The notion of Hopf algebra is an abstraction of the properties of
Coalgebra
A coalgebra is just like a unital algebra, but with all the structure maps reversed. Precisely, a coalgebra
is a vector space over a field , equipped with a comultiplication and a counit , satisfying the following left/right co-unitality laws and co-associativity laws
where
is the tensor product, and are the left and right unit isomorphisms of the monoidal category of vector spaces.
Hopf Algebra
An
-bialgebra with multiplication , comultiplication , unit and counit is called a Hopf algebra if there exists a linear map , called the antipode or coinverse, such that . Moreover, if the antipode is an anti-involution, i.e. , then is called an involutive Hopf algebra or Hopf *-algebra.
Remark
Just like an inverse in most cases is unique, the antipode is also unique when it exists.
Group-like & Lie-like Elements
A group-like element in a Hopf algebra
is a non-zero element such that . An element is called Lie-like or primitive if .
Milnor–Moore Theorem