Contractions, deformations and curvature

Abstract

The role of curvature in relation with Lie algebra contractions of the pseudo-ortogonal algebras so(p,q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley-Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the curvature of such "quantum" spaces.

17 pages. Based on the talk given in the Oberwolfach workshop: Deformations and Contractions in Mathematics and Physics (Germany, january 2006) organized by M. de Montigny, A. Fialowski, S. Novikov and M. Schlichenmaier

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Cited by in corpus (7)

• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
• Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups
• Recurrence approach and higher rank cubic algebras for the $N$-dimensional superintegrable systems
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Universal constants and natural systems of units in a spacetime of arbitrary dimension
• A perspective on the Magic Square and the 'special unitary' realizations of simple Lie algebras
• Superintegrable potentials on 3D Riemannian and Lorentzian spaces with non-constant curvature