Hopf Algebra - 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 Algebra

Famous quotes containing the words analogy and/or groups:

    The whole of natural theology ... resolves itself into one simple, though somewhat ambiguous proposition, That the cause or causes of order in the universe probably bear some remote analogy to human intelligence.
    David Hume (1711–1776)

    Under weak government, in a wide, thinly populated country, in the struggle against the raw natural environment and with the free play of economic forces, unified social groups become the transmitters of culture.
    Johan Huizinga (1872–1945)