A Characteristic Map for Symplectic Manifolds
We construct a local characteristic map to a symplectic manifold M via certain cohomology groups of Hamiltonian vector fields. For each p in M, the Leibniz cohomology of the Hamiltonian vector fields on R^{2n} maps to the Leibniz cohomology of all Ha…
Authors: Jerry Lodder
A Characteristic Map for Symplectic Manifolds Jerry M. Lo dder Mathematic al Scienc es, Dept. 3MB Box 30001 New Mexic o State University L as Cruc es NM, 88 0 03, U.S.A. e-mail: jlo dder@nmsu.e du Abstract. W e construct a lo cal c haracteristic map to a symple ctic manifold M via certain cohomolo gy groups of Hamiltonian v ector fields. F o r eac h p ∈ M , the Leibniz cohomolo gy of the Hamiltonian v ector fields on R 2 n maps to the Leibniz cohomology of all Hamiltonian v ector fields on M . F or a particular extens ion g n of the symplectic Lie a lgebra, the Leibniz cohomology of g n is s ho wn to b e an ex terior algebra o n the canonical sy mplectic t w o- form. The Leibniz homology o f g n then maps to the Leibniz homology of Hamiltonian v ector fields on R 2 n . Mathematics Sub ject Classifications (2000): 17B56, 53D05, 17A32. Key W ords: Sy mplectic top ology , Leibniz homology , symplectic in v ariants. 1 In tro duction W e construct a lo cal c hara cteris tic map to a symplectic manifold M via certain cohomology gro ups of Hamiltonian vec tor fields. Recall that the group of affine symplectomorphisms, i.e., the a ffine symplectic group AS p n , is giv en by a ll transformations ψ : R 2 n → R 2 n of the form ψ ( z ) = Az + z 0 , where A is a 2 n × 2 n symplectic matrix and z 0 a fixed elemen t of R 2 n [5, p. 55]. Let g n denote the Lie algebra o f AS p n , referred to as the affine symplectic 1 Lie algebra. Then g n is the larg es t finite dimensional Lie subalgebra of the Hamiltonian v ector fields on R 2 n , and serv es as our p oin t of departure for calculations. Particular a t ten tion is dev oted to the Leibniz homolog y of g n , i.e., H L ∗ ( g n ; R ), and pro ve n is t hat H L ∗ ( g n ; R ) ≃ Λ ∗ ( ω n ) , where ω n = P n i =1 ∂ ∂ x i ∧ ∂ ∂ y i and Λ ∗ denotes the exterior algebra. D ually , for cohomology , H L ∗ ( g n ; R ) ≃ Λ ∗ ( ω ∗ n ) , where ω ∗ n = P n i =1 dx i ∧ dy i . F o r p ∈ M , the lo cal c har acteristic map acquires the form H L ∗ ( X H ( R 2 n ); R )( p ) → H L ∗ ( X H ( M ); C ∞ ( M )) , where X H denotes the Lie algebra of Hamiltonian v ector fields, and C ∞ ( M ) is the ring of C ∞ real-v alued functions on M . Using previous w ork of the author [4], there is a natural map H ∗ dR ( M ; R ) → H L ∗ ( X ( M ); C ∞ ( M )) , where H ∗ dR denotes deRham cohomolog y . Comp osing with H L ∗ ( X ( M ); C ∞ ( M )) → H L ∗ ( X H ( M ); C ∞ ( M )) , w e hav e H ∗ dR ( M ; R ) → H L ∗ ( X H ( M ); C ∞ ( M )) . The inclusion of Lie a lg ebras g n ֒ → X H ( R 2 n ) induces a linear map H L ∗ ( g n ; R ) → H L ∗ ( X H ( R 2 n ); R ) and H L ∗ ( X H ( R 2 n ); R ) con tains a copy of H L ∗ ( g n ; R ) as a direct summand. The calculational to ols for H L ∗ ( g n ) include the Ho c hsc hild-Serre sp ectral sequence for Lie-a lgebra (co)homolo gy , the Pirash vili spectral seq uence for Leibniz homo lo gy , and the identification of certain symplectic in v arian ts of g n whic h app ear in the app endix. 2 2 The Affine Symplect ic Lie Algeb r a As a p oin t of departure, consider a C ∞ Hamiltonian function H : R 2 n → R with the asso ciated Hamiltonian vec tor field X H = n X i =1 ∂ H ∂ x i ∂ ∂ y i − n X i =1 ∂ H ∂ y i ∂ ∂ x i , where R 2 n is giv en co ordinates { x 1 , x 2 , . . . , x n , y 1 , y 2 , . . . , y n } , and ∂ ∂ x i , ∂ ∂ y i are the unit v ector fields parallel to the x i and y i axes resp ectiv ely . The v ector field X H is then ta ngen t to the lev el curve s (o r hyper-surfaces) of H . Res tricting H to a quadratic function in { x 1 , x 2 , . . . , x n , y 1 , y 2 , . . . , y n } , yields a fa mily of v ector fields isomorphic to the r eal symplectic Lie a lgebra sp n . F or i , j , k ∈ { 1 , 2 , 3 , . . . , n } , an R -v ector space ba sis, B 1 , fo r sp n is giv en by the families: (1) x k ∂ ∂ y k (2) y k ∂ ∂ x k (3) x i ∂ ∂ y j + x j ∂ ∂ y i , i 6 = j (4) y i ∂ ∂ x j + y j ∂ ∂ x i , i 6 = j (5) y j ∂ ∂ y i − x i ∂ ∂ x j , ( i = j p ossible). It follo ws that dim R ( sp n ) = 2 n 2 + n . Let I n denote the Ab elian Lie algebra of Hamiltonian v ector fields ar ising from t he linear (affine) functions H : R 2 n → R . Then I n has an R -v ector space basis giv en by B 2 = n ∂ ∂ x 1 , ∂ ∂ x 2 , . . . , ∂ ∂ x n , ∂ ∂ y 1 , ∂ ∂ y 2 , . . . , ∂ ∂ y n o . 3 The affine symplectic Lie algebra, g n , has an R - v ector space basis B 1 ∪ B 2 . There is a short exact sequence of Lie algebras 0 − − − → I n i − − − → g n π − − − → sp n − − − → 0 , where i is the inclusion map and π is the pro jection g n → ( g n /I n ) ≃ sp n . In fact, I n is an Ab elian ideal of g n with I n acting on g n via the brac k et of v ector fields. 3 The Lie Algebra Homolog y o f g n F o r an y Lie algebra g o v er a ring k , the Lie algebra homolo g y of g , written H Lie ∗ ( g ; k ), is the homolog y of the c hain complex Λ ∗ ( g ), namely k 0 ← − − − g [ , ] ← − − − g ∧ 2 ← − − − . . . ← − − − g ∧ ( n − 1) d ← − − − g ∧ n ← − − − . . . , where d ( g 1 ∧ g 2 ∧ . . . ∧ g n ) = X 1 ≤ i
Original Paper
Loading high-quality paper...
Comments & Academic Discussion
Loading comments...
Leave a Comment