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# Geometric Models for Noncommutative Algebras

## By Da Silva, Ana Cannas

Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: Poisson Maps; Characterization of Poisson Maps; Complete Poisson Maps; Symplectic Realizations; Coisotropic Calculus; Poisson Quotients; and Poisson Submanifolds.

Contents Preface xi Introduction xiii I Universal Enveloping Algebras 1 1 Algebraic Constructions 1 1.1 Universal Enveloping Algebras . . . . . . . . . . . . . . . . . . . . . 1 1.2 Lie Algebra Deformations . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 The Graded Algebra of U(g) . . . . . . . . . . . . . . . . . . . . . . . 3 2 The Poincar e-Birkho -Witt Theorem 5 2.1 Almost Commutativity of U(g) . . . . . . . . . . . . . . . . . . . . . 5 2.2 Poisson Bracket on Gr U(g) . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 The Role of the Jacobi Identity . . . . . . . . . . . . . . . . . . . . . 7 2.4 Actions of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Proof of the Poincar e-Birkho -Witt Theorem . . . . . . . . . . . . . 9 II Poisson Geometry 11 3 Poisson Structures 11 3.1 Lie-Poisson Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Almost Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Structure Functions and Canonical Coordinates . . . . . . . . . . . . 13 3.5 Hamiltonian Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . 14 3.6 Poisson Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Normal Forms 17 4.1 Lie's Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 A Faithful Representation of g . . . . . . . . . . . . . . . . . . . . . 17 4.3 The Splitting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 Special Cases of the Splitting Theorem . . . . . . . . . . . . . . . . . 20 4.5 Almost Symplectic Structures . . . . . . . . . . . . . . . . . . . . . . 20 4.6 Incarnations of the Jacobi Identity . . . . . . . . . . . . . . . . . . . 21 5 Local Poisson Geometry 23 5.1 Symplectic Foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Transverse Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.3 The Linearization Problem . . . . . . . . . . . . . . . . . . . . . . . 25 5.4 The Cases of su(2) and sl(2;R) . . . . . . . . . . . . . . . . . . . . . 27 III Poisson Category 29

Book Id: WPLBN0000661979
Format Type: PDF eBook
File Size: 3.35 MB
Reproduction Date: 2005
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Title: Geometric Models for Noncommutative Algebras
Author: Da Silva, Ana Cannas
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Language: English
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