Bialgebra

Coassociativity and counit

The K-linear map Δ: B → B ⊗ B is coassociative if ${\displaystyle ({\mathrm{i}\mathrm{d}}_B\otimes \mathrm{\Delta })\circ \mathrm{\Delta }=(\mathrm{\Delta }\otimes {\mathrm{i}\mathrm{d}}_B)\circ \mathrm{\Delta }}$.

The K-linear map ε: B → K is a counit if ${\displaystyle ({\mathrm{i}\mathrm{d}}_B\otimes ϵ)\circ \mathrm{\Delta }={\mathrm{i}\mathrm{d}}_B=(ϵ\otimes {\mathrm{i}\mathrm{d}}_B)\circ \mathrm{\Delta }}$.

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

${\displaystyle {\eta }_2:=(\eta \otimes \eta ):K\otimes K\equiv K\to (B\otimes B)}$

${\displaystyle {\mathrm{\nabla }}_2:=(\mathrm{\nabla }\otimes \mathrm{\nabla })\circ (id\otimes \tau \otimes id):(B\otimes B)\otimes (B\otimes B)\to (B\otimes B)}$,

so that ${\displaystyle {\mathrm{\nabla }}_2((x_1\otimes x_2)\otimes (y_1\otimes y_2))=\mathrm{\nabla }(x_1\otimes y_1)\otimes \mathrm{\nabla }(x_2\otimes y_2)}$ or, omitting ∇ and writing multiplication as juxtaposition, ${\displaystyle (x_1\otimes x_2)(y_1\otimes y_2)=x_1{y}_1\otimes x_2{y}_2}$;

similarly, (K, Δ₀, ε₀) is a coalgebra in an obvious way and B ⊗ B is a coalgebra with counit and comultiplication

${\displaystyle {ϵ}_2:=(ϵ\otimes ϵ):(B\otimes B)\to K\otimes K\equiv K}$

${\displaystyle {\mathrm{\Delta }}_2:=(id\otimes \tau \otimes id)\circ (\mathrm{\Delta }\otimes \mathrm{\Delta }):(B\otimes B)\to (B\otimes B)\otimes (B\otimes B)}$.

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)

${\displaystyle \mathrm{\Delta }\circ \mathrm{\nabla }={\mathrm{\nabla }}_2\circ (\mathrm{\Delta }\otimes \mathrm{\Delta }):(B\otimes B)\to (B\otimes B)}$, or simply Δ(xy) = Δ(x) Δ(y),

${\displaystyle \mathrm{\Delta }\circ \eta ={\eta }_2:K\to (B\otimes B)}$, 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, η₀):

${\displaystyle ϵ\circ \mathrm{\nabla }={\mathrm{\nabla }}_0\circ (ϵ\otimes ϵ):(B\otimes B)\to K}$, or simply ε(xy) = ε(x) ε(y)

${\displaystyle ϵ\circ \eta ={\eta }_0:K\to K}$, 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, Δ₂, ε₂) and (B, Δ, ε):

${\displaystyle \mathrm{\nabla }\otimes \mathrm{\nabla }\circ {\mathrm{\Delta }}_2=\mathrm{\Delta }\circ \mathrm{\nabla }:(B\otimes B)\to (B\otimes B),}$

${\displaystyle {\mathrm{\nabla }}_0\circ {ϵ}_2=ϵ\circ \mathrm{\nabla }:(B\otimes B)\to K}$;

diagrams 3 and 4 say that η: K → B is a homomorphism of (counital coassociative) coalgebras (K, Δ₀, ε₀) and (B, Δ, ε):

${\displaystyle {\eta }_2\circ {\mathrm{\Delta }}_0=\mathrm{\Delta }\circ \eta :K\to (B\otimes B),}$

${\displaystyle {\eta }_0\circ {ϵ}_0=ϵ\circ \eta :K\to K}$,

where

${\displaystyle {ϵ}_0=i{d}_K={\eta }_0}$.

Examples

Group bialgebra

An example of a bialgebra is the set of functions from a finite group G (or more generally, any finite monoid) to ${\displaystyle \mathbb{R}}$, which we may represent as a vector space ${\displaystyle {\mathbb{R}}^G}$ consisting of linear combinations of standard basis vectors e_g 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

${\displaystyle \mathrm{\Delta }({\mathbf{e}}_g)={\mathbf{e}}_g\otimes {\mathbf{e}}_g,}$

which represents making a copy of a random variable (which we extend to all ${\displaystyle {\mathbb{R}}^G}$ by linearity), and

${\displaystyle \epsilon ({\mathbf{e}}_g)=1,}$

(again extended linearly to all of ${\displaystyle {\mathbb{R}}^G}$) 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:

1. η is an operator preparing a normalized probability distribution which is independent of all other random variables;
2. The product ∇ maps a probability distribution on two variables to a probability distribution on one variable;
3. Copying a random variable in the distribution given by η is equivalent to having two independent random variables in the distribution η;
4. 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

${\displaystyle \mathrm{\nabla }({\mathbf{e}}_g\otimes {\mathbf{e}}_h)={\mathbf{e}}_{gh},}$

again extended to all ${\displaystyle {\mathbb{R}}^G\otimes {\mathbb{R}}^G}$ by linearity; this produces a normalized probability distribution from a distribution on two random variables, and has as a unit the delta-distribution ${\displaystyle \eta ={\mathbf{e}}_i,}$ where i ∈ G denotes the identity element of the group G.

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.