Hopf algebra - Knowlpedia

Definition

Let $k$ be a commutative ring. A Hopf algebra over $k$ is a bialgebra $(H,m,u,\Delta,\varepsilon)$ together with a $k$ -linear map $S:H\to H$ , called the antipode, such that

m(S\otimes\operatorname{id})\Delta =u\varepsilon =m(\operatorname{id}\otimes S)\Delta.

Equivalently, $S$ is the two-sided inverse of $\operatorname{id}_H$ for the convolution product

f*g=m(f\otimes g)\Delta \quad\text{on}\quad \operatorname{End}_k(H).

Meaning of the antipode

The unit for convolution is $u\varepsilon$ , so the antipode identities read, in Sweedler notation,

\sum S(h_{(1)})h_{(2)} =\varepsilon(h)1_H =\sum h_{(1)}S(h_{(2)}).

They are the algebraic counterpart of the equations $g^{-1}g=e=gg^{-1}$ in a group. When an antipode exists it is unique.

Basic properties

The antipode reverses multiplication and comultiplication:

S(ab)=S(b)S(a), \qquad \Delta(S(h))=(S\otimes S)\tau\Delta(h),

where $\tau$ swaps tensor factors. These conclusions follow from uniqueness of convolution inverses. Bijectivity of $S$ , however, is not automatic for an arbitrary infinite-dimensional Hopf algebra.

A Hopf-algebra homomorphism is a bialgebra homomorphism; it automatically commutes with antipodes by their uniqueness.

Examples

The group algebra $k[G]$ is a Hopf algebra with $S(g)=g^{-1}$ .
For a Lie algebra $\mathfrak g$ , its universal enveloping algebra has the cocommutative Hopf structure $\Delta(x)=x\otimes1+1\otimes x,\quad \varepsilon(x)=0,\quad S(x)=-x \qquad(x\in\mathfrak g).$
Coordinate rings of affine group schemes are commutative Hopf algebras. For affine formal groups, ordinary tensor products are replaced by completed tensor products.

Variance

Functions pull back. Consequently, a homomorphism of affine groups $G\to H$ induces a Hopf-algebra homomorphism $\mathcal O(H)\to\mathcal O(G)$ in the opposite direction. The same contravariance appears for affine formal groups and their complete coordinate Hopf algebras.

References

Moss E. Sweedler, Hopf Algebras, W. A. Benjamin, 1969. Relevant: Chapters 1–4.
Christian Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer, 1995. Publisher record. Relevant: Chapter III.
William C. Waterhouse, Introduction to Affine Group Schemes, Graduate Texts in Mathematics 66, Springer, 1979. Publisher record. Relevant: Chapters 1–3.