1,502 research outputs found
Courant algebroids and Poisson Geometry
Given a manifold M with an action of a quadratic Lie algebra d, such that all
stabilizer algebras are co-isotropic in d, we show that the product M\times d
becomes a Courant algebroid over M. If the bilinear form on d is split, the
choice of transverse Lagrangian subspaces g_1, g_2 of d defines a bivector
field on M, which is Poisson if (d,g_1,g_2) is a Manin triple. In this way, we
recover the Poisson structures of Lu-Yakimov, and in particular the Evens-Lu
Poisson structures on the variety of Lagrangian Grassmannians and on the de
Concini-Procesi compactifications. Various Poisson maps between such examples
are interpreted in terms of the behaviour of Lagrangian splittings under
Courant morphisms
On Deformation Quantization of Poisson-Lie Groups and Moduli Spaces of Flat Connections
We give simple explicit formulas for deformation quantization of Poisson-Lie
groups and of similar Poisson manifolds which can be represented as moduli
spaces of flat connections on surfaces. The star products depend on a choice of
Drinfe\v{l}d associator and are obtained by applying certain monoidal functors
(fusion and reduction) to commutative algebras in Drinfe\v{l}d categories. From
a geometric point of view this construction can be understood as a quantization
of the quasi-Poisson structures on moduli spaces of flat connections.Comment: 11 page
Selective Categories and Linear Canonical Relations
A construction of Wehrheim and Woodward circumvents the problem that
compositions of smooth canonical relations are not always smooth, building a
category suitable for functorial quantization. To apply their construction to
more examples, we introduce a notion of highly selective category, in which
only certain morphisms and certain pairs of these morphisms are "good". We then
apply this notion to the category of linear canonical
relations and the result of our version of the WW
construction, identifying the morphisms in the latter with pairs
consisting of a linear canonical relation and a nonnegative integer. We put a
topology on this category of indexed linear canonical relations for which
composition is continuous, unlike the composition in itself.
Subsequent papers will consider this category from the viewpoint of derived
geometry and will concern quantum counterparts
Quasi-Hamiltonian groupoids and multiplicative Manin pairs
We reformulate notions from the theory of quasi-Poisson g-manifolds in terms
of graded Poisson geometry and graded Poisson-Lie groups and prove that
quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids. We then
interpret this result within the theory of Dirac morphisms and multiplicative
Manin pairs, to connect our work with more traditional approaches, and also to
put it into a wider context suggesting possible generalizations.Comment: 39 page
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