- Supergroup (physics)
The concept of

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

**Lie supergroup**is asupermanifold G together with amorphism $cdot\; :G\; imes\; G\; ightarrow\; G$ which makes G agroup object in the category of supermanifolds. This generalises the notion of aLie group . The algebra ofsupercommutative functions over the supergroup can be turned into a**Z**_{2}-graded Hopf algebra. The representations of this Hopf algebra turn out to becomodule s. 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 theLie 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)".

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