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Homotopy quantum symmetries, monoidal categories and formality

Objective

This project will be hosted at the MIT (outgoing host) and at the University of Zurich (return
host). Its main objectives are :
(I) to develop a theory of homotopy quantum groups. This can be understood as the natural the-
ory that should sit at the intersection of four important disciplines of mathematics and physics :
monoidal categories, homotopy theory, quantum groups and higher categories. This fact gives a
clear multidisciplinary aspect to the proposal.
(II) to prove the formality of the homotopy Lie algebra governing simultaneous deformations of a
Poisson manifold and its coisotropic submanifolds. This is the key step in solving the problem of
quantization of symmetries from the point of view of deformation quantization, giving interdisciplinary applications. These two objectives organize themselves into subobjectives :
I1 ) define homotopy braided monoidal categories (∞-braided ∞-monoidal category)
I2 ) define homotopy quasi-triangular quasi-Hopf algebras (∞-triangular ∞-Hopf algebras)
I3 ) define monodromy for higher connections
I4 ) give examples of higher Drinfeld associators
I5 ) define and give examples of homotopy quantum groups
II1 ) construct the homotopy Lie algebra governing simultaneous deformations of a
Poisson manifold and its coisotropic submanifolds
II2 ) prove the formality of this homotopy Lie algebra.
Interdisciplinary aspects come also from tools used which are borrowed from physics (higher
holonomies, branes, quantization) or from new rewriting techniques in computer science and operads (Gröbner basis).

Field of science

  • /natural sciences/mathematics/pure mathematics/algebra
  • /natural sciences/computer and information sciences

Call for proposal

FP7-PEOPLE-2010-IOF
See other projects for this call

Funding Scheme

MC-IOF - International Outgoing Fellowships (IOF)

Coordinator

University of Zurich
Address
Ramistrasse 71
8006 Zurich
Switzerland
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 225 233,60
Administrative Contact
Alberto Cattaneo (Prof.)