Mathematics – Quantum Algebra
Scientific paper
2009-03-13
Russ. J. Math. Phys., 16, No. 3 (2009) 371
Mathematics
Quantum Algebra
27 pages. Based on the talk presented at the conference "Geometry and Operators Theory," Ancona (Italy), September 2007
Scientific paper
10.1134/S1061920809030054
We consider noncommutative geometries obtained from a triangular Drinfeld twist. This allows to construct and study a wide class of noncommutative manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms. This way symmetry principles can be implemented. We review two main examples [15]-[18]: a) general covariance in noncommutative spacetime. This leads to a noncommutative gravity theory. b) Symplectomorphims of the algebra of observables associated to a noncommutative configuration space. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e., we establish a noncommutative correspondence principle from *-Poisson brackets to *-commutators. New results concerning noncommutative gravity include the Cartan structural equations for the torsion and curvature tensors, and the associated Bianchi identities. Concerning scalar field theories the deformed algebra of classical and quantum observables has been understood in terms of a twist within the algebra.
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