Bialgebra
Encyclopedia
In 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.
Compatibility means that the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, that the multiplication and the unit of the algebra both be coalgebra morphisms: these statements are equivalent in that they are expressed by the same diagrams. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism.
As reflected in the symmetry of the diagrams, the definition of bialgebra is self-dual
, so if one can define a dual
of B (which is always possible if B is finite-dimensional), then it is automatically a bialgebra.
is a bialgebra over K iff it has the following properties:
is coassociative
if
The K-linear map
is a counit if
.
Coassociativy and counit are expressed by the commutativity
of the following two diagrams with B in place of C (they are the duals of the diagrams expressing associativity and unit of an algebra):
s of algebras" or, equivalently, "multiplication and unit are homomorphism
s of coalgebras".
These statements are meaningful once we explicit the natural structures of algebra and coalgebra in all the vector spaces involved besides
: 
. 
is a unital associative algebra in an obvious way and 
is a unital associative algebra with unit and multiplication
,
so that
or, omitting 
and writing multiplication as juxtaposition, 
;
similarly,
is a coalgebra in an obvious way and 
is a coalgebra with counit and comultiplication
.
Then, diagrams 1 and 3 say that
is a homomorphism of unital (associative) algebras 
and 
:
, or simply 
and
, or simply 
;
diagrams 2 and 4 say that
is a homomorphism of unital (associative) algebras 
and 
:
, or simply 
and
, or simply 
.
Equivalently, diagrams 1 and 2 say that
is a homomorphism of (counital coassociative) coalgebras 
and 
:
and
;
diagrams 3 and 4 say that
is a homomorphism of (counital coassociative) coalgebras 
and 
:
and
.
s. Similar structures with different compatibility between the product and coproduct, or different types of product and coproduct, include Lie bialgebra
s and Frobenius algebra
s. Additional examples are given in the article on coalgebra
s.
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...
, a bialgebra over a field
Field (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...
over K which is both a 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...
and a coalgebra
Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...
, such that these structures are compatible.
Compatibility means that the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, that the multiplication and the unit of the algebra both be coalgebra morphisms: these statements are equivalent in that they are expressed by the same diagrams. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism.
As reflected in the symmetry of the diagrams, the definition of bialgebra is self-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...
, so if one can define a dual
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of B (which is always possible if B is finite-dimensional), then it is automatically a bialgebra.
Formal definition
is a bialgebra over K iff it has the following properties:
- B is a vector space over K;
- there are K-linear maps (multiplication)
(equivalent to K-multilinear map 
) and (unit) 
, such that 
is a unital associative algebraAlgebra over a fieldIn mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
; - there are K-linear maps (comultiplication)
and (counit) 
, such that 
is a (counital coassociative) coalgebraCoalgebraIn mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...
; - compatibility conditions expressed by the following commutative diagramCommutative diagramIn 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:
- 1 multiplication
and comultiplication 
- where
is the linear map defined by 
for all x and y in B,
- where
- 2 multiplication
and counit 
- 3 comultiplication
and unit 
- 4 unit
and counit 
Coassociativity and counit
The K-linear maps is coassociativeCoalgebra
In mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...
if
The K-linear map
is a counit if.Coassociativy and counit are expressed by the commutativity
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...
of the following two diagrams with B in place of C (they are the duals of the diagrams expressing associativity and unit of an algebra):
Compatibility conditions
The four commutative diagrams can be read either as "comultiplication and counit are homomorphismHomomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s of algebras" or, equivalently, "multiplication and unit are homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s of coalgebras".
These statements are meaningful once we explicit the natural structures of algebra and coalgebra in all the vector spaces involved besides
: . is a unital associative algebra in an obvious way and is a unital associative algebra with unit and multiplication,so that
or, omitting and writing multiplication as juxtaposition, ;similarly,
is a coalgebra in an obvious way and is a coalgebra with counit and comultiplication.Then, diagrams 1 and 3 say that
is a homomorphism of unital (associative) algebras and :, or simply and, or simply ;diagrams 2 and 4 say that
is a homomorphism of unital (associative) algebras and :, or simply and, or simply .Equivalently, diagrams 1 and 2 say that
is a homomorphism of (counital coassociative) coalgebras and : and;diagrams 3 and 4 say that
is a homomorphism of (counital coassociative) coalgebras and : and.Examples
Examples of bialgebras include the Hopf algebraHopf 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. Similar structures with different compatibility between the product and coproduct, or different types of product and coproduct, include Lie bialgebra
Lie bialgebra
In 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 and Frobenius algebra
Frobenius algebra
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...
s. Additional examples are given in the article on coalgebra
Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...
s.