Dirac structures TJ Courant: Dirac Manifolds, Trans. AMS, 319 (1990 ...

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Dirac structures
Juan Carlos Marrero
XVIII International Workshop on Geometric Methods in
Theoretical Mechanics, Jaca 2003, August 2003
T.J. Courant: Dirac Manifolds, Trans. A.M.S.,
319 (1990), 631-661.
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Scheme of the talk
I LINEAR DIRAC STRUCTURES
I.1 Linear Dirac structures on a vector space
I.2 Basis of a linear Dirac structure
I.3 Induced linear Dirac structures
II SMOOTH DIRAC STRUCTURES
II.1 Lie algebroids
II.2 Almost-Dirac structures on manifolds
II.3 Dirac structures on manifolds
II.4 The Poisson bracket on admissible functions
II.5 Dirac structures as Poisson structures on
quotient manifolds
II.6 Darboux coordinates around a regular point
II.7 Induced Dirac structures
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Linear Dirac Structures
I.1 Dirac structures on a vector space
V a real vector space of finite dimension
< , > + the natural symmetric pairing on V ⊕ V ∗
< (x, α), (y, β) > + = 1 ∗
{α(y)+β(x)}, (x, α), (y, β) ∈ V ⊕V
2
< , > − the natural skew-symmetric pairing on V ⊕ V ∗
< (x, α), (y, β) > − = 1 ∗
{α(y)−β(x)}, (x, α), (y, β) ∈ V ⊕V
2
L a subspace of V ⊕ V ∗
L a linear Dirac structure on V
⇕
L is maximally isotropic under < , > +
L a subspace of V ⊕ V ∗
dim L = r dim L ∩ V ∗ = r − s
⇓
{(e 1 , α 1 ), . . . , (e s , α s ), (0, α s+1 ), . . . , (0, α r )} a basis of L
and {e 1 , . . . , e s } are linearly independent
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Theorem: L a subspace of V ⊕ V ∗
L a linear Dirac structure on V ⇔ L = L ⊥
⇔ L is isotropic and dim L = dim V
Examples:
i) A : V → V ∗ a linear skew-symmetric map
⇓
graphA = {(x, A(x)) ∈ V ⊕ V ∗ /x ∈ V }
linear Dirac structure on V
ii) B : V ∗ → V a linear skew-symmetric map
⇓
graphB = {(B(x), α) ∈ V ⊕ V ∗ /α ∈ V ∗ }
linear Dirac structure on V
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Characteristic equations of a linear Dirac structure
ρ : V ⊕V ∗ → V, ρ ∗ : V ⊕V ∗ → V ∗ canonical projections
L a linear Dirac structure
L ∩ V ∗ ⊆ ρ(L) 0 , dim ρ(L) 0 = dim L ∩ V ∗
⇓
ρ(L) 0 = L ∩ V ∗
ρ ∗ (L) ⊆ (L ∩ V ) 0 , dim ρ ∗ (L) = dim(L ∩ V ) 0
⇓
ρ ∗ (L) = (L ∩ V ) 0
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Theorem:
A Dirac structure on a vector space V
is equivalently defined as a subspace together with
a skew-symmetric 2-form on it
• E a subspace of V, Ω E : E × E → R a 2-form on E
⇓
L = {(x, α) ∈ V ⊕ V ∗ /x ∈ E, α |E = i(x)Ω E }
linear Dirac structure on V
• L a Dirac structure on V
⇓
E = ρ(L)
Ω E : E × E → R a 2-form on E
Ω E (x, y) = α(y) if (x, α) ∈ L
Remark:
L a Dirac structure on V
⇓
ker Ω ρ(L) = L ∩ V
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L ⊆ V ⊕ V ∗ a linear Dirac structure on V
⇓
π L a 2-vector on V/L ∩ V
(V/L ∩ V ) ∗ ∼ = (L ∩ V ) 0 = ρ ∗ (L)
π L : ρ ∗ (L) × ρ ∗ (L) → R
π L (α, β) = β(u),
if (u, α) ∈ L
π L (α, β) = 0, ∀β ⇔ α(v) = 0, ∀v ∈ ρ(L) ⇔ α ∈ ρ(L) 0 =L ∩ V ∗
ker π L = L ∩ V ∗
rank π L = dim(V/V ∩ L) ∗ − dim(L ∩ V ∗ )
