Supergroup (physics)

The concept of supergroup is a generalization of a that of group.

[That means every group is a supergroup but not every supergroup is a group.]

First, let's define what a Hopf superalgebra is. Recall that a Hopf algebra is defined category theoretically over the category K-Vect. Similarly, a Hopf superalgebra is defined category theoretically over the category K-Z_2Vect, which is a Z₂-graded category. The definitions are the same together with the additional requirement that the morphisms η, <math>\nabla<math>, ε, Δ and S are all even morphisms.

A Lie supergroup is a supermanifold G together with a morphism <math>\cdot :G \times G\rightarrow G<math> which makes G a group object in the category of supermanifolds. This is a generalization of a Lie group.