Lectures on the Geometry of Poisson Manifolds by Izu VaismanLectures on the Geometry of Poisson Manifolds by Izu Vaisman

Lectures on the Geometry of Poisson Manifolds

byIzu Vaisman

Paperback | October 23, 2012

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Everybody having even the slightest interest in analytical mechanics remembers having met there the Poisson bracket of two functions of 2n variables (pi, qi) f g <_28_8f8g 8="" _29_="" _28_0.129_="" _7b_f2c_g7d_="L..."><_5b_ji -=""><_20_2c_3b_3d_120_p2c_20_q20_q20_p2c_20_and20_the20_fundamental20_role20_it20_plays20_in20_that20_field.20_in20_modern20_works2c_20_this20_bracket20_is20_derived20_from20_a20_symplectic20_structure2c_20_and20_it20_appears20_as20_one20_of20_the20_main20_inc2ad_20_gredients20_of20_symplectic20_manifolds.20_in20_fact2c_20_it20_can20_even20_be20_taken20_as20_the20_defining20_clement20_of20_the20_structure20_28_e.g.2c_20_5b_til5d_29_.20_but2c_20_the20_study20_of20_some20_mechanical20_sysc2ad_20_tems2c_20_particularly20_systems20_with20_symmetry20_groups20_or20_constraints2c_20_may20_lead20_to20_more20_general20_poisson20_brackets.20_therefore2c_20_it20_was20_natural20_to20_define20_a20_mathematical20_structure20_where20_the20_notion20_of20_a20_poisson20_bracket20_would20_be20_the20_primary20_notion20_of20_the20_theory2c_20_and2c_20_from20_this20_viewpoint2c_20_such20_a20_theory20_has20_been20_developed20_since20_the20_early20_197082c_20_by20_a.20_lichnerowicz2c_20_a.20_weinstein2c_20_and20_many20_other20_authors20_28_see20_the20_references20_at20_the20_end20_of20_the20_book29_.20_but2c_20_it20_has20_been20_remarked20_by20_weinstein20_5b_we35d_20_that2c_20_in20_fact2c_20_the20_theory20_can20_be20_traced20_back20_to20_s.20_lie20_himself20_5b_lie5d_. _2c_3b_="1" _p2c_="" q="" and="" the="" fundamental="" role="" it="" plays="" in="" that="" field.="" modern="" _works2c_="" this="" bracket="" is="" derived="" from="" a="" symplectic="" _structure2c_="" appears="" as="" one="" of="" main="" _inc2ad_="" gredients="" manifolds.="" _fact2c_="" can="" even="" be="" taken="" defining="" clement="" structure="" _28_e.g.2c_="" _5b_til5d_29_.="" _but2c_="" study="" some="" mechanical="" _sysc2ad_="" _tems2c_="" particularly="" systems="" with="" symmetry="" groups="" or="" _constraints2c_="" may="" lead="" to="" more="" general="" poisson="" brackets.="" _therefore2c_="" was="" natural="" define="" mathematical="" where="" notion="" would="" primary="" _theory2c_="" _and2c_="" _viewpoint2c_="" such="" theory="" has="" been="" developed="" since="" early="" _197082c_="" by="" a.="" _lichnerowicz2c_="" _weinstein2c_="" many="" other="" authors="" _28_see="" references="" at="" end="" _book29_.="" remarked="" weinstein="" _5b_we35d_="" _that2c_="" traced="" back="" s.="" lie="" himself="">
Title:Lectures on the Geometry of Poisson ManifoldsFormat:PaperbackDimensions:206 pagesPublished:October 23, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034896492

ISBN - 13:9783034896498


Table of Contents

0 Introduction.- 1 The Poisson bivector and the Schouten-Nijenhuis bracket.- 1.1 The Poisson bivector.- 1.2 The Schouten-Nijenhuis bracket.- 1.3 Coordinate expressions.- 1.4 The Koszul formula and applications.- 1.5 Miscellanea.- 2 The symplectic foliation of a Poisson manifold.- 2.1 General distributions and foliations.- 2.2 Involutivity and integrability.- 2.3 The case of Poisson manifolds.- 3 Examples of Poisson manifolds.- 3.1 Structures on ?n. Lie-Poisson structures.- 3.2 Dirac brackets.- 3.3 Further examples.- 4 Poisson calculus.- 4.1 The bracket of 1-forms.- 4.2 The contravariant exterior differentiations.- 4.3 The regular case.- 4.4 Cofoliations.- 4.5 Contravariant derivatives on vector bundles.- 4.6 More brackets.- 5 Poisson cohomology.- 5.1 Definition and general properties.- 5.2 Straightforward and inductive computations.- 5.3 The spectral sequence of Poisson cohomology.- 5.4 Poisson homology.- 6 An introduction to quantization.- 6.1 Prequantization.- 6.2 Quantization.- 6.3 Prequantization representations.- 6.4 Deformation quantization.- 7 Poisson morphisms, coinduced structures, reduction.- 7.1 Properties of Poisson mappings.- 7.2 Reduction of Poisson structures.- 7.3 Group actions and momenta.- 7.4 Group actions and reduction.- 8 Symplectic realizations of Poisson manifolds.- 8.1 Local symplectic realizations.- 8.2 Dual pairs of Poisson manifolds.- 8.3 Isotropic realizations.- 8.4 Isotropic realizations and nets.- 9 Realizations of Poisson manifolds by symplectic groupoids.- 9.1 Realizations of Lie-Poisson structures.- 9.2 The Lie groupoid and symplectic structures of T*G.- 9.3 General symplectic groupoids.- 9.4 Lie algebroids and the integrability of Poisson manifolds.- 9.5 Further integrability results.- 10 Poisson-Lie groups.- 10.1 Poisson-Lie and biinvariant structures on Lie groups.- 10.2 Characteristic properties of Poisson-Lie groups.- 10.3 The Lie algebra of a Poisson-Lie group.- 10.4 The Yang-Baxter equations.- 10.5 Manin triples.- 10.6 Actions and dressing transformations.- References.

Editorial Reviews

    "The book serves well as an introduction and an overview of the subject and a long list of references helps with further study."      -- Zbl. Math.        "The book is well done...should be an essential purchase for mathematics libraries and is likely to be a standard reference for years to come, providing an introduction to an attractive area of further research."  --   Mathematical Reviews        "The importance and actuality of the subject, as well as the very rigorous and didactic presentation of the content, make out of this book a valuable contribution to current mathematics. The book is intended first of all to mathematicians, but it can be interesting also for a wide circle of readers, including mechanicists and physicists."    -- Mathematica