Hopf Algebra - Formal Definition

Formal Definition

Formally, a Hopf algebra is a (associative and coassociative) bialgebra H over a field K together with a K-linear map S: HH (called the antipode) such that the following diagram commutes:

Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. In the sumless Sweedler notation, this property can also be expressed as

As for algebras, one can replace the underlying field K with a commutative ring R in the above definition.

The definition of Hopf algebra is self-dual (as reflected in the symmetry of the above diagram), so if one can define a dual of H (which is always possible if H is finite-dimensional), then it is automatically a Hopf algebra.

Read more about this topic:  Hopf Algebra

Famous quotes containing the words formal and/or definition:

    That anger can be expressed through words and non-destructive activities; that promises are intended to be kept; that cleanliness and good eating habits are aspects of self-esteem; that compassion is an attribute to be prized—all these lessons are ones children can learn far more readily through the living example of their parents than they ever can through formal instruction.
    Fred Rogers (20th century)

    It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.”
    Jane Adams (20th century)