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An Isaac Newton Institute Workshop

Noncommutative Geometry and Physics: Fundamental Structure of Space and Time

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

Author: Bieliavsky, P (Brussels)


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.