Bialgebra - Knowlpedia

Definition

Let $k$ be a commutative ring. A $k$ -bialgebra is a unital associative $k$ -algebra $(H,m,u)$ and a $k$ -coalgebra $(H,\Delta,\varepsilon)$ on the same module such that $\Delta:H\to H\otimes_k H$ and $\varepsilon:H\to k$ are unital algebra homomorphisms. The tensor-product algebra structure used here is

(a\otimes b)(c\otimes d)=ac\otimes bd.

Equivalently, $m$ and $u$ are coalgebra homomorphisms.

Compatibility in formulas

The algebra-homomorphism formulation means

\Delta(ab)=\Delta(a)\Delta(b),\qquad \Delta(1_H)=1_H\otimes1_H,

and

\varepsilon(ab)=\varepsilon(a)\varepsilon(b),\qquad \varepsilon(1_H)=1_k.

Together with coassociativity and the counit identities, these conditions say that $H$ is simultaneously a monoid and a comonoid, with each structure compatible with the other, in the symmetric monoidal category of $k$ -modules.

Morphisms

A bialgebra homomorphism is a unital algebra homomorphism that is also a coalgebra homomorphism. Bialgebras may be commutative as algebras, cocommutative as coalgebras, both, or neither; these are independent conditions.

Examples

For a group $G$ , the group algebra $k[G]$ has $\Delta(g)=g\otimes g$ and $\varepsilon(g)=1$ , extended linearly and multiplicatively. It is cocommutative, but it is commutative exactly when $G$ is abelian.
The polynomial algebra $k[x]$ becomes a commutative and cocommutative bialgebra with $\Delta(x)=x\otimes1+1\otimes x$ and $\varepsilon(x)=0$ .

Why an antipode is extra

A bialgebra encodes multiplication and comultiplication but need not have an operation representing inversion. A bialgebra equipped with such a convolution inverse is a Hopf algebra.

References

Christian Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer, 1995. Publisher record. Relevant: Chapter III, §§1–2.
Susan Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series 82, American Mathematical Society, 1993. Publisher record. Relevant: Chapter 1.