Bialgebra

(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:

1. Multiplication ∇ and comultiplication Δ[1]

   

   where τ: B ⊗ B → B ⊗ B is the linear map defined by τ(x ⊗ y) = y ⊗ x for all x and y in B,

2. Multiplication ∇ and counit ε

   

3. Comultiplication Δ and unit η[2]

   

4. Unit η and counit ε

   

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

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

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):

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

${\displaystyle \eta _{2}:=(\eta \otimes \eta ):K\otimes K\equiv K\to (B\otimes B)}$

${\displaystyle \nabla _{2}:=(\nabla \otimes \nabla )\circ (id\otimes \tau \otimes id):(B\otimes B)\otimes (B\otimes B)\to (B\otimes B)}$,

so that ${\displaystyle \nabla _{2}((x_{1}\otimes x_{2})\otimes (y_{1}\otimes y_{2}))=\nabla (x_{1}\otimes y_{1})\otimes \nabla (x_{2}\otimes y_{2})}$ 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

${\displaystyle \epsilon _{2}:=(\epsilon \otimes \epsilon ):(B\otimes B)\to K\otimes K\equiv K}$

${\displaystyle \Delta _{2}:=(id\otimes \tau \otimes id)\circ (\Delta \otimes \Delta ):(B\otimes B)\to (B\otimes B)\otimes (B\otimes B)}$.

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)

${\displaystyle \Delta \circ \nabla =\nabla _{2}\circ (\Delta \otimes \Delta ):(B\otimes B)\to (B\otimes B)}$, or simply Δ(xy) = Δ(x) Δ(y),

${\displaystyle \Delta \circ \eta =\eta _{2}:K\to (B\otimes B)}$, 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):

${\displaystyle \epsilon \circ \nabla =\nabla _{0}\circ (\epsilon \otimes \epsilon ):(B\otimes B)\to K}$, or simply ε(xy) = ε(x) ε(y)

${\displaystyle \epsilon \circ \eta =\eta _{0}:K\to K}$, 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):

${\displaystyle \nabla \otimes \nabla \circ \Delta _{2}=\Delta \circ \nabla$ :(B\otimes B)\to (B\otimes B),}

${\displaystyle \nabla _{0}\circ \epsilon _{2}=\epsilon \circ \nabla$ :(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):

${\displaystyle \eta _{2}\circ \Delta _{0}=\Delta \circ \eta :K\to (B\otimes B),}$

${\displaystyle \eta _{0}\circ \epsilon _{0}=\epsilon \circ \eta :K\to K}$,

where

${\displaystyle \epsilon _{0}=\epsilon \circ \eta }$.

• Quasi-bialgebra

• Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Șerban (2001), Hopf Algebras: An introduction, Pure and Applied Mathematics, 235 (1st ed.), Marcel Dekker, ISBN 0-8247-0481-9.