Supergroup (physics)
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The concept of supergroup is a generalization of that of group. In other words, every group is a supergroup but not every supergroup is a group.
First, let us recall what a Hopf superalgebra is. 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, 
, ε, Δ and S are all even morphisms.
A Lie supergroup is a supermanifold G together with a morphism 
which makes G a group object in the category of supermanifolds. This is a generalization of a Lie group. The algebra of supercommutative functions over the supergroup can be turned into a Z2-graded Hopf algebra. The representations of this Hopf algebra turn out to be comodules. This Hopf algebra gives the global properties of the supergroup.
There is another related Hopf algebra which is the dual of the previous Hopf algebra. This only gives the local properties of the symmetries (i.e., they only give the information about infinitesimal supersymmetry transformations). The representations of this Hopf algebra are modules. And this Hopf algebra is the universal enveloping algebra of the Lie superalgebra.
Let us try to understand supergroups in a less abstract way. There are many possible supergroups but the only ones in which we are interested are the ones which generalize the Poincare group and the conformal group. Thus, interesting groups are the orthosymplectic groups Osp(N/M) and the superconformal groups SU(N/M).
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