Supergroup (physics)

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

Related Topics:
Generalization - Group

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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, abla, ε, Δ and S are all even morphisms.

Related Topics:
Hopf algebra - Category theoretically - K-Vect - K-Z_2Vect - Graded category - Morphism

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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 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.

Related Topics:
Supermanifold - Morphism - Group object - Category - Lie group - Supercommutative - Comodule

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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.

Related Topics:
Module - Universal enveloping algebra - Lie superalgebra

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