nLab super Poincaré group

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.

Related concepts

• Poincaré group, Poincaré spinor group

• R-symmetry group

• super Poincaré Lie algebra

• super translation algebra

• unitary representation of the super Poincaré group, spin representation, BPS state

group	symbol	universal cover	symbol	higher cover	symbol
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

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

• Pierre Deligne, Daniel Freed, section 1.1 of Supersolutions (arXiv:hep-th/9901094)

  in Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey, David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, (eds.), Quantum Fields and Strings

• Daniel Freed, lecture 6 of Classical field theory and Supersymmetry, IAS/Park City Mathematics Series Volume 11 (2001) (pdf)

• Daniel Freed, Lecture 3 of Five lectures on supersymmetry

• Veeravalli Varadarajan, section 7.5 (draft: 6.5) of Supersymmetry for mathematicians: An introduction, Courant lecture notes in mathematics, American Mathematical Society (2004) [doi;10.1090/cln/011, pdf]

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