Differential calculus on ISO-q(N), quantum Poincare algebra and q gravity

Castellani, Leonardo

doi:10.1007/BF02099276

Differential calculus on ISO-q(N), quantum Poincare algebra and q gravity

Leonardo Castellani
(
- INFN, Turin and
- Turin U.
)

Dec, 1993

25 pages

Published in:

Commun.Math.Phys. 171 (1995) 383-404

e-Print:

hep-th/9312179 [hep-th]

DOI:

10.1007/BF02099276

Report number:

DFTT-70-93

View in:

ADS Abstract Service

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Abstract: (arXiv)

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

ISO_q(N)

as a consistent projection of the (bicovariant)

q

-algebra

SO_q(N+2)

. This projection works for particular multiparametric deformations of

SO(N+2)

, the so-called ``minimal" deformations. The case of

ISO_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 Poincar\'e 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

-Poincar\'e algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.

quantum gravity
differential forms
operator: algebra
algebra: Poincare
algebra: deformation
quantum algebra
bibliography

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