Polyvector Super-Poincare Algebras

TL;DR

This paper classifies a broad class of super-Poincare algebras extended with polyvector generators across all spacetime dimensions and signatures, revealing their structure as spinor modules and invariant bilinear forms.

Contribution

It provides a comprehensive classification of polyvector super-Poincare algebras in arbitrary dimensions and signatures, linking algebraic structures to invariant bilinear forms on spinors.

Findings

Classified all such Lie (super)algebras for all dimensions and signatures.

Established that W_0 consists of polyvectors, i.e., submodules of mbda V.

Reduced the classification problem to invariant bilinear forms on spinor modules.

Abstract

A class of Z_2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form g = g_0 + g_1, with g_0 = so(V) + W_0 and g_1 = W_1, where the algebra of generalized translations W = W_0 + W_1 is the maximal solvable ideal of g, W_0 is generated by W_1 and commutes with W. Choosing W_1 to be a spinorial so(V)-module (a sum of an arbitrary number of spinors and semispinors), we prove that W_0 consists of polyvectors, i.e. all the irreducible so(V)-submodules of W_0 are submodules of \Lambda V. We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of so(V)-invariant \Lambda^k V-valued bilinear forms on the spinor module S.