Coalgebra
Encyclopedia
In mathematics
, coalgebras or cogebras are structures that are dual
(in the sense of reversing arrows) to unital associative algebra
s. The axiom
s of unital associative algebras can be formulated in terms of commutative diagram
s. Turning all arrows around, one obtains the axioms of coalgebras.
Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions (see below).
Coalgebras occur naturally in a number of contexts (for example, universal enveloping algebra
s and group scheme
s).
There are also F-coalgebra
s, with important applications in computer science
.
K is a vector space
C over K together with K-linear maps
and 
such that
(Here
and 
refer to the tensor product
over K and id is the identity function
.)
Equivalently, the following two diagrams commute
:
In the first diagram we silently identify
with 
; the two are naturally isomorphic. Similarly, in the second diagram the naturally isomorphic spaces 
, 
and 
are identified.
The first diagram is the dual of the one expressing associativity
of algebra multiplication (called the coassociativity of the comultiplication); the second diagram is the dual of the one expressing the existence of a multiplicative identity
. Accordingly, the map Δ is called the comultiplication (or coproduct) of C and ε is the of C.
The key point is that in finite dimensions,
.
To distinguish these: in general, algebra and coalgebra are dual notions (meaning that their axioms are dual: reverse the arrows), while for finite dimensions, they are dual objects (meaning that a coalgebra is the dual object of an algebra and conversely).
If A is a finite-dimension
al unital associative K-algebra, then its K-dual A* consisting of all K-linear maps from A to K is a coalgebra. The multiplication of A can be viewed as a linear map
, which when dualized yields a linear map 
. In the finite-dimensional case, 
is naturally isomorphic to 
, so we have defined a comultiplication on A*. The counit of A* is given by evaluating linear function
als at 1.
In Sweedler's notation, this is abbreviated to
The fact that ε is a counit can then be expressed with the following formula
The coassociativity of Δ can be expressed as
In Sweedler's notation, both of these expressions are written as
Some authors omit the summation symbols as well; in this sumless Sweedler notation, we may write
and
Whenever a variable with lowered and parenthesized index is encountered in an expression of this kind, a summation symbol for that variable is implied.
is called co-commutative if 
, where 
is the K-linear map defined by 
for all c,d in C. In Sweedler's sumless notation, C is co-commutative if and only if
for all c in C. (It's important to understand that the implied summation is significant here: we are not requiring that all the summands are pairwise equal, only that the sums are equal, a much weaker requirement.)
If
and 
are two coalgebras over the same field K, then a coalgebra morphism from 
to 
is a K-linear map 
such that 
and 
.
In Sweedler's sumless notation, the first of these properties may be written as:
The composition of two coalgebra morphisms is again a coalgebra morphism, and the coalgebras over K together with this notion of morphism form a category
.
A linear subspace
I in C is called a coideal if I⊆ker(ε) and Δ(I)⊆I⊗C + C⊗I. In that case, the quotient space
C/I becomes a coalgebra in a natural fashion.
A subspace D of C is called a subcoalgebra if Δ(D)⊆D⊗D; in that case, D is itself a coalgebra, with the restriction of ε to D as counit.
The kernel
of every coalgebra morphism f : C1 → C2 is a coideal in C1, and the image
is a subcoalgebra of C2. The common isomorphism theorem
s are valid for coalgebras, so for instance C1/ker(f) is isomorphic to im(f).
If A is a finite-dimensional unital associative K-algebra, then
A* is a finite-dimensional coalgebra, and indeed every finite-dimensional coalgebra arises in this fashion from some finite-dimensional algebra (namely from the coalgebra's K-dual). Under this correspondence, the commutative finite-dimensional algebras correspond to the cocommutative finite-dimensional coalgebras. So in the finite-dimensional case, the theories of algebras and of coalgebras are dual; studying one is equivalent to studying the other. However, things diverge in the infinite-dimensional case: while the K-dual of every coalgebra is an algebra, the K-dual of an infinite-dimensional algebra need not be a coalgebra.
Every coalgebra is the sum of its finite-dimensional coalgebras, something that's not true for algebras. In a certain sense then, coalgebras are generalizations of (duals of) finite-dimensional unital associative algebras.
Corresponding to the concept of representation
for algebras is a corepresentation or comodule
.
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...
, coalgebras or cogebras are structures that are dual
Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
(in the sense of reversing arrows) to 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...
s. The axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s of unital associative algebras can be formulated in terms of commutative diagram
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
s. Turning all arrows around, one obtains the axioms of coalgebras.
Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions (see below).
Coalgebras occur naturally in a number of contexts (for example, universal enveloping algebra
Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
s and group scheme
Group scheme
In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...
s).
There are also F-coalgebra
F-coalgebra
In mathematics, specifically in category theory, an F-coalgebra is a structure defined according to a functor F. For both algebra and coalgebra, a functor is a convenient and general way of organizing a signature...
s, with important applications in computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
.
Formal definition
Formally, a coalgebra 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 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...
C over K together with K-linear maps
and such that-
-
.
(Here
and refer to the tensor productTensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps to be carried out in terms of linear maps...
over K and id is the identity function
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
.)
Equivalently, the following two diagrams commute
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
:
In the first diagram we silently identify
with ; the two are naturally isomorphic. Similarly, in the second diagram the naturally isomorphic spaces , and are identified.The first diagram is the dual of the one expressing associativity
Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
of algebra multiplication (called the coassociativity of the comultiplication); the second diagram is the dual of the one expressing the existence of a multiplicative identity
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
. Accordingly, the map Δ is called the comultiplication (or coproduct) of C and ε is the of C.
