Supergroup (physics)

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 define a Hopf superalgebra. A Hopf algebra can be defined category-theoretically as an object in the category of vector spaces together with a collection of morphisms (∇, Δ, η, ε, "S") satisfying certain commutativity axioms. A Hopf superalgebra can be defined in a completely analogous manner in the category of super vector spaces. The amounts to the additional requirement that the morphisms ∇, Δ, η, ε, "S" are all even.

A Lie supergroup is a supermanifold G together with a morphism $cdot\; :G\; imes\; G\; ightarrow\; G$ which makes G a group object in the category of supermanifolds. This generalises the notion 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.

There are many possible supergroups. The ones of most interest in theoretical physics are the ones which extend the Poincaré group or the conformal group. In this setup, one is particularly interested with the orthosymplectic groups "Osp(N/M)" and the superconformal groups "SU(N/M)".