= dim V − (dim L ∩ L + dim L ∩ V ∗ )
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I.2 Basis of a linear Dirac structure
L ⊆ V ⊕ V ∗ a linear Dirac structure on V
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a : R n → V,
dim V = n
A basis of L
⇓
b : R n → V ∗ linear maps
{(ae 1 , be 1 ), . . . , (ae n , be n )} basis of L
Isotropic character of L
⇓
dim L = n
⇓
a ∗ ◦ b + b ◦ a ∗ = 0 ker a ∩ ker b = {0} (1)
a : R n → V, b : R n → V ∗ linear maps which satisfy (1)
L = {(ax, bx)/x ∈ R n } a linear Dirac structure on V
An application:
V a real vector space of finite dimension
< , >: V × V → R an scalar product onV
♭ < , > : V → V ∗ linear isomorphism
I(V ) = {φ : V → V/φ is a linear isometry}
⇓
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Theorem: There exists a one-to-one correspondence
between the set of linear Dirac structures
on V and I(V )
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Proof: ” ← ”) φ : V → V a linear isometry
⇓
L = {(x + φ(x), ♭ < , > (x) − ♭ < , > (φ(x))) ∈ V × V ∗ /x ∈ V }
is a linear Dirac structure on V
” → ” L ⊆ V ⊕ V ∗ a linear Dirac structure on V
a : R n → V,
b : R n → V ∗ a basis of L
¯b = ♭< , > ◦ b : R n → V ∗ → V
⇓
a + ¯b, a − ¯b : R n → V are linear isomorphies
φ = (a + ¯b) ◦ (ā − ¯b) −1 : V → V is a linear isometry
φ does not depend on the basis (a, b) of L
Corollary: There exists a one-to-one correspondence
between linear Dirac structures on R n
and the orthogonal group O(n) of order n
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I.3 Induced linear Dirac structures
V a real vector space of finite dimension
W a subspace of V
L a Dirac structure on V
E = ρ(L), Ω E : E × E → R the corresponding 2-form
E W = E ∩ W = ρ(L) ∩ W, Ω EW = (Ω E ) |E∩W
⇓
L W ≡ (E W , Ω EW )) a linear Dirac structure on W
An alternative description
L W = L ∩ (W ⊕ V ∗ )
L ∩ ({0} ⊕ W 0 )
ker Ω EW = L W ∩ W ∼ = L ∩ (W ⊕ W 0 )
L ∩ ({0} ⊕ W 0 )
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II. Smooth Dirac Structures
II.1 Lie algebroids
A → P a real vector bundle over P
([ , ], ρ) a Lie algebroid structure on A
i) [ , ] : Γ(A) × Γ(A) → Γ(A) is a Lie braket
ii) ρ : A → T P a homomorphism of vector
bundles and
[X, fY ] = f [X, Y ] + ρ(X)(f)Y,
X, Y ∈ Γ(A), f ∈ C ∞ (P, R)
Remark: i) + ii) ⇒ ρ : Γ(A) → X(P ) is a Lie algebra
homomorphism
Theorem: ([ , ], ρ) Lie algebroid structure on A
⇒ x ∈ M → ρ(A)(x) ⊆ T x P defines a generalized
foliation on P
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Examples: i) P a smooth manifold
⇓
(T P, [ , ], Id) is a Lie algebroid on P
ii) (P, π) a Poisson manifold(⇔ π ∈ V 2 (M), [π, π] SN = 0)
⇓
(T ∗ P, [ , ] π , # π ) is a Lie algebroid
[α, β ] π = L #π (α)β − L #π (β)α − d(π(α, β))
β(# π (α)) = π(α, β),
α, β ∈ Ω 1 (M)
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II.2 Almost-Dirac structures on manifolds
P a smooth manifold (dim P = m)
Natural symmetric and skew-symmetric pairings on
T P ⊕ T ∗ P
< (X, α), (Y, β) > + = 1 (α(Y ) + β(X))
2
< (X, α), (Y, β) > − = 1 (α(Y ) − β(X))
2
X, Y ∈ T p P, α, β ∈ T ∗ p P
L a vector subbundle of T P ⊕ T ∗ P
Definition: L is an almost-Dirac structure on P ⇔
