A Characteristic Map for Symplectic Manifolds

0.15 MB·

by  Jerry Lodder

A Characteristic Map for Symplectic Manifolds
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0 Jerry M. Lodder
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a
J Mathematical Sciences, Dept. 3MB

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Box 30001
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New Mexico State University

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Las Cruces NM, 88003, U.S.A.
G
S e-mail: [email protected]
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Abstract. We construct a local characteristic map to a symplectic manifold
t
a
M via certain cohomology groups of Hamiltonian vector fields. For each
m
p ∈ M, the Leibniz cohomology of the Hamiltonian vector fields on R2n
[
maps to the Leibniz cohomology of all Hamiltonian vector fields on M. For a
1
v particular extension gn ofthe symplectic Liealgebra, theLeibniz cohomology
6 of g is shown to be an exterior algebra on the canonical symplectic two-
n
4
form. The Leibniz homology of g then maps to the Leibniz homology of
4 n
3 Hamiltonian vector fields on R2n.
.
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0 Mathematics Subject Classifications (2000): 17B56, 53D05, 17A32.
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Key Words: Symplectic topology, Leibniz homology, symplectic invariants.
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1 Introduction
r
a
We construct a local characteristic map to a symplectic manifold M via
certain cohomology groups of Hamiltonian vector fields. Recall that the
group of affine symplectomorphisms, i.e., the affine symplectic group ASp ,
n
is given by all transformations ψ : R2n → R2n of the form
ψ(z) = Az +z ,
0
whereAisa2n×2nsymplectic matrixandz afixedelement ofR2n [5,p.55].
0
Let g denote the Lie algebra of ASp , referred to as the affine symplectic
n n
1
Lie algebra. Then g is the largest finite dimensional Lie subalgebra of the
n
Hamiltonian vector fields on R2n, and serves as our point of departure for
calculations. Particular attention is devoted to the Leibniz homology of g ,
n
i.e., HL (g ; R), and proven is that
∗ n
HL (g ; R) ≃ Λ∗(ω ),
∗ n n
where ω = n ∂ ∧ ∂ and Λ∗ denotes the exterior algebra. Dually, for
n i=1 ∂xi ∂yi
cohomology,P
HL∗(g ; R) ≃ Λ∗(ω∗),
n n
where ω∗ = n dxi ∧dyi.
n i=1
For p ∈ MP, the local characteristic map acquires the form
HL∗(X (R2n); R)(p) → HL∗(X (M); C∞(M)),
H H
where X denotes the Lie algebra of Hamiltonian vector fields, and C∞(M)
H
is the ring of C∞ real-valued functions on M. Using previous work of the
author [4], there is a natural map
H∗ (M; R) → HL∗(X(M); C∞(M)),
dR
where H∗ denotes deRham cohomology. Composing with
dR
HL∗(X(M); C∞(M)) → HL∗(X (M); C∞(M)),
H
we have
H∗ (M; R) → HL∗(X (M); C∞(M)).
dR H
The inclusion of Lie algebras g ֒→ X (R2n) induces a linear map
n H
HL (g ; R) → HL (X (R2n); R)
∗ n ∗ H
and HL∗(X (R2n); R) contains a copy of HL∗(g ; R) as a direct summand.
H n
The calculational tools for HL (g ) include the Hochschild-Serre spectral
∗ n
sequence for Lie-algebra (co)homology, the Pirashvili spectral sequence for
Leibniz homology, and the identification of certain symplectic invariants of
g which appear in the appendix.
n
2
2 The Affine Symplectic Lie Algebra
As a point of departure, consider a C∞ Hamiltonian function H : R2n → R
with the associated Hamiltonian vector field
n n
∂H ∂ ∂H ∂
X = − ,
H
∂x ∂yi ∂y ∂xi
Xi=1 i Xi=1 i
where R2n is given coordinates
{x , x , ..., x , y , y , ..., y },
1 2 n 1 2 n
and ∂ , ∂ aretheunitvectorfieldsparalleltothex andy axesrespectively.
∂xi ∂yi i i
The vector field X is then tangent to the level curves (or hyper-surfaces) of
H
H. Restricting H to a quadratic function in
{x , x , ..., x , y , y , ..., y },
1 2 n 1 2 n
yields a family of vector fields isomorphic to the real symplectic Lie algebra
sp . For i, j, k ∈ {1, 2, 3, ..., n}, an R-vector space basis, B , for sp is
n 1 n
given by the families:
(1) x ∂
k∂yk
(2) y ∂
k∂xk
(3) x ∂ +x ∂ , i 6= j
i∂yj j∂yi
(4) y ∂ +y ∂ , i 6= j
i∂xj j∂xi
(5) y ∂ −x ∂ , (i = j possible).
j∂yi i∂xj
It follows that dimR(spn) = 2n2 +n.
Let I denote the Abelian Lie algebra of Hamiltonian vector fields arising
n
from the linear (affine) functions H : R2n → R. Then I has an R-vector
n
space basis given by
∂ ∂ ∂ ∂ ∂ ∂
B = , , ..., , , , ..., .
2 ∂x1 ∂x2 ∂xn ∂y1 ∂y2 ∂yn
n o
3
The affine symplectic Lie algebra, g , has an R-vector space basis B ∪B .
n 1 2
There is a short exact sequence of Lie algebras
i π
0 −−−→ I −−−→ g −−−→ sp −−−→ 0,
n n n
where i is the inclusion map and π is the projection
g → (g /I ) ≃ sp .
n n n n
In fact, I is an Abelian ideal of g with I acting on g via the bracket of
n n n n
vector fields.
3 The Lie Algebra Homology of g
n
For any Lie algebra g over a ring k, the Lie algebra homology of g, written
HLie(g; k), is the homology of the chain complex Λ∗(g), namely
∗
k ←−0−− g ←[−,−]− g∧2 ←−−− ... ←−−− g∧(n−1) ←−d−− g∧n ←−−− ...,
where
d(g ∧g ∧ ... ∧g ) =
1 2 n
(−1)j (g ∧ ... ∧g ∧[g , g ]∧g ∧ ... gˆ ... ∧g ).
1 i−1 i j i+1 j n
1≤Xi<j≤n
For actual calculations in this paper, k = R. Additionally, Lie algebra
homology with coefficients in the adjoint representation, written HLie(g; g),
∗
is the homology of the chain complex g⊗Λ∗(g), i.e.,
g ←− g⊗g ←− g⊗g∧2 ←− ... ←− g⊗g∧(n−1)←d−g⊗g∧n ←− ...,
where
n+1
d(g ⊗g ∧g ... ∧g ) = (−1)i([g , g ]⊗g ∧ ... gˆ ... ∧g )
