Frobenius algebra
Encyclopedia
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 and especially in his . Dieudonné
used this to characterize Frobenius algebras in his where he called this property of Frobenius algebras a perfect duality. Frobenius algebras were generalized to quasi-Frobenius ring
s, those Noetherian ring
s whose right regular representation
is injective
. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory
.
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: 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 λ(a·b) = λ(b·a).
There is also a different, mostly unrelated notion of the symmetric algebra
of a vector space
.
, the notion of Frobenius object is an abstract definition of a Frobenius algebra in a category. A Frobenius object
in a monoidal category
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
is strict).
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.
, and have contributed to the study of number theory
, algebraic geometry
, and combinatorics
. They have been used to study Hopf algebra
s, coding theory
, and cohomology ring
s of compact
oriented
manifold
s.
. 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 cobordism
s between 1-dimensional manifolds) to VectK (the category of vector space
s over K).
The correspondence between TQFTs and Frobenius algebras is given as :
satisfying the bimodule condition 
for all 
and 
, together with elements in A denoted 
and 
such that 
and 
for all 
.
The map E is sometimes referred to as a Frobenius homomorphism and the elements
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
is a Frobenius extension A | K with 
. 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 Nakayama 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
be left coset representatives, so that G is a disjoint union of the cosets 
. 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 
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 
for 
. (The orthonormality condition 
follows.) The dual base is given by 
, since 
. The other dual base equation may be derived from the observation that G is also a disjoint union of the right cosets 
.
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 
, 
in A satisfying 
for each g in G: whence also 
. Then A is a Frobenius extension of B with 
defined by 
which satisfies the equations for all x in A, 
(Furthermore an example of a separable algebra
extension since
is a separability element satisfying ea = ae for all a in A as well as 
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
(if M is a left module) and co-induced module 
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 
is a Frobenius extension, then so is 
where the mapping 
is given by 
and 
for each 
. Endomorphism ring theorems and converses were investigated later by Mueller, Morita, Onodera and others.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the fields of representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
and module theory, a Frobenius algebra is a finite dimensional
Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...
unital associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
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
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...
and Nesbitt
Cecil J. Nesbitt
Cecil James Nesbitt, Ph.D., F.S.A., M.A.A.A. was a mathematician who was a Ph.D. student of Richard Brauer and wrote many influential papers in the early history of modular representation theory. He taught actuarial mathematics at the University of Michigan from 1938 to 1980. Nesbitt was born in...
and were named after Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
. Nakayama
Tadashi Nakayama (mathematician)
was a mathematician who made important contributions to representation theory. He received his degrees from Tokyo University and Osaka University and held permanent positions at Osaka University and Nagoya University. He had visiting positions at Princeton University, Illinois University, and...
discovered the beginnings of a rich duality theory in his and especially in his . Dieudonné
Jean Dieudonné
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
used this to characterize Frobenius algebras in his where he called this property of Frobenius algebras a perfect duality. Frobenius algebras were generalized to quasi-Frobenius ring
Quasi-Frobenius ring
In ring theory, the class of Frobenius rings and their generalizations are the extension of work done on Frobenius algebras. Perhaps the most important generalization is that of quasi-Frobenius rings , which are in turn generalized by right pseudo-Frobenius rings and right finitely...
s, those Noetherian ring
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
s whose right regular representation
Regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation....
is injective
Injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory
Topological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
.
Definition
A finite dimensional, unital, associative algebra A defined over a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
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: 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
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
λ:A→k such that the kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of λ contains no nonzero left ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
of A.
A Frobenius algebra is called symmetric if σ is symmetric
Symmetric bilinear form
A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
, or equivalently λ satisfies λ(a·b) = λ(b·a).
There is also a different, mostly unrelated notion of the symmetric algebra
Symmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
of a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
.
Examples
- Any matrix algebraMatrix ringIn abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings...
defined over a field k is a Frobenius algebra with Frobenius form σ(a,b)=tr(a·b) where tr denotes the traceTrace (linear algebra)In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
. - 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 ringGroup ringIn algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
of a finite groupFinite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
over a field is a Frobenius algebra, with Frobenius form σ(a,b) the coefficient of the identity element in a·b. This is a special case of example 2. - For a field k, the four-dimensional k-algebra k[x,y]/ (x2, y2) is a Frobenius algebra in the sense of the second example.
