nLab super Poincaré group

Contents

Context

Super-Geometry

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The super Poincaré group is the Lie integration in supergeometry of the super Poincaré Lie algebra. This is a super Lie group-extension of the ordinary Poincaré group.

In physics an action/symmetry of the super Poincaré group is also called a supersymmetry.

The coset (super Klein geometry) of a super Poincaré group by the Spin group/Pin group inside it is a super translation group, whose underlying supermanifold is a super Minkowski spacetime.

groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
conformal groupO(n+1,t+1)\mathrm{O}(n+1,t+1)\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,
superconformal group

References

These references speak of the super Poincaré group but tend to focus on its super Poincaré Lie algebra:

Detailed construction of the actual super Lie group-structure are rare, but see the discussion at least of the super translation subgroup: there.

Last revised on August 27, 2024 at 15:58:45. See the history of this page for a list of all contributions to it.