Mini-Workshop

Symmetry & Reduction in Poisson & Related Geometries

May 20-24, 2024

Department of Mathematics, University of Salerno

Description

This mini-workshop aims at discussing the most recent results around coisotropic reduction in Poisson and related geometries, and their algebraic counterparts, both in the classical and the quantum setting. The bulk of the event consists of three mini-courses on different aspects of the topic. The mini-courses will be complemented by invited talks. Notice that this is the only event during the intensive period that will take place at the University of Salerno rather than in Napoli! 

Speakers

Schedule

You can find here a PDF file with the complete program of the miniworkshop.

Mini-courses

Nicola Ciccoli (University of Perugia)

Title: TBA

Abstract: TBA

Leonid Ryvkin (University Claude Bernard Lyon 1) & Antonio Michele Miti (Sapienza Università di Roma)

Title: Reduction of (multi)-symplectic observables

Abstract: Let M be a manifold with a geometric structure and sufficiently nice G a symmetry group, often the geometric structure can be transferred to M/G. In (multi-)symplectic geometry, reduction procedures permit to transfer the differential form to an even smaller space. However, all approaches working directly on the space have very strong regularity requirements. We present an approach to reducing the algebra of (multi-)symplectic observables for general (covariant) moment maps, without any regularity assumptions on the level sets (and the symmetries). Even in the well-studied symplectic case, this construction is distinct from pre-existing ones. Based on joint work with Casey Blacker.

Stefan Waldmann (University of Wurzburg)

Title: TBA

Abstract: TBA

Talks

Sylvain Lavau (Ruđer Bošković Institute Zagreb)

Title: The Bott connection generates gauge transformations

Abstract: This talk is about connecting the Bott connection associated to regular foliations and the gauge generators of coisotropic (first-class) submanifolds. The annihilator bundle of a smooth involutive regular distribution is a coisotropic submanifold (constraint surface) of the cotangent bundle. One observes that the generators of this involutive distribution (vector fields tangent to the foliation) can be lifted to first-class constraints defining the constraint surface. Then, we show that the gauge orbits generated by these constraints correspond to the sections of the annihilator bundle which are horizontal with respect to the Bott connection. This allows to characterize the zero-th group of the foliated cohomology as the gauge invariant linear functions on the annihilator bundle.

Peter Crooks (Utah State University)

Title: Generalized Hamiltonian reduction and Moore-Tachikawa varieties

Abstract: Marsden and Weinstein's approach to Hamiltonian reduction is a mainstay of modern research at the interface of Poisson geometry, representation theory, and theoretical physics. Generalizations of this approach include Mikami-Weinstein reduction, symplectic cutting, symplectic implosion, and Ginzburg-Kazhdan reduction by abelian group schemes. The last of these was motivated by conjectural solutions to the decade-old Moore-Tachikawa conjecture on topological quantum field theories (TQFTs).

Praful Rahangdale (Paderborn University)

Title: Poisson Lie groups, Lie bialgebras, the Yang-Baxter equation, and Manin triples

Abstract: In this talk, I will present the work by V. G. Drinfeld (1983) which gives the one-to-one correspondence between  finite dimensional connected simply connected Poisson Lie groups and finite dimensional Lie bialgebras. Then, I will show the correspondence between Lie bialgebras, the classical Yang-Baxter equation, and Manin triples. In the end, I  will discuss attempts made and difficulties arisen in the context of infinite dimensional Poisson Lie groups (modeled on locally convex topological vector spaces).

Matthias Schötz (IMPAN Warsaw)

Title: Symmetry Reduction of States

Abstract: We develop a general theory of symmetry reduction of states on (possibly non-commutative) ∗-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra 𝔤. The key idea here is that the correct notion of positivity on a ∗-algebra A is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares, but can be a more general one that depends on the example at hand, like pointwise positivity on ∗-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A, and the notion of positivity on the reduced algebra should be such that its states are obtained as reductions of certain states on A. After laying out the general setup I will then discuss several examples, like smooth functions on a Poisson manifold M, the reduction of the Weyl algebra, and an example from deformation quantization. In this talk, I will present the work by V. G. Drinfeld (1983) which gives the one-to-one correspondence between finite-dimensional connected simply connected Poisson Lie groups and finite-dimensional Lie bialgebras. Then, I will show the correspondence between Lie bialgebras, the classical Yang-Baxter equation, and Manin triples. In the end, I  will discuss attempts made and difficulties arisen in the context of infinite-dimensional Poisson Lie groups (modeled on locally convex topological vector spaces).

Mikhail Vasilev (University of Glasgow)

Title: Ruijsenaars duality for (B, C, D) Toda chains from Hamiltonian reduction

Abstract: We use the Hamiltonian reduction method to construct the Ruijsenaars dual systems to generalized Toda chains associated with the classical Lie algebras of types B, C, D. The dual systems turn out to be the B, C and D analogues of the rational Goldfish model, which is, as in the type A case, the strong coupling limit of rational Ruijsenaars systems. We explain how both types of systems emerge in the reduction of the cotangent bundle of a Lie group and provide the formulae for dual Hamiltonians and action-angle duality maps. We compute explicitly the higher Hamiltonians of Goldfish models using Cauchy-Binet theorem.

Registration

There is no registration fee but registration is required. To register send an e-mail to mdippell@unisa.it (Marvin Dippell).

Participants

TBA

More Info

Here is a map with some relevant places in and outside the Campus of the University of Salerno, where the mini-workshop will take place. For more information contact us at mdippell@unisa.it (Marvin Dippell).