# nLab super Poincaré group

Contents

supersymmetry

## Applications

#### Lie theory

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 group$\mathrm{O}(n)$Pin group$Pin(n)$Tring group$Tring(n)$
special orthogonal group$SO(n)$Spin group$Spin(n)$String group$String(n)$
Lorentz group$\mathrm{O}(n,1)$$\,$$Spin(n,1)$$\,$$\,$
anti de Sitter group$\mathrm{O}(n,2)$$\,$$Spin(n,2)$$\,$$\,$
conformal group$\mathrm{O}(n+1,t+1)$$\,$
Narain group$O(n,n)$
Poincaré group$ISO(n,1)$Poincaré spin group$\widehat {ISO}(n,1)$$\,$$\,$
super Poincaré group$sISO(n,1)$$\,$$\,$$\,$$\,$
superconformal group

## References

Last revised on March 7, 2024 at 08:59:31. See the history of this page for a list of all contributions to it.