Differential Calculus on Iso_q (N), Quantum Poincare' Algebra and q - Gravity

Abstract

We present a general method to deform the inhomogeneous algebras of the $B_n,C_n,D_n$ type, and find the corresponding bicovariant differential calculus. The method is based on a projection from $B_{n+1},C_{n+1},D_{n+1}$. For example we obtain the (bicovariant) inhomogeneous $q$-algebra $IS{O}_q(N)$ as a consistent projection of the (bicovariant) $q$-algebra $S{O}_q(N+2)$. This projection works for particular multiparametric deformations of $SO(N+2)$, the so-called ``minimal" deformations. The case of $IS{O}_q(4)$ is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameter $q$. The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains the {\sl classical} Lorentz algebra. Only the commutation relations involving the momenta depend on $q$. Finally, we discuss a $q$-deformation of gravity based on the ``gauging" of this $q$-Poincaré algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.

26 pp., LaTeX. (various misprints corrected; the q-εtensor in the q-gravity lagrangian is given explicitly)

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• An introduction to non-commutative differential geometry on quantum groups
• Differential Geometry on Linear Quantum Groups
• The Lagrangian of q-Poincare' Gravity
• Gauge theories of quantum groups
• Bicovariant Differential Calculus on the Quantum D=2 Poincare Group
• Inhomogeneous quantum groups IGL_{q,r}(N): Universal enveloping algebra and differential calculus

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