L P
is a linear Dirac structure on T p P , for all p ∈ P
Corollary: The following conditions are equivalent:
(i)L is an almost-Dirac structure on P
(ii) L is a maximally isotropic vector subbundle
of T P ⊕ T ∗ P under < , > +
(iii) L = L ⊥
(iv) L is a isotropic vector subbundle of
T P ⊕ T ∗ P of rank m
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L an almost-Dirac structure on P
ρ : T P ⊕ T ∗ P → T P,
canonical projections
ρ ∗ : T P ⊕ T ∗ P → T ∗ P
Characteristic equations of L
ρ(L) 0 = L ∩ T ∗ P, ρ ∗ (L) = (L ∩ T P ) 0
✩
The 2-form Ω L : ρ(L) → ρ(L) ∗
ker Ω L = L ∩ T P
Examples: i) Poisson structures
π : T ∗ P → T P a Poisson structure on P
⇓
L π = graphπ = {(# π (α), α) ∈ T P ⊕ T ∗ P/α ∈ T ∗ P }
almost-Dirac structure on P
ρ(L π ) = im(π) symplectic foliation associated with π
L π ∩ T P = {0} ⇒ ρ ∗ (L π ) = T ∗ P
Ω Lπ
is a non-degenerate 2-form
S p the symplectic leaf passing through p
⇓
Ω Lπ |S p
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is the symplectic 2-form on S p
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ii) Presymplectic structures
Ω : T P → T ∗ P a closed 2-form on P
⇓
L Ω = graphΩ = {(x, i(x)Ω)/x ∈ T P }
an almost-Dirac structure on P
ρ(Ω) = T P
⇓
The foliation associated with L Ω has an only leaf: P
L Ω ∩ T P = ker Ω
Additional structures in i) and ii)
[π, π] SN = 0
dΩ = 0
⇓
Dirac structures on manifolds
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II.3 Dirac structures on manifolds
P a smooth manifold
The Courant bracket on Γ(T P ⊕ T ∗ P ) ∼ = X(P ) × Ω 1 (P )
[(X, α), (Y, β)] = ([X, Y ], L X β − L Y α + d <
(X, α), (Y, β) > − )
L an almost-Dirac structure on P
T L (e 1 , e 2 , e 3 ) =< [e 1 , e 2 ], e 3 > + , e i ∈ Γ(L)
Define T ∈ Γ(∧ 3 (T P ⊕ T ∗ P ) ∗ )
T ((X, α), (Y, β), (Z, γ)) = −{dα(Y, Z)+dβ(Z, X)+dγ(X, Y )
+X(< (Y, β), (Z, γ) > − ) + Y (< (Z, γ), (X, α) > − )+
Z(< (X, α), (Y, β) > − )}
Proposition T L = T |Γ(L)×Γ(L)×Γ(L) . In particular,
T L ∈ Γ(∧ 3 L ∗ )
Proposition: (X, α), (Y, β), (Z, γ) ∈ Γ(L), f ∈ C ∞ (P, R)
⇓
• [(X, α), f(Y, β)] = f[(X, α), (Y, β)] + X(f)(Y, β)
• [[(X, α), (Y, β)], (Z, γ)] + [[(Y, β), (Z, γ)], (X, α)] +
[[(Z, γ), (X, α)], (Y, β)] = (0, dT L ((X, α), (Y, β), (Z, γ))))
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Definition: L a Dirac structure on P ⇔
Γ(L) is closed under the bracket [ , ]
Theorem: If L is an almost-Dirac structure
on P , then the following conditions are
equivalent:
(i) L is a Dirac structure on P
(ii) T L identically vanishes
(iii) (L, [ , ] |Γ(L)×Γ(L) , ρ |L ) is a Lie algebroid
Corollary: L a Dirac structure on P ⇒
ρ(L) generates a singular foliation on P
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L a Dirac structure on P
F a leaf of ρ(L)
⇓
Ω F a 2-form on F
y ∈ F → Ω F (y) = Ω L (y) : T y F × T y F → R
X 1 (y), X 2 (y), X 3 (y) ∈ T y F/X i (y) = ρ L (e i (y))
e i (y) ∈ L y , ∀i ∈ {1, 2, 3}
⇓
dΩ F (y)(X 1 (y), X 2 (y), X 3 (y)) =
T L (y)(e 1 (y), e 2 (y), e 3 (y)) = 0
Theorem: If L is a Dirac structure on P
then there exits a generalized foliation
on P whose leaves are presymplectic
manifolds.
