What compatibility means
A bialgebra is a vector space B that is both an algebra (m, u) and a coalgebra (Δ, ε), with the two structures compatible. There are two equivalent ways to state compatibility, and it is worth seeing that they coincide. Version A: Δ: B → B⊗B and ε: B → k are algebra homomorphisms (where B⊗B carries the tensor-product algebra structure). Version B: m: B⊗B → B and u: k → B are coalgebra homomorphisms. The symmetry is the point — multiplication and comultiplication are on equal footing.
Unwinding Version A on elements: Δ(ab) = Δ(a)Δ(b), where the product on the right is taken in B⊗B, namely (a₍₁₎⊗a₍₂₎)(b₍₁₎⊗b₍₂₎) = a₍₁₎b₍₁₎ ⊗ a₍₂₎b₍₂₎. Also Δ(1) = 1⊗1, ε(ab) = ε(a)ε(b), and ε(1) = 1. That is the whole definition. Everything later — the antipode, quantum groups — lives inside a bialgebra.
Why this is the right axiom
Representation theory explains the choice. If B = k[G] and V, W are modules — i.e. representations — their tensor product V⊗W is again a representation via g·(v⊗w) = gv⊗gw. Look closely: that formula is exactly Δ(g) = g⊗g acting. The comultiplication is what tells you how a single element acts on a tensor product of representations. Compatibility (Δ a homomorphism) is precisely what makes V⊗W a module over B and not merely over B⊗B. The counit makes the trivial 1-dimensional representation. So a bialgebra is exactly the structure on B that makes its category of modules a tensor category.
The convolution product
A bialgebra C (we only need the coalgebra side here) and any algebra A together turn the space of linear maps Hom(C, A) into an algebra. Given f, g: C → A, define their convolution product by (f * g)(c) = m_A(f⊗g)Δ(c) = Σ f(c₍₁₎) g(c₍₂₎). Coassociativity makes * associative; the unit is u_A ∘ ε, i.e. the map c ↦ ε(c)1_A. This single construction is the engine of the next guide: the antipode will be defined as the convolution inverse of the identity map.
Convolution on End(C) = Hom(C, C), with C a bialgebra.
Product: (f * g)(c) = f(c_(1)) . g(c_(2)) (sum over Sweedler indices)
Unit: e(c) = eps(c) 1_C
Associativity, checked by coassociativity:
((f * g) * h)(c)
= (f * g)(c_(1)) . h(c_(2))
= f(c_(1)) g(c_(2)) h(c_(3)) [coassoc: regroup the three legs]
= f(c_(1)) . (g * h)(c_(2))
= (f * (g * h))(c).
Unit law:
(e * f)(c) = e(c_(1)) f(c_(2)) = eps(c_(1)) 1 . f(c_(2))
= f( eps(c_(1)) c_(2) ) = f(c). [counit axiom]
So (End(C), *, e) is a genuine associative unital algebra.
Next guide: an antipode S is the *-inverse of id, i.e. S * id = id * S = e.