Structure constants of shs$[\lambda]$: the deformedoscillator point of view
Abstract
We derive and spell out the structure constants of the $\mathbb{Z}_2$graded algebra $\mathfrak{shs}[\lambda]\,$ by using deformedoscillators techniques in $Aq(2;\nu)\,$, the universal enveloping algebra of the Wignerdeformed Heisenberg algebra in 2 dimensions. The use of Weyl ordering of the deformed oscillators is made throughout the paper, via the symbols of the operators and the corresponding associative, noncommutative star product. The deformed oscillator construction was used by Vasiliev in order to construct the higher spin algebras in three spacetime dimensions. We derive an expression for the structure constants of $\mathfrak{shs}[\lambda]\,$ and show that they must obey a recurrence relation as a consequence of the associativity of the star product. We solve this condition and show that the $\mathfrak{hs}[\lambda]\,$ structure constants are given by those postulated by Pope, Romans and Shen for the Lone Star product.
 Publication:

arXiv eprints
 Pub Date:
 April 2016
 DOI:
 10.48550/arXiv.1604.04510
 arXiv:
 arXiv:1604.04510
 Bibcode:
 2016arXiv160404510B
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 22 pages, no figures. Contribution to the proceedings of the workshop "About various kinds of interactions", 45 June 2015 at UMONS, in honour of Philippe Spindel