nLab
super Poincaré group

Context

Super-Geometry

Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

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 of a super Poincaré group by the orthogonal group/Lorentz 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)\,\,
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)\,\,\,\,

References

Section 1.1 of

Section 7.5 of

Also

  • Super spacetimes and super Poincaré-group (pdf)

Revised on August 17, 2014 22:38:02 by Urs Schreiber (89.204.138.55)