Hopf Algebra

Coalgebras, Bialgebras and Hopf Algebras

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.

e.g.

• The group algebra of a group is a Hopf algebra with comultiplication defined by for all , counit defined by for all , and antipode defined by for all .
• Any universal enveloping algebra of a Lie algebra is a Hopf algebra, as shown here.

Remark

Just like an inverse in most cases is unique, the antipode is also unique when it exists.

Hopf Algebra Morphism

A Hopf algebra morphism between two Hopf algebras and is a linear map that is both an algebra homomorphism and a coalgebra homomorphism, and is compatible with the antipodes, i.e., .

The Theorem of Milnor—Moore

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 .

Proposition

The set of all Lie-like elements in a Hopf algebra , denoted , forms a Lie algebra under the commutator bracket .

Proof First, we verify that is a vector space. For any and , we have so . Next, we check that the commutator bracket is closed in . For any , we have so . Finally, the commutator bracket satisfies bilinearity, antisymmetry, and the Jacobi identity, making a Lie algebra.

We’ve already showed in here that the universal enveloping algebra of a Lie algebra is a Hopf algebra. The Milnor–Moore theorem states that, under certain conditions, the converse is also true:

Milnor–Moore Theorem

Given a graded, cocommutative Hopf algebra that is connected over a field of characteristic (i.e., is the ground field), suppose for all , then is isomorphic to the universal enveloping algebra of its Lie algebra of Lie-like elements.

Although it is beyond the scope of this article to provide a full proof, we outline the ideas for a much weaker version of the theorem, where we assume is generated by its Lie-like elements (then is automatically connected). Let be the inclusion. By the universal property of universal enveloping algebra, extends uniquely to a Hopf map . Since is generated by its Lie-like elements, then any element can be written as a linear combination of monomials in , so is surjective by the Poincaré–Birkhoff–Witt Theorem. To show that is an isomorphism, it suffices to prove that it is injective. For this, we use the filtration structure. Recall that has a standard filtration where is the subspace spanned by monomials of length at most in . Let with induced filtration . Assume , then there exists a smallest such that . Take any non-zero , then its image in the associated graded algebra is non-zero. Pick a nonzero element . Observe that cannot be zero or one. is the ground field , acts as identity on it; , and restricted to is just the inclusion. So . Let . Suppose , then so is the unique term in coming from picking the first part of every . Similarly, is also the unique term in . Therefore, . Since is a Hopf map, we have . So . Combing the above inclusions, we get where the last equality follows from the minimality of . Thus, , i.e., is Lie-like, , which means has degree , contradicting . Therefore, , and is an isomorphism.