29 research outputs found
Kontsevich's Universal Formula for Deformation Quantization and the Campbell-Baker-Hausdorff Formula, I
We relate a universal formula for the deformation quantization of arbitrary
Poisson structures proposed by Maxim Kontsevich to the Campbell-Baker-Hausdorff
formula. Our basic thesis is that exponentiating a suitable deformation of the
Poisson structure provides a prototype for such universal formulae.Comment: 48 pages, over 90 (small) epsf figures, uses some ams-latex package
Formal symplectic groupoid
The multiplicative structure of the trivial symplectic groupoid over associated to the zero Poisson structure can be expressed in terms of a
generating function. We address the problem of deforming such a generating
function in the direction of a non-trivial Poisson structure so that the
multiplication remains associative. We prove that such a deformation is unique
under some reasonable conditions and we give the explicit formula for it. This
formula turns out to be the semi-classical approximation of Kontsevich's
deformation formula. For the case of a linear Poisson structure, the deformed
generating function reduces exactly to the CBH formula of the associated Lie
algebra. The methods used to prove existence are interesting in their own right
as they come from an at first sight unrelated domain of mathematics: the
Runge--Kutta theory of the numeric integration of ODE's.Comment: 28 pages, 4 figure
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Aesthetical entanglements in mathematics education
In this study, we develop a perspective on the diverse aesthetics historically associated with mathematics, inspired by Rancière's approach to aesthetics and politics. We call “Silencing Aesthetics” a dominant aesthetic that Rota has characterized as a “copout (…) intended to keep our formal description of mathematics as close as possible to the description of a mechanism”. The challenge this study attempts to explore is how to question silencing aesthetics to make space for inclusive ones. Our efforts have focused on setting up and studying inclusive and pluralist “Studios”, gathering craftworkers, anthropologists, mathematics educators, and mathematics enthusiasts. We include here a case study based on a conversation amongst basket weavers, anthropologists, and mathematics educators focused on the artisanal and mathematical nature of knots. We discuss the implications of aesthetical entanglements, such as those in our case study, for mathematics learning
Heisenberg double versus deformed derivatives
Two approaches to the tangent space of a noncommutative space whose
coordinate algebra is the enveloping algebra of a Lie algebra are known: the
Heisenberg double construction and the approach via deformed derivatives,
usually defined by procedures involving orderings among noncommutative
coordinates or equivalently involving realizations via formal differential
operators. In an earlier work, we rephrased the deformed derivative approach
introducing certain smash product algebra twisting a semicompleted Weyl
algebra. We show here that the Heisenberg double in the Lie algebra case, is
isomorphic to that product in a nontrivial way, involving a datum
parametrizing the orderings or realizations in other approaches. This way, we
show that the two different formalisms, used by different communities, for
introducing the noncommutative phase space for the Lie algebra type
noncommutative spaces are mathematically equivalent
Gauge theories and non-commutative geometry
It is shown that a -dimensional classical SU(N) Yang-Mills theory can be
formulated in a -dimensional space, with the extra two dimensions forming
a surface with non-commutative geometry. In this paper we present an explicit
proof for the case of the torus and the sphere.Comment: 12 page
New realizations of Lie algebra kappa-deformed Euclidean space
We study Lie algebra -deformed Euclidean space with undeformed
rotation algebra and commuting vectorlike derivatives. Infinitely
many realizations in terms of commuting coordinates are constructed and a
corresponding star product is found for each of them. The -deformed
noncommutative space of the Lie algebra type with undeformed Poincar{\'e}
algebra and with the corresponding deformed coalgebra is constructed in a
unified way.Comment: 30 pages, Latex, accepted for publication in Eur.Phys.J.C, some typos
correcte
Star Product and Invariant Integration for Lie type Noncommutative Spacetimes
We present a star product for noncommutative spaces of Lie type, including
the so called ``canonical'' case by introducing a central generator, which is
compatible with translations and admits a simple, manageable definition of an
invariant integral. A quasi-cyclicity property for the latter is shown to hold,
which reduces to exact cyclicity when the adjoint representation of the
underlying Lie algebra is traceless. Several explicit examples illuminate the
formalism, dealing with kappa-Minkowski spacetime and the Heisenberg algebra
(``canonical'' noncommutative 2-plane).Comment: 21 page
Graph complexes in deformation quantization
Kontsevich's formality theorem and the consequent star-product formula rely
on the construction of an -morphism between the DGLA of polyvector
fields and the DGLA of polydifferential operators. This construction uses a
version of graphical calculus. In this article we present the details of this
graphical calculus with emphasis on its algebraic features. It is a morphism of
differential graded Lie algebras between the Kontsevich DGLA of admissible
graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between
polyvector fields and polydifferential operators. Kontsevich's proof of the
formality morphism is reexamined in this light and an algebraic framework for
discussing the tree-level reduction of Kontsevich's star-product is described.Comment: 39 pages; 3 eps figures; uses Xy-pic. Final version. Details added,
mainly concerning the tree-level approximation. Typos corrected. An abridged
version will appear in Lett. Math. Phy
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Aesthetical Entanglements in Mathematics Learning
In this paper we develop a perspective on the diverse aesthetics historically associated with mathematics, inspired by Rancière’s approach to aesthetics and politics. We call ‘Silencing Aesthetics’ a dominant aesthetic that Rota has characterized as a “copout (...) intended to keep our formal description of mathematics as close as possible to the description of a mechanism.” The challenge this paper attempts to explore is how to question silencing aesthetics to make space for generative ones. Our efforts have focused on setting up and studying inclusive and pluralist ‘Studios’, gathering craftworkers, anthropologists, mathematics educators and mathematics enthusiasts. We include here a case study based on a conversation among basket weavers, anthropologists, and mathematics educators, focused on the artisanal and mathematical nature of knots