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In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit 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 diagrams.)
Contents
- Formal definition
- Coassociativity and counit
- Compatibility conditions
- Group bialgebra
- Other examples
- References
Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a linear map 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, 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:
- Multiplication ∇ and comultiplication Δ
- Multiplication ∇ and counit ε
- Comultiplication Δ and unit η
- 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
Coassociativy 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 homomorphisms of algebras" or, equivalently, "multiplication and unit are homomorphisms 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, η0) is a unital associative algebra in an obvious way and (B ⊗ B, ∇2, η2) is a unital associative algebra with unit and multiplication
so that
similarly, (K, Δ0, ε0) 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, η2)
diagrams 2 and 4 say that ε: B → K is a homomorphism of unital (associative) algebras (B, ∇, η) and (K, ∇0, η0):
Equivalently, diagrams 1 and 2 say that ∇: B ⊗ B → B is a homomorphism of (counital coassociative) coalgebras (B ⊗ B, Δ2, ε2) and (B, Δ, ε):
diagrams 3 and 4 say that η: K → B is a homomorphism of (counital coassociative) coalgebras (K, Δ0, ε0) and (B, Δ, ε):
Group bialgebra
An example of a bialgebra is the set of functions from a group G (or more generally, any monoid) to
which represents making a copy of a random variable (which we extend to all
(again extended linearly to all of
- 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 operator
again extended to all
Other examples
Other examples of bialgebras include the tensor algebra, 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 bialgebras and Frobenius algebras. Additional examples are given in the article on coalgebras.