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Examples: i) Poisson manifolds
π a Poisson structure on P
⇓
L π = {(# π (α), α) ∈ T P ⊕ T ∗ P/α ∈ T ∗ P } a Dirac
structure on P
L π → T ∗ P,
((# π )(α), α) → α
a Lie algebroid isomorphism
ρ(L π )(y) = # π (T ∗ y P ),
∀y ∈ P
Ω Lπ is symplectic on each leaf of ρ(L π )
⇓
Characteristic foliation of L π
Symplectic foliation of (P, π)
ii) Presymplectic structures
Ω a closed 2-form on P
⇓
L Ω = {(X, i(X)Ω) ∈ T P ⊕ T ∗ P/X ∈ T P }
a Dirac structure on P
L Ω → T P,
(X, i(X)Ω) → X
a Lie algebroid isomorphism
The characteristic foliation of L Ω has a unique
leaf: the manifold P
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II.4 The Poisson bracket on admissible functions
L a Dirac structure on P
f ∈ C ∞ (P, R)
Definition: f is admissible ⇔ df ∈ ρ ∗ (L) ⇔
∃X f ∈ X(P )/e f = (X f , df) ∈ Γ(L)
f admissible ⇒ X f the hamiltonian vector field of f
C ∞ a (P, R) = {f ∈ C ∞ (P, R)/f is admissible}
f, g ∈ C ∞ a (P, R)
e f = (X f , df) ∈ Γ(L),
e g = (X g , dg) ∈ Γ(L)
⇓
{f, g} L = X f (g) = Ω L (X f , X g )
Theorem: (C ∞ a (P, R), { , } L ) is a Poisson algebra
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Proof: f, g, h ∈ C ∞ a (P, R)
e f = (X f , df) ∈ Γ(L) , e g = (X g , dg) ∈ Γ(L), e h =
(X h , dh) ∈ Γ(L)
i) [e f , e g ] = ([X f , X g ], d{f, g} L ) ∈ Γ(L)
⇓
{f, g} L ∈ C ∞ a (P, R)
ii) fe g + ge f = (fX g + gX f , d(fg)) ∈ Γ(L)
⇓
fg ∈ C ∞ a (P, R), X fg = fX g + gX f
⇓
{fg, h} L = f{g, h} L + g{f, h} L
iii) 0 = T L (e f , e g , e h ) = +
= {f, {g, h} L } L + {g, {h, f} L } L + {h, {f, g} L } L
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II.5 Dirac structures as Poisson structures
on quotient manifolds
L a Dirac structure on P , dim P = m
Assumption 1: L ∩ T P is a vector subbundle of T P
of rank r ⇔ dim L p ∩ T p P = r, ∀p ∈ P
✩
ker Ω L = L ∩ T P
⇓
L ∩ T P defines a foliation F on P
Assumption 2: F is a regular foliation on P in such
a way that the space of leaves ˆP = P/F is a smooth
manifold and the canonical projection τ : P → ˆP is a
submersion
Theorem:
is a Poisson manifold
The quotient space ˆP = P/F
Proof:
ρ ∗ (L) = (L ∩ T P ) 0
⇓
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τ ∗ : C ∞ ( ˆP , R) → C ∞ a (P, R) is a bijection
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ˆπ L Poisson 2-vector on ˆP
p ∈ P ⇒ The 2-vector π L (p) on the vector space
T p P/F p ≡ T τ(p) ˆP ≡ Tτ(p) (P/F)
⇓
ˆπ L (τ(p)) = π L (p)
rank (ˆπ L (τ(p))) = m − r − dim(L p ∩ T ∗ p P )
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II.5 Darboux coordinates around a regular
point
L a Dirac structure on P
x ∈ P
Definition: x is regular ⇔ ∃ an open
subset U of P / x ∈ U and
dim ρ(L)(x ′ ) = c,
dim ρ ∗ (L)(x ′ ) = k
for all x ′ ∈ U
Theorem (Darboux theorem): x ∈ P a
regular point ⇒ ∃ an open subset U such
that x ∈ U and we may find coordinates
(x i , q j , p j , c k ) on U in such a way that
{( ∂
∂x i , 0), ( ∂
∂p j
, dq j ), (− ∂
∂q j , dp j ), (0, dc k )}
is a local basis of L in U
(x i , q j , p j , c k ) ≡ Darboux coordinates for L
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Proof: ∃ an open subset U of P/x ∈ U and
i) Û = U/F U is a smooth manifold and
τ U : U → Û is a submersion, where
x ∈ U → F U (x) = L(x) ∩ T x U
ii)
Û is a regular Poisson manifold
Darboux Theorem (Weinstein) for regular Poisson
structures
+
L is a maximally isotropic vector subbundle of
T P ⊕ T ∗ P
The existence of Darboux coordinates for L
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II.7 Induced Dirac structures
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L an almost-Dirac structure on P
Q a submanifold of P
x ∈ Q ⇒ L Q (x) = L(x) ∩ (T xQ ⊕ T ∗ x P )
L(x) ∩ (T x Q) 0
L Q (x) is a linear Dirac structure of T x Q
Corollary: The following conditions are
equivalent:
i) dim(L(x)) ∩ (T x Q ⊕ T ∗ x P )) is constant, for
all x ∈ Q
ii) dim((L(x) ∩ T x Q) 0 ) is constant, for all
x ∈ Q
i) and ii) ⇒ L Q is an almost-Dirac structure
on Q
Definition: Q is clean with respect to L
if i) or (ii) holds
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Q is clean with respect to L
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Γ(L Q ) ≡ Γ(L |Q ∩ (T Q ⊕ T ∗ P )) modulo Γ(L |Q ∩ T Q 0 )
( ˜X, ˜α), (Ỹ , ˜β), ( ˜Z, ˜γ) ∈ Γ(L)
⇓
(X, α), (Y, β), (Z, γ) ∈ Γ(L Q )
X = ˜X |Q , Y = Ỹ|Q, Z = ˜Z |Q
α = i ∗ ˜α, β = i ∗ ˜β, γ = i∗˜γ
T L (( ˜X, ˜α), (Ỹ , ˜β), ( ˜Z