Hopf algebra

A Hopf algebra is a bialgebra with an antipode morphism satisfying

.

Hopf algebras are also called quantum groups because they can be interpreted as "quantizations" of groups.

See also Algebra/set analogy.

Examples

1. Given a topological group G, we can form two different Hopf algebras over it. The first is the algebra of continuous functions from G to K whose product is the pointwise product. ε acting on a function gives its value at the identity and (Δ f)(x,y)=f(xy) for all x and y in G. (Sf)(x)=f(x^-1). The coaction of this Hopf algebra upon noncommutative spaces is as a left (right) comodule. The other Hopf algebra we can construct is the convolution product algebra of distributions over G. This time, the action of this Hopf algebra upon noncommutative spaces is as a left (right) module.
2. If, in addition, G is a Lie group, it has a Lie algebra g. Its universal enveloping algebra can be turned into a Hopf algebra by εx=0, Δx=x⊗1+1⊗x and Sx=-x for all elements of the Lie algebra. There's an injective homomorphism from this Hopf algebra to the Hopf algebra of convolutions over G such that the image of this homomorphism is the subalgebra generated by the Dirac delta distribution and its derivatives over the identity of G.

See also superalgebra, anyonic Lie algebra

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