# Universal deformation formulae for the `ax+b' group, and noncommutative BTZ black holes

Author: Bieliavsky, P (Brussels)

### Abstract

Rieffel's ``Deformation Quantization for actions of \$R^d\$'' provides a simple machinery for defining a class of noncommutative Riemannian manifolds within the \$C^\star\$-algebraic framework i.e. within a non-formal context. The basic idea coinsists in viewing Weyl symbol composition formula --- in the context of Weyl's quantization of \$R^d\$--- as a {\sl Universal deformation formula}, that is, a formula which defines a \$C^\star\$- deformation of any \$C^\star\$-algebra endowed with an action of the Abelian Lie group \$R^d\$. Typical examples obtained from this machinery are noncommutative tori and related noncommutative manifolds.

Of course, in many geometrical situations where curvature is involved, one disposes of no action of \$R^d\$, but rather of actions of {\sl non-Abelian} Lie groups. The above observation motivates the search of more general universal deformation formulae.