POISSON STRUCTURES AND THEIR NORMAL FORMS

By SARAH LEE

CONTENTS POISSON BRACKETS POISSON TENSORS POISSON MORPHISMS LOCAL CANONICAL COORDINATES SINGULAR SYMPLECTIC FOLIATIONS TRANSVERSE POISSON STRUCTURES GROUP ACTIONS AND REDUCTION THE SCHOUTEN BRACKET SCHOUTEN BRACKET OF MULTI-VECTOR ?ELDS SCHOUTEN BRACKET ON LIE ALGEBRAS COMPATIBLE POISSON STRUCTURES SYMPLECTIC REALIZATIONS POISSON COHOMOLOGY DE?NITION OF POISSON COHOMOLOGY INTERPRETATION OF POISSON COHOMOLOGY POISSON COHOMOLOGY VERSUS DE RHAM COHOMOLOGY OTHER VERSIONS OF POISSON COHOMOLOGY COMPUTATION OF POISSON COHOMOLOGY NORMAL FORMS OF POISSON STRUCTURES COHOMOLOGY OF LIE ALGEBRAS CHEVALLEY–EILENBERG COMPLEXES COHOMOLOGY OF LINEAR POISSON STRUCTURES RIGID LIE ALGEBRAS SPECTRAL SEQUENCES SPECTRAL SEQUENCE OF A ?LTERED COMPLEX LERAY SPECTRAL SEQUENCE HOCHSCHILD–SERRE SPECTRAL SEQUENCE SPECTRAL SEQUENCE FOR POISSON COHOMOLOGY POISSON COHOMOLOGY IN DIMENSION 2 SIMPLE SINGULARITIES COHOMOLOGY OF POISSON GERMS SOME EXAMPLES AND REMARKS THE CURL OPERATOR DE?NITION OF THE CURL OPERATOR SCHOUTEN BRACKET VIA CURL OPERATOR THE MODULAR CLASS THE CURL OPERATOR OF AN A?NE CONNECTION POISSON HOMOLOGY FORMAL LEVI DECOMPOSITION LEVI DECOMPOSITION OF POISSON STRUCTURES CONSTRUCTION OF LEVI DECOMPOSITION NORMED VANISHING OF COHOMOLOGY PROOF OF ANALYTIC LEVI DECOMPOSITION THEOREM THE SMOOTH CASE NONDEGENERATE LIE ALGEBRAS LINEARIZATION OF LOW-DIMENSIONAL POISSON STRUCTURES TWO-DIMENSIONAL CASE THREE-DIMENSIONAL CASE FOUR-DIMENSIONAL CASE POISSON GEOMETRY OF REAL SEMISIMPLE LIE ALGEBRAS NONDEGENERACY OF AFF(N) SOME OTHER LINEARIZATION RESULTS EQUIVARIANT LINEARIZATION LINEARIZATION OF POISSON–LIE TENSORS POISSON STRUCTURES WITH A HYPERBOLIC RK-ACTION TRANSVERSE POISSON STRUCTURES TO COADJOINT ORBITS FINITE DETERMINACY OF POISSON STRUCTURES MULTIPLICATIVE TENSORS POISSON–LIE GROUPS AND R-MATRICES THE DUAL AND THE DOUBLE OF A POISSON-LIE GROUP ACTIONS OF POISSON–LIE GROUPS POISSON ACTIONS OF POISSON–LIE GROUPS DRESSING TRANSFORMATIONS MOMENTUM MAPS R-MATRICES AND QUADRATIC POISSON STRUCTURES LINEAR CURL VECTOR ?ELDS QUADRATIZATION OF POISSON STRUCTURES NONHOMOGENEOUS QUADRATIC POISSON STRUCTURES NAMBU BRACKETS AND NAMBU TENSORS INTEGRABLE DI?ERENTIAL FORMS FROBENIUS WITH SINGULARITIES LINEAR NAMBU STRUCTURES KUPKA’S PHENOMENON LINEARIZATION OF NAMBU STRUCTURES DECOMPOSABILITY OF ? FORMAL LINEARIZATION OF THE ASSOCIATED FOLIATION THE ANALYTIC CASE FORMAL LINEARIZATION OF ? THE SMOOTH ELLIPTIC CASE INTEGRABLE 1-FORMS WITH A NON-ZERO LINEAR PART QUADRATIC INTEGRABLE 1-FORMS POISSON STRUCTURES IN DIMENSION 3 SOME BASIC NOTIONS ON GROUPOIDS DE?NITIONS AND ?RST EXAMPLES LIE GROUPOIDS GERMS AND SLICES OF LIE GROUPOIDS ACTIONS OF GROUPOIDS HAAR SYSTEMS MORITA EQUIVALENCE PROPER LIE GROUPOIDS DE?NITION AND ELEMENTARY PROPERTIES SOURCE-LOCAL TRIVIALITY ORBIFOLD GROUPOIDS LINEARIZATION OF LIE GROUPOIDS LINEARIZATION OF LIE GROUP ACTIONS LOCAL LINEARIZATION OF LIE GROUPOIDS SLICE THEOREM FOR LIE GROUPOIDS SYMPLECTIC GROUPOIDS DE?NITION AND BASIC PROPERTIES PROPER SYMPLECTIC GROUPOIDS HAMILTONIAN ACTIONS OF SYMPLECTIC GROUPOIDS SOME GENERALIZATIONS SOME BASIC DE?NITIONS AND PROPERTIES DE?NITION AND SOME EXAMPLES THE LIE ALGEBROID OF A LIE GROUPOID ISOTROPY ALGEBRAS CHARACTERISTIC FOLIATION OF A LIE ALGEBROID LIE PSEUDOALGEBRAS FIBER-WISE LINEAR POISSON STRUCTURES LIE ALGEBROID MORPHISMS LIE ALGEBROID ACTIONS AND CONNECTIONS SPLITTING THEOREM AND TRANSVERSE STRUCTURES COHOMOLOGY OF LIE ALGEBROIDS LINEARIZATION OF LIE ALGEBROIDS INTEGRABILITY OF LIE BRACKETS RECONSTRUCTION OF GROUPOIDS FROM THEIR ALGEBROIDS INTEGRABILITY CRITERIA INTEGRABILITY OF POISSON MANIFOLDS APPENDIX A 1 MOSER’S PATH METHOD A 2 DIVISION THEOREMS A 3 REEB STABILITY A 4 ACTION-ANGLE VARIABLES A 5 NORMAL FORMS OF VECTOR ?ELDS POINCAR´E–DULAC NORMAL FORMS BIRKHO?NORMAL FORMS TORIC CHARACTERIZATION OF NORMAL FORMS SMOOTH NORMAL FORMS A 6 NORMAL FORMS ALONG A SINGULAR CURVE A 7 THE NEIGHBORHOOD OF A SYMPLECTIC LEAF GEOMETRIC DATA AND COUPLING TENSORS LINEAR MODELS A 8 DIRAC STRUCTURES A 9 DEFORMATION QUANTIZATION