In many of the quantum algebras studied in last years modular automorphisms play a relevant role. Recently, in the context of deformation quantization, Dolgushev showed how to relate the van den Bergh automorphism, carrying informations on duality between Hochschild homology and cohomology, with the Poisson modular class of the semiclassical limit. We will introduce his results and exhibit some interesting examples were trivial and non trivial modular automorphisms can be expected. In particular we will consider standard Poisson-Lie groups and their homogeneous spaces, which are easily shown to be non unimodular, in comparison with theta-Poisson manifolds which, under mild hypothesis are shown to be unimodular and argue how this can explain the different behaviours of such structures under quantization.
The modular class and its quantization - a minicourse
CICCOLI, Nicola
2009
Abstract
In many of the quantum algebras studied in last years modular automorphisms play a relevant role. Recently, in the context of deformation quantization, Dolgushev showed how to relate the van den Bergh automorphism, carrying informations on duality between Hochschild homology and cohomology, with the Poisson modular class of the semiclassical limit. We will introduce his results and exhibit some interesting examples were trivial and non trivial modular automorphisms can be expected. In particular we will consider standard Poisson-Lie groups and their homogeneous spaces, which are easily shown to be non unimodular, in comparison with theta-Poisson manifolds which, under mild hypothesis are shown to be unimodular and argue how this can explain the different behaviours of such structures under quantization.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