1 2 3 n+1 1 i 2 i n+1
Xi=2
+ (−1)j(g ⊗g ∧ ... ∧g ∧[g , g ]∧g ∧ ... gˆ ... ∧g ).
1 2 i−1 i j i+1 j n+1
2≤i<Xj≤n+1
4
The canonical projection g⊗Λ∗(g) → Λ∗+1(g) given by g⊗g∧n → g∧(n+1) is
a map of chain complexes and induces a k-linear map on homology
HLie(g; g) → HLie (g; k).
n n+1
Given a (right) g-module M, the module of invariants Mg is defined as
Mg = {m ∈ M | [m, g] = 0 ∀g ∈ g}.
Note that sp acts on I and on the affine symplectic Lie algebra g via the
n n n
bracket of vector fields. The action is extended to I∧k by
n
k
[α ∧α ∧ ... ∧α , X] = α ∧α ∧ ... ∧[α , X]∧ ... ∧α
1 2 k 1 2 i k
Xi=1
for α ∈ I , X ∈ sp , and similarly for the sp action on g ⊗I∧k. The main
i n n n n n
result of this section of the following.
Lemma 3.1. There are natural vector space isomorphisms
HLie(g ; R) ≃ HLie(sp ; R)⊗[Λ∗(I )]spn,
∗ n ∗ n n
HLie(g ; g ) ≃ HLie(sp ; R)⊗[g ⊗Λ∗(I )]spn
∗ n n ∗ n n n
Proof. The lemma follows essentially from the Hochschild-Serre spectral se-
quence [2], the application of which we briefly outline to aid in the identi-
fication of representative homology cycles, and to reconcile the lemma with
its cohomological version in [2]. Consider the filtration F , m ≥ −1, of the
m
complex Λ∗(g ) given by
n
F = {0},
−1
F = Λ∗(I ), Fk = I∧k, k = 0, 1, 2, 3, ...,
0 n 0 n
Fk = {g ∧ ... ∧g ∈ g∧(k+m) | at most m-many g ’s ∈/ I }.
m 1 k+m n i n
Then each F is a chain complex, and F is a subcomplex of F . For
m m m+1
m ≥ 0, we have
E0 = Fk/Fk ≃ I∧k ⊗(g /I )∧m.
m,k m m−1 n n n
Since I is Abelian and the action of I on g /I is trivial, it follows that
n n n n
E1 ≃ I∧k ⊗(g /I )∧m.
m,k n n n
5
Using the isomorphism g /I ≃ sp , we have
n n n
E2 ≃ H (sp ; I∧k).
m,k m n n
Now, sp is a simple Lie algebra and as an sp -module
n n
I∧k ≃ (I∧k)spn ⊕M,
n n
where M ≃ M ⊕M ⊕...⊕M is a direct sum of simple modules on which
1 2 t
sp acts non-trivially. Hence
n
H∗(spn; In∧k) ≃ H∗(spn; (In∧k)spn)⊕H∗(spn; M).
Clearly,
H∗(spn; (In∧k)spn) ≃ H∗(spn; R)⊗(In∧k)spn
t
H (sp ; M) ≃ H (sp ; M ) ≃ 0,
∗ n ∗ n i
Xi=1
where the latter isomorphism holds since each M is simple with non-trivial
i
sp action. See [1, Prop. VII.5.6] for more details.
n
Let θ be a cycle in Λm(sp ) representing an element of H (sp ; R), and
n m n
let z ∈ (In∧k)spn. Then z∧θ ∈ g∧n(m+k) represents an absolute cycle in Λ∗(gn),
since, if θ is a sum of elements of the form s ∧s ∧ ... ∧s , then [z, s ] = 0
1 2 m i
for each s ∈ sp . Thus, E2 ≃ E∞ , and
i n m,k m,k
H∗(gn; R) ≃ H∗(spn; R)⊗[Λ∗(In)]spn.
By a similar filtration and spectral sequence argument for g ⊗ Λ∗(g ), we
n n
have
H∗(gn; gn) ≃ H∗(spn; R)⊗[gn ⊗Λ∗(In)]spn.
Let ωn = ni=1 ∂∂xi ∧ ∂∂yi ∈ In∧2. One checks that ωn ∈ (In∧2)spn against the
basis for sp Pgiven in §2. It follows that
n
ω∧k ∈ [I∧2k]spn.
n n
Letting Λ∗(ω ) denote the exterior algebra generated by ω , we prove in the
n n
appendix that
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Lemma 3.2. There are isomorphisms
[Λ∗(I )]spn ≃ Λ∗(ω ) := Λk(ω ),
n n n
Xk≥0
[g ⊗Λ∗(I )]spn ≃ Λ¯∗(ω ) := Λk(ω )
n n n n
Xk≥1
where the first is an isomorphism of algebras, and the second is an isomor-
phism of vector spaces.
Combining this with Lemma (3.1), we have
Lemma 3.3. There are vector space isomorphisms
HLie(g ; R) ≃ H (sp ; R)⊗Λ∗(ω ),
∗ n ∗ n n
HLie(g ; g ) ≃ H (sp ; R)⊗Λ¯∗(ω ).
∗ n n ∗ n n
It is known that for cohomology,
H∗ (sp ; R) ≃ Λ∗(u , u , u , ..., u ),
Lie n 3 7 11 4n−1
where u is a class in dimension i. Also,
i
HLie(sp ; R) ≃ Hk (sp ; R).
k n Lie n
See the reference [8, p. 343] for the homology of the symplectic Lie group.
4 The Leibniz Homology of g
n
Recall that for a Lie algebra g over a ring k, and more generally for a Leibniz
algebra g [3], the Leibniz homology of g, written HL (g; k), is the homology
∗
of the chain complex T(g):
k ←−0−− g ←[−,−]− g⊗2 ←−−− ... ←−−− g⊗(n−1) ←−d−− g⊗n ←−−− ...,
where
d(g , g , ..., g ) =
1 2 n
(−1)j(g , g , ..., g , [g , g ], g , ... gˆ ..., g ),
1 2 i−1 i j i+1 j n
1≤Xi<j≤n
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and (g , g , ..., g ) denotes the element g ⊗g ⊗ ... ⊗g ∈ g⊗n.
1 2 n 1 2 n
The canonical projection π : g⊗n → g∧n, n ≥ 0, is a map of chain
1
complexes, T(g) → Λ∗(g), and induces a k-linear map on homology
HL (g; k) → HLie(g; k).
∗ ∗
Letting
(kerπ ) [2] = ker[g⊗(n+2) → g∧(n+2)], n ≥ 0,
1 n
Pirashvili [7] defines the relative theory Hrel(g) as the homology of the com-
plex
Crel(g) = (kerπ ) [2],
n 1 n
and studies the resulting long exact sequence relating Lie and Leibniz ho-
mology:
··· −−−∂→ Hrel (g) −−−→ HL (g) −−−→ HLie(g) −−−∂→ Hrel (g) −−−→
n−2 n n n−3
··· −−−∂→ Hrel(g) −−−→ HL (g) −−−→ HLie(g) −−−→ 0
0 2 2
0 −−−→ HL (g) −−−→ HLie(g) −−−→ 0
1 1
0 −−−→ HL (g) −−−→ HLie(g) −−−→ 0.
0 0
An additional exact sequence is required for calculations of HL . Con-
∗
sider the projection
π : g⊗g∧n → g∧(n+1), n ≥ 0,
2
and the resulting chain map
π : g⊗Λ∗(g) → Λ∗+1(g).
2
Let HR (g) denote the homology of the complex
n
CR (g) = (kerπ ) [1] = ker[g⊗g∧(n+1) → g∧(n+2)], n ≥ 0.
n 2 n
There is a resulting long exact sequence
··· −−−∂→ HR (g) −−−→ HLie(g; g) −−−→ HLie (g) −−−∂→
n−1 n n+1
··· −−−∂→ HR (g) −−−→ HLie(g; g) −−−→ HLie(g) −−−∂→
0 1 2
0 −−−→ HLie(g; g) −−−→ HLie(g) −−−→ 0.
0 1
8
The projection π : g⊗(n+1) → g∧(n+1) can be written as the composition of
1
projections
g⊗(n+1) −→ g⊗g∧n −→ g∧(n+1),
which leads to a natural map between exact sequences
Hrel (g) −−−→ HL (g) −−−→ HLie (g) −−−∂→ Hrel (g)
n−1 n+1 n+1 n−2
1
   