Examples
- Take an arbitrary set S and form the K-vector space with basisBasis (linear algebra)In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
S. The elements of this vector space are those functions from S to K that map all but finitely many elements of S to zero; we identify the element s of S with the function that maps s to 1 and all other elements of S to 0. We will denote this space by C. We define
By linearity, both Δ and ε can then uniquely be extended to all of C. The vector space C becomes a coalgebra with comultiplication Δ and counit ε (checking this is a good way to get used to the axioms).
As a second example, consider the polynomial ringPolynomial ringIn mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
K[X] in one indeterminateIndeterminateIndeterminate has a variety of meanings in mathematics:* Indeterminate * Indeterminate system* Indeterminate equation* Statically indeterminate* Indeterminate formIt is also a term in botany and gardening:*Indeterminate growth...
X. This becomes a coalgebra (the divided power coalgebra) if we define
and
for all
Again, because of linearity, this suffices to define Δ and ε uniquely on all of K[X]. Now K[X] is both a unital associative algebra and a coalgebra, and the two structures are compatible. Objects like this are called 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....
s, and in fact most of the important coalgebras considered in practice are bialgebras. Examples include 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...
s and Lie bialgebraLie bialgebraIn mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: its a set with a Lie algebra and a Lie coalgebra structure which are compatible....
s.
In some cases the singular homologySingular homologyIn algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
of a topological space forms a coalgebra.
If C is the K-vector space with basisBasis (linear algebra)In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
{s, c}, consider
is given by
and
is given by
In this situation,
is a coalgebra known as trigonometric coalgebra.
Finite dimensions
In finite dimensions, the duality between algebras and coalgebras is closer: the dual of a finite-dimensional (unital associative) algebra is a coalgebra, while the dual of a finite-dimensional coalgebra is a (unital associative) algebra. In general, the dual of an algebra may not be a coalgebra.The key point is that in finite dimensions,
.To distinguish these: in general, algebra and coalgebra are dual notions (meaning that their axioms are dual: reverse the arrows), while for finite dimensions, they are dual objects (meaning that a coalgebra is the dual object of an algebra and conversely).
If A is a finite-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al unital associative K-algebra, then its K-dual A* consisting of all K-linear maps from A to K is a coalgebra. The multiplication of A can be viewed as a linear map
, which when dualized yields a linear map . In the finite-dimensional case, is naturally isomorphic to , so we have defined a comultiplication on A*. The counit of A* is given by evaluating linear functionLinear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
als at 1.
Sweedler notation
When working with coalgebras, a certain notation for the comultiplication simplifies the formulas considerably and has become quite popular. Given an element c of the coalgebra (C,Δ,ε), we know that there exist elements c(1)(i) and c(2)(i) in C such thatIn Sweedler's notation, this is abbreviated to
The fact that ε is a counit can then be expressed with the following formula
The coassociativity of Δ can be expressed as
In Sweedler's notation, both of these expressions are written as
Some authors omit the summation symbols as well; in this sumless Sweedler notation, we may write
and
Whenever a variable with lowered and parenthesized index is encountered in an expression of this kind, a summation symbol for that variable is implied.
Further concepts and facts
A coalgebra is called co-commutative if , where is the K-linear map defined by for all c,d in C. In Sweedler's sumless notation, C is co-commutative if and only iffor all c in C. (It's important to understand that the implied summation is significant here: we are not requiring that all the summands are pairwise equal, only that the sums are equal, a much weaker requirement.)
If
and are two coalgebras over the same field K, then a coalgebra morphism from to is a K-linear map such that and .In Sweedler's sumless notation, the first of these properties may be written as:
The composition of two coalgebra morphisms is again a coalgebra morphism, and the coalgebras over K together with this notion of morphism form a category
Category 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...
.
A linear subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
I in C is called a coideal if I⊆ker(ε) and Δ(I)⊆I⊗C + C⊗I. In that case, the quotient space
Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....
C/I becomes a coalgebra in a natural fashion.
A subspace D of C is called a subcoalgebra if Δ(D)⊆D⊗D; in that case, D is itself a coalgebra, with the restriction of ε to D as counit.
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 every coalgebra morphism f : C1 → C2 is a coideal in C1, and the image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
is a subcoalgebra of C2. The common isomorphism theorem
Isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures...
s are valid for coalgebras, so for instance C1/ker(f) is isomorphic to im(f).
If A is a finite-dimensional unital associative K-algebra, then
A* is a finite-dimensional coalgebra, and indeed every finite-dimensional coalgebra arises in this fashion from some finite-dimensional algebra (namely from the coalgebra's K-dual). Under this correspondence, the commutative finite-dimensional algebras correspond to the cocommutative finite-dimensional coalgebras. So in the finite-dimensional case, the theories of algebras and of coalgebras are dual; studying one is equivalent to studying the other. However, things diverge in the infinite-dimensional case: while the K-dual of every coalgebra is an algebra, the K-dual of an infinite-dimensional algebra need not be a coalgebra.
Every coalgebra is the sum of its finite-dimensional coalgebras, something that's not true for algebras. In a certain sense then, coalgebras are generalizations of (duals of) finite-dimensional unital associative algebras.
Corresponding to the concept of representation
Representation of an algebra
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a ring...
for algebras is a corepresentation or comodule
Comodule
In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.-Formal definition:...
.
External links
- William Chin: A brief introduction to coalgebra representation theory