Kavli IPMU-FMSP Lectures

Speaker: Alan Weinstein (UC Berkeley)
Title: Special subspaces in symplectic vector spaces
Date (JST): Mon, Apr 11, 2016, 15:30 - 17:00
Place: Lecture Hall
Abstract: In a vector space V carrying a symplectic (i.e. nondegenerate, skew-symmetric) bilinear form \omega, each subspace A has a symplectic orthogonal space A^\omega consisting of those elements v for which \ omega(v,w)=0 whenever w\in A. Subspaces for which the intersection of A with A^\omega is equal to A, A^\omega, or {0} are especially important; they are called isotropic, coisotropic, or symplectic respectively. When both of the first conditions hold, the subspace is called lagrangian.

Linear maps (W,\omega_W) -> (V,\omega_V) whose graphs in the product (V \times W,\omega_V \times -\omega_W) are isotropic, coisotropic, or lagrangian also have a special importance; they are symplectic embeddings, Poisson submersions, or symplectic isomorphisms respectively. It is also useful to look at subspaces of these types in the product which may not be graphs of mappings. Those which are lagrangian are called canonical relations or lagrangian correspondences} and are of particular importance as the morphisms in symplectic categories.

In a manifold M carrying a closed non- degenerate 2-form, submanifolds whose tangent spaces are of the distinguished types above are the subject of many interesting problems, both solved and unsolved. Even at the linear level, there are still important problems in both finite and infinite-dimensional (for field theory) cases, having to do with
classification of (k-tuples of) subspaces, composition of relations, and quantization.

All of the problems above can be formulated as cases of the general problem of understanding representations of quivers (objects consisting of vertices and arrows between them) by symplectic vector spaces and
special relations. It would be interesting to develop this general representation theory.
Remarks: <Reference>

David Li-Bland and Alan Weinstein

Selective Categories and Linear Canonical Relations

SIGMA 10 (2014), 100, 31 pages
Contribution to the Special Issue on Poisson Geometry in Mathematics and Physics

arXiv:1401.7302 http://dx.doi.org/10.3842/SIGMA.2014.100

Jonathan Lorand and Alan Weinstein
(Co)isotropic Pairs in Poisson and Presymplectic Vector Spaces

SIGMA 11 (2015), 072, 10 pages

Contribution to the Special Issue on Poisson Geometry in Mathematics and Physics

arXiv:1503.00169 http://dx.doi.org/10.3842/SIGMA.2015.072

Jonathan Lorand and Alan Weinstein
Decomposition of (Co)isotropic Relations