nLab
super Poincare Lie algebra

Idea

A super Lie algebra extension of a Poincare Lie algebra.

The corresponding super Lie group is the super Euclidean group (except for the signature of the metric).

Properties

Lie algebra cohomology

The super Poincaré Lie algebra has, on top of the Lie algebra cocycles that it inherits from $\mathrm{𝔰𝔬}(n)$, a discrete number of special cocycles bilinear in the spinors, on the super translation algebra, that exist only in very special dimensions.

For the statement of the following theorem, we use the relation between division algebra and supersymmetry, as described there.

Theorem

• In dimensional $d=3,4,6,10$, $\mathrm{𝔰𝔦𝔰𝔬}(d-1,1)$ has a nontrivial 3-cocycle given by

  $$(\psi ,\varphi ,A)\mapsto g(\psi \cdot \varphi ,A)$$(\psi, \phi, A) \mapsto g(\psi \cdot \phi, A)

  for spinors $\psi ,\varphi \in 𝒮$ and vectors $A\in 𝒯$, and 0 otherwise.

• In dimensional $d=4,5,7,11$, $\mathrm{𝔰𝔦𝔰𝔬}(d-1,1)$ has a nontrivial 4-cocycle given by

  $$(\Psi ,\Phi ,𝒜,ℬ)\mapsto ⟨\Psi ,(𝒜ℬ-ℬ𝒜)\Phi ⟩$$(\Psi, \Phi, \mathcal{A}, \mathcal{B}) \mapsto \langle \Psi , (\mathcal{A}\mathcal{B}- \mathcal{B} \mathcal{A})\Phi \rangle

  for spinors $\Psi ,\Phi \in 𝒮$ and vectors $𝒜,ℬ\in 𝒱$, with the commutator taken in the Clifford algebra.

The 4-cocycle in $d=11$ is the one that induces the supergravity Lie 3-algebra.

Extensions

Super $L_{\mathrm{\infty }}$-algebra extensions

The super L-infinity algebra infinity-Lie algebra cohomology of the super Poincaré Lie algebra corresponding to the above cocycles involves

supergravity Lie 6-algebra $\to$ supergravity Lie 3-algebra $\to$ super-Poincaré Lie algebra

Polyvector extensions

The super-Poincaré Lie algebra has a class of super Lie algebra extensions called polyvector extensions , because they involve additional generators that transforn as skew-symmetric tensors. A complete classification is in (ACDP).

As automorphism Lie algebras of Lie $n$-superalgebras

At least some of the polyvector extensions of the super Poincaré Lie algebra arise as the automorphism super Lie algebras of the Lie n-algebra extensions classified by the cocycles discussed above.

For instance the automorphisms of the supergravity Lie 3-algebra gives the “M-theory Lie algebra”-extension of super-Poincaré in 11-dimensions. This is discussed here.

Related concepts

• super Poincaré group

• supersymmetry

References

Polyvector extensions

Some standard exposition is for instance in

• Super spacetimes and super Poincaré-group (pdf)

A comprehensive account and classification of the polyvector extensions of the super Poincaré Lie algebras is in

• D. V. Alekseevsky, V. Cortés, C. Devchand, A. Van Proeyen, Polyvector Super-Poincaré Algebras Commun.Math.Phys. 253 (2004) 385-422 (arXiv:hep-th/0311107)

See also

• C. Chrysso‌malakos, J. A. de Azcárraga1, J. M. Izquierdo and C. Pérez Bueno, The geometry of branes and extended superspaces (pdf)

Super Lie algebra cohomology and division algebras

The exceptional fermionic cocycles on the super Poincaré Lie algebra, and their construction using normed division algebras, are discussed here:

• John Baez, John Huerta, Division algebras and supersymmetry I (arXiv:0909.0551)

• John Baez, John Huerta, Division algebras and supersymmetry II (arXiv:1003.34360)

• John Huerta, Division Algebras, Supersymmetry and Higher Gauge Theory, (arXiv:1106.3385)

For more see division algebra and supersymmetry.

This subsumes some of the results in

• J. A. de Azcárraga and P. K. Townsend, Superspace geometry and classification of supersymmetric extended objects, Phys. Rev. Lett. 62, 2579–2582 (1989)

The cohomology of the super Poincare Lie algebra in low dimensions $\le 5$ is analyzed in

• Friedemann Brandt?,

  Supersymmetry algebra cohomology I: Definition and general structure J. Math. Phys.51:122302, 2010, arXiv

  Supersymmetry algebra cohomology II: Primitive elements in 2 and 3 dimensions J. Math. Phys. 51 (2010) 112303 (arXiv)

  Supersymmetry algebra cohomology III: Primitive elements in four and five dimensions (arXiv)

and in some higher dimensions with the help of a computer algebra system in:

• Michael Movshev, Albert Schwarz, Renjun Xu, Homology of Lie algebra of supersymmetries (arXiv)

A direct constructions of ordinary (Lie algebraic) extensions of the super Poincare Lie algebra by means of division algebras is in

• Jerzy Lukierski, Francesco Toppan, Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory (pdf)