HRny−1(g) −−−→ HnLiey(g; g) −−−→ HnL+iye1(g) −−−∂→ HRny−2(g)
and an articulation of their respective boundary maps ∂.
Lemma 4.1. For the affine symplectic Lie algebra g , there is a natural
n
isomorphism
≃
H (sp ; R) −→ HR (g ; R), k ≥ 3,
k n k−3 n
that factors as the composition
Lie ≃ ≃
H (sp ; R) −→ HR (sp ; R) −→ HR (g ;R),
k n k−3 n k−3 n
∂
and the latter isomorphism is induced by the inclusion sp ֒→ g .
n n
Proof. Since sp is a simple Lie algebra, from [1, Prop. VII.5.6] we have
n
HLie(sp ; sp ) = 0, k ≥ 0.
k n n
From the long exact sequence
··· −→ HR (sp ; R) −→ HLie(sp ; sp ) −→ HLie (sp ; R) −∂→ ··· ,
k−1 n k n n k+1 n
it follows that ∂ : HLie(sp ; R) → HR (sp ; R) is an isomorphism for
k n k−3 n
k ≥ 3. The inclusion of Lie algebras sp ֒→ g induces a map of exact
n n
sequences
−−−→ HR (sp ; R) −−−→ HLie(sp ; sp ) −−−→ HLie (sp ; R) −−−∂→
k−1 n k n n k+1 n
  
