In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative 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.)
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: B is a vector space over K; there are K-linear maps (multiplication) ∇: B ⊗ B → B (equivalent to K-multilinear map ∇: B × B → B) and (unit) η: K → B, such that (B, ∇, η) is a unital associative algebra; 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; compatibility conditions expressed by the following commutative diagrams:
Multiplication ∇ and comultiplication Δ
where τ: B ⊗ B → B ⊗ B is the linear map defined by τ(x ⊗ y) = y ⊗ x for all x and y in B, 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 .
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 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 or, omitting ∇ and writing multiplication as juxtaposition, ;
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)
, or simply Δ(xy) = Δ(x) Δ(y), , or simply Δ(1B) = 1B ⊗ B;
diagrams 2 and 4 say that ε: B → K is a homomorphism of unital (associative) algebras (B, ∇, η) and (K, ∇0, η0):
, or simply ε(xy) = ε(x) ε(y) , or simply ε(1B) = 1K.
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):
, where .
Examples
Group bialgebra
An example of a bialgebra is the set of functions from a group G (or more generally, any monoid) to , which we may represent as a vector space consisting of linear combinations of standard basis vectors eg for each g ∈ G, which may represent a probability distribution 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 (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 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: