nLab Poincaré-Weyl algebra

Idea

An extension of the Poincaré Lie algebra by the dilatation generator.

Definition

The Poincaré-Weyl algebra is defined by generators $\{M_{\mu \nu}, P_{\lambda}, D\}$ satisfying the relations

$$[M_{\mu\nu}, M_{\rho \sigma} ] = i (\eta_{\mu \rho} M_{\nu \sigma} - \eta_{\mu \sigma} M_{\nu \rho} + \eta_{\nu \rho} M_{\mu \sigma} - \eta_{\nu \sigma} M_{\mu \rho} )$$

$$[M_{\mu \nu}, P_{\lambda}] = i(\eta_{\mu \lambda} P_{\nu} - \eta_{\nu \lambda} P_{\mu}) )$$

$$[D, M_{\mu \nu} ] = [P_{\mu}, P_{\nu} ] = 0$$

$$[D,P_{\mu} ] = i P_{\mu}$$

(see e.g. Eq. 2.10-14 in Charap & Tait 1974).

References

• J. M. Charap and W. Tait. A gauge theory of the Weyl group. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 340, no. 1622 (1974): 249-262. (doi)