−−−→ HRk−1y(gn; R) −−−→ HkLie(gyn; gn) −−−→ HkL+ie1(ygn; R) −−−∂→
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From Lemma (3.3)
HLie(g ; R) ≃ H (sp ; R)⊗Λ∗(ω )
∗ n ∗ n n
HLie(g ; g ) ≃ H (sp ; R)⊗Λ¯∗(ω ).
∗ n n ∗ n n
The map HLie(g ; g ) → HLie (g ; R) is an inclusion on homology with
∗ n n ∗+1 n
cokernel HLie (sp ; R). The result now follows from the map between exact
∗+1 n
sequences and a knowledge of the generators of HLie(g ; R) gleaned from
∗ n
Lemma (3.1).
Theorem 4.2. There is an isomorphism of vector spaces
HL (g ; R) ≃ Λ∗(ω )
∗ n n
and an algebra isomorphism
n
HL∗(g ; R) ≃ Λ∗(ω∗), ω∗ = dxi ∧dyi,
n n n
Xi=1
where HL∗ is afforded the shuffle algebra.
Proof. Consider the Pirashvili filtration [7] of the complex
Crel(g) = ker(g⊗(n+2) → g∧(n+2)), n ≥ 0,
n
given by
Fk(g) = g⊗k ⊗ker(g⊗(m+2) → g∧(m+2)), m ≥ 0, k ≥ 0.
m
Then F∗ is a subcomplex of F∗ and the resulting spectral sequence con-
m m+1
verges to Hrel(g). From [7] we have
∗
E2 ≃ HL (g)⊗HR (g), m ≥ 0, k ≥ 0.
m,k k m
From the proof of Lemma (4.1), there is an isomorphism
∂ : HLie(g ; R) −≃→ HR (g ; R) ≃ R.
3 n 0 n
From the long exact sequence relating Lie and Leibniz homology, it follows
that HL (g ; R) → HLie(g ; R) is an isomorphism. Since
2 n 2 n
n
1 ∂ ∂ ∂ ∂
ω˜ = ⊗ − ⊗
n 2 (cid:18)∂xi ∂yi ∂yi ∂xi(cid:19)
Xi=1
10