TY - BOOK

T1 - Group actions on deformation, quantizations and an equivariant algebraic index theorem

AU - de Kleijn, Niek

PY - 2016

Y1 - 2016

N2 - Group actions on algebras obtained by formal deformation quantization are the main topic of thisthesis. We study these actions in order to obtain an equivariant algebraic index theorem that leads toexplicit formulas in terms of equivariant characteristic classes. The Fedosov construction, as realizedin a deformed version of Gelfand's formal geometry, is used to obtain the results.We describe the main points of Gelfand's formal geometry in the deformed case and show how itleads to Fedosov connections and the well-known classication of formal deformation quantization inthe direction of a symplectic structure.A group action on a deformation quantization induces an action on the underlying symplecticmanifold. We consider the lifting problem of nding group actions inducing a given action by symplectomorphisms.We reformulate some known sucient conditions for existence of a lift and showthat they are not necessary. Given a particular lift of an action by symplectomorphisms to the deformationquantization, we obtain a classication of all such lifts satisfying a certain technical condition.The classication is in terms of a rst non-Abelian group cohomology. We supply tools for computingthese sets in terms of a commuting diagram with exact rows and columns. Finally we consider someexamples to formulate vanishing and non-vanishing results.In joint work with A. Gorokhovsky and R. Nest, we prove an equivariant algebraic index theorem.The equivariant algebraic index theorem is a formula expressing the trace on the crossed productalgebra of a deformation quantization with a group in terms of a pairing with certain equivariantcharacteristic classes. The equivariant characteristic classes are viewed as classes in the periodic cycliccohomology of the crossed product by using the inclusion of Borel equivariant cohomology due toConnes.

AB - Group actions on algebras obtained by formal deformation quantization are the main topic of thisthesis. We study these actions in order to obtain an equivariant algebraic index theorem that leads toexplicit formulas in terms of equivariant characteristic classes. The Fedosov construction, as realizedin a deformed version of Gelfand's formal geometry, is used to obtain the results.We describe the main points of Gelfand's formal geometry in the deformed case and show how itleads to Fedosov connections and the well-known classication of formal deformation quantization inthe direction of a symplectic structure.A group action on a deformation quantization induces an action on the underlying symplecticmanifold. We consider the lifting problem of nding group actions inducing a given action by symplectomorphisms.We reformulate some known sucient conditions for existence of a lift and showthat they are not necessary. Given a particular lift of an action by symplectomorphisms to the deformationquantization, we obtain a classication of all such lifts satisfying a certain technical condition.The classication is in terms of a rst non-Abelian group cohomology. We supply tools for computingthese sets in terms of a commuting diagram with exact rows and columns. Finally we consider someexamples to formulate vanishing and non-vanishing results.In joint work with A. Gorokhovsky and R. Nest, we prove an equivariant algebraic index theorem.The equivariant algebraic index theorem is a formula expressing the trace on the crossed productalgebra of a deformation quantization with a group in terms of a pairing with certain equivariantcharacteristic classes. The equivariant characteristic classes are viewed as classes in the periodic cycliccohomology of the crossed product by using the inclusion of Borel equivariant cohomology due toConnes.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122141861905763

M3 - Ph.D. thesis

SN - 978-87-7078-949-3

BT - Group actions on deformation, quantizations and an equivariant algebraic index theorem

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -