In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a bialgebra over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K'' is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over ''K'' which is both a
unital associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
and a
counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the
comultiplication In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
and the
counit In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagram ...
are both unital algebra
homomorphisms, or equivalently, the multiplication and the unit of the algebra both are
coalgebra morphisms. (These statements are equivalent since they are expressed by the same
commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
s.)
Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
that is both an algebra and a coalgebra homomorphism.
As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is
self-dual
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...
, so if one can define a
dual of ''B'' (which is always possible if ''B'' is finite-dimensional), then it is automatically a bialgebra.
Formal definition
(''B'', ∇, η, Δ, ε) is a bialgebra over ''K'' if it has the following properties:
* ''B'' is a vector space over ''K'';
* there are ''K''-
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s (multiplication) ∇: ''B'' ⊗ ''B'' → ''B'' (equivalent to ''K''-
multilinear map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
:f\colon V_1 \times \cdots \times V_n \to W\text
where V_1,\ldots,V_n and W are ...
∇: ''B'' × ''B'' → ''B'') and (unit) η: ''K'' → ''B'', such that (''B'', ∇, η) is a unital associative
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
;
* there are ''K''-linear maps (comultiplication) Δ: ''B'' → ''B'' ⊗ ''B'' and (counit) ε: ''B'' → ''K'', such that (''B'', Δ, ε) is a (counital coassociative)
coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) 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 diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
s:
# Multiplication ∇ and comultiplication Δ
#::
![Bialgebra2](https://upload.wikimedia.org/wikipedia/commons/d/da/Bialgebra2.svg)
#: where τ: ''B'' ⊗ ''B'' → ''B'' ⊗ ''B'' is the
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
defined by τ(''x'' ⊗ ''y'') = ''y'' ⊗ ''x'' for all ''x'' and ''y'' in ''B'',
# Multiplication ∇ and counit ε
#::
![Bialgebra3](https://upload.wikimedia.org/wikipedia/commons/b/ba/Bialgebra3.svg)
# Comultiplication Δ and unit η
#::
![Bialgebra4a](https://upload.wikimedia.org/wikipedia/commons/7/7b/Bialgebra4a.svg)
# Unit η and counit ε
#::
Coassociativity and counit
The
''K''-linear map Δ: ''B'' → ''B'' ⊗ ''B'' is
coassociative if
.
The ''K''-linear map ε: ''B'' → ''K'' is a counit if
.
Coassociativity and counit are expressed by the commutativity of the following two diagrams (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
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s of algebras" or, equivalently, "multiplication and unit are
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s of coalgebras".
These statements are meaningful once we explain the natural structures of algebra and coalgebra in all the vector spaces involved besides ''B'': (''K'', ∇
0, η
0) is a unital associative algebra in an obvious way and (''B'' ⊗ ''B'', ∇
2, η
2) is a unital associative algebra with unit and multiplication
:
:
,
so that
or, omitting ∇ and writing
multiplication as juxtaposition,
;
similarly, (''K'', Δ
0, ε
0) is a coalgebra in an obvious way and ''B'' ⊗ ''B'' is a coalgebra with counit and comultiplication
:
:
.
Then, diagrams 1 and 3 say that Δ: ''B'' → ''B'' ⊗ ''B'' is a homomorphism of unital (associative) algebras (''B'', ∇, η) and (''B'' ⊗ ''B'', ∇
2, η
2)
:
, or simply Δ(''xy'') = Δ(''x'') Δ(''y''),
:
, or simply Δ(1
''B'') = 1
''B'' ⊗ ''B'';
diagrams 2 and 4 say that ε: ''B'' → ''K'' is a homomorphism of unital (associative) algebras (''B'', ∇, η) and (''K'', ∇
0, η
0):
:
, or simply ε(''xy'') = ε(''x'') ε(''y'')
:
, or simply ε(1
''B'') = 1
''K''.
Equivalently, diagrams 1 and 2 say that ∇: ''B'' ⊗ ''B'' → ''B'' is a homomorphism of (counital coassociative) coalgebras (''B'' ⊗ ''B'', Δ
2, ε
2) and (''B'', Δ, ε):
:
:
;
diagrams 3 and 4 say that η: ''K'' → ''B'' is a homomorphism of (counital coassociative) coalgebras (''K'', Δ
0, ε
0) and (''B'', Δ, ε):
:
:
,
where
:
.
Examples
Group bialgebra
An example of a bialgebra is the set of functions from a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''G'' (or more generally, any
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
) to
, which we may represent as a vector space
consisting of linear combinations of standard basis vectors e
''g'' for each ''g'' ∈ ''G'', which may represent a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
over ''G'' in the case of vectors whose coefficients are all non-negative and sum to 1. An example of suitable comultiplication operators and counits which yield a counital coalgebra are
:
which represents making a copy of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
(which we extend to all
by linearity), and
:
(again extended linearly to all of
) which represents "tracing out" a random variable — ''i.e.,'' forgetting the value of a random variable (represented by a single tensor factor) to obtain a
marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
on the remaining variables (the remaining tensor factors).
Given the interpretation of (Δ,ε) in terms of probability distributions as above, the bialgebra consistency conditions amount to constraints on (∇,η) as follows:
# η is an operator preparing a normalized probability distribution which is independent of all other random variables;
# The product ∇ maps a probability distribution on two variables to a probability distribution on one variable;
# Copying a random variable in the distribution given by η is equivalent to having two independent random variables in the distribution η;
# Taking the product of two random variables, and preparing a copy of the resulting random variable, has the same distribution as preparing copies of each random variable independently of one another, and multiplying them together in pairs.
A pair (∇,η) which satisfy these constraints are the
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
operator
:
again extended to all
by linearity; this produces a normalized probability distribution from a distribution on two random variables, and has as a unit the delta-distribution
where ''i'' ∈ ''G'' denotes the identity element of the group ''G''.
Other examples
Other examples of bialgebras include the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
, which can be made into a bialgebra by adding the appropriate comultiplication and counit; these are worked out in detail in that article.
Bialgebras can often be extended to
Hopf algebras, if an appropriate antipode can be found. Thus, all Hopf algebras are examples of bialgebras.
Similar structures with different compatibility between the product and comultiplication, or different types of multiplication and comultiplication, include
Lie bialgebra In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible.
It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacob ...
s and
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 dualit ...
s. Additional examples are given in the article on
coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
s.
See also
*
Quasi-bialgebra
In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by having coassociativity replaced by an invertible ...
Notes
References
* .
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Coalgebras
Monoidal categories