctors on $C$ (with monoidal product given by endofunctor composition), and if we have a pair of adjunctions $F \dashv U$ and $U \dashv F$, then $M = U F$ carries a monad structure and a comonad structure and the Frobenius laws are satisfied, a fact most easily seen by using string diagrams.

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:

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:

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:

The comonoid laws look like this:

The Frobenius laws look like this:

and the commutative law looks like this:

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 $m \circ \delta = 1$ says that

Classification of 2d TQFT

2d TQFT (“TCFT”)	coefficients	algebra structure on space of quantum states
open topological string	Vect${}_k$	Frobenius algebra $A$	folklore+(Abrams 96)
open topological string with closed string bulk theory	Vect${}_k$	Frobenius algebra $A$ with trace map $B \to Z(A)$ and Cardy condition	(Lazaroiu 00, Moore-Segal 02)
non-compact open topological string	Ch(Vect)	Calabi-Yau A-∞ algebra	(Kontsevich 95, Costello 04)
non-compact open topological string with various D-branes	Ch(Vect)	Calabi-Yau A-∞ category	“
non-compact open topological string with various D-branes and with closed string bulk sector	Ch(Vect)	Calabi-Yau A-∞ category with Hochschild cohomology	“
local closed topological string	2Mod(Vect${}_k$) over field $k$	separable symmetric Frobenius algebras	(SchommerPries 11)
non-compact local closed topological string	2Mod(Ch(Vect))	Calabi-Yau A-∞ algebra	(Lurie 09, section 4.2)
non-compact local closed topological string	2Mod$(\mathbf{S})$ for a symmetric monoidal (∞,1)-category $\mathbf{S}$	Calabi-Yau object in $\mathbf{S}$	(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 $\mu:A\otimes A \to A$ but $\delta :A \to A \parr A$. 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

$$A\otimes A \xrightarrow{1_A \otimes \delta} A \otimes (A \parr A) \xrightarrow{lin-distrib} (A \otimes A) \parr A \xrightarrow{\mu \parr 1_A} A\parr A$$

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

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

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 $\overset{n}{\overbrace{(A,A,\dots,A)}} \to \overset{m}{\overbrace{(A,A,\dots,A)}}$ 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 $(2,1)$ and $(0,1)$, while the comonoid structure is the morphisms of arity $(1,2)$ and $(1,0)$, and the Frobenius relations say that three ways to compose these to produce a morphism of arity $(2,2)$ are equal. (The morphism of arity $(0,0)$ is the composite $\epsilon\eta$; 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.

Related concepts

• Frobenius monad

• Calabi-Yau algebra

• hypergraph category

• Frobenius pseudomonoid

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)

• Andrew Baker, Frobenius algebras 2010 (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)

• Aaron Lauda, Hendryk Pfeiffer (2005), Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras, Topology Appl. 155, 623-666. (arXiv:0510664)

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, 2010 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)

• Craig Pastro and Ross Street, Weak Hopf monoids in braided monoidal categories, 2008, arxiv