Hopf Algebras - Analogy With Groups

Analogy With Groups

Groups can be axiomatized by the same diagrams (equivalently, operations) as a Hopf algebra, where G is taken to be a set instead of a module. In this case:

  • the field K is replaced by the 1-point set
  • there is a natural counit (map to 1 point)
  • there is a natural comultiplication (the diagonal map)
  • the unit is the identity element of the group
  • the multiplication is the multiplication in the group
  • the antipode is the inverse

In this philosophy, a group can be thought of as a Hopf algebra over the "field with one element".

Read more about this topic:  Hopf Algebras

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