- For a field k not of characteristic 2, the three-dimensional k-algebra k[x,y]/ (x, y)2 is not a Frobenius algebra in the sense of the second example.
Properties
- The direct productProduct of ringsIn mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product Πi in I Ri can be turned into a ring by defining the operations coordinatewise, i.e...
and tensor productTensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of Frobenius algebras are Frobenius algebras. - A finite-dimensional commutativeCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
localLocal ringIn abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
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-dimensionalKrull dimensionIn commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
local Gorenstein ringGorenstein ringIn commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition....
s containing their residue fieldResidue fieldIn mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field...
and finite dimensional over it. - The right regular representationRegular representationIn mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation....
of a Frobenius algebra is always injectiveInjective moduleIn mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
. - 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 scalarsRestriction of scalarsIn abstract algebra, restriction of scalars is a procedure of creating a module over a ring R from a module over another ring S, given a homomorphism f : R \to S between them...
, 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 moduleSimple moduleIn mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
s M have the same dimension as their A-duals, HomA(M,A). Amongst these algebras, the A-duals of simple modules are always simple.
Category-theoretical definition
In category theoryCategory theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, the notion of Frobenius object is an abstract definition of a Frobenius algebra in a category. A Frobenius object
in a monoidal categoryMonoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
consists of an object A of C together with four morphismssuch that
-
is a monoid object in 
,
-
is a comonoid object in 
,
- the diagrams
and
commute (for simplicity the diagrams are given here in the case where the monoidal category
is strict).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.
Applications
Frobenius algebras originally were studied as part of an investigation into the representation theory of finite groupsRepresentation theory of finite groups
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction...
, and have contributed to the study of number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, and combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
. They have been used to study Hopf algebra
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
s, coding theory
Coding theory
Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...
, and cohomology ring
Cohomology ring
In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is...
s of compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
oriented
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s.
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 theoryTopological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
. 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
Equivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics...
to the category of symmetric strong monoidal functors from 2-Cob (the category of 2-dimensional cobordism
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
s between 1-dimensional manifolds) to VectK (the category of vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s over K).
The correspondence between TQFTs and Frobenius algebras is given as :
- 1-dimensional manifolds are disjoint unions of circles: a TQFT associates a vector space with a circle, and the tensor product of vector spaces with a disjoint union of circles,
- a TQFT associates (functorially) to each cobordism between manifolds a map between vector spaces,
- the map associated with a pair of pantsPair of pantsIn mathematics, a pair of pants is a simple two-dimensional surface resembling a pair of pants: topologically, it is a sphere with three holes in it...
(a cobordism between 1 circle and 2 circles) gives a product map
or a coproduct map 
depending on how the boundary components are grouped – which is commutative or cocommutative, and - the map associated with a disk gives a counit (trace) or unit (scalars), depending on grouping of boundary.
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, which is a Frobenius extension if there is a linear mapping satisfying the bimodule condition for all and , together with elements in A denoted and such that and for all .The map E is sometimes referred to as a Frobenius homomorphism and the elements
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
is a Frobenius extension A | K with . 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 Nakayama automorphism of the overalgebra.The details of the group ring
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
example are the following application of elementary notions in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
. Let G be a group and H a subgroup of finite index n in G; let
be left coset representatives, so that G is a disjoint union of the cosets . 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 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 for . (The orthonormality condition follows.) The dual base is given by , since . The other dual base equation may be derived from the observation that G is also a disjoint union of the right cosets .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 , in A satisfying for each g in G: whence also . Then A is a Frobenius extension of B with defined by which satisfies the equations for all x in A, (Furthermore an example of a separable algebraSeparable algebra
A separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.- Definition and First Properties :Let K be a field...
extension since
is a separability element satisfying ea = ae for all a in A as well as 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
(if M is a left module) and co-induced module 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 is a Frobenius extension, then so is where the mapping is given by and for each . Endomorphism ring theorems and converses were investigated later by Mueller, Morita, Onodera and others.See also
- BialgebraBialgebraIn mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a coalgebra, such that these structures are compatible....
- Frobenius norm
- Frobenius inner product
- Hopf algebraHopf algebraIn mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
- Quasi-Frobenius Lie algebraQuasi-Frobenius Lie algebraIn mathematics, a quasi-Frobenius Lie algebraover a field k is a Lie algebraequipped with a nondegenerate skew-symmetric bilinear form...
External links
- Ross Street, Frobenius algebras and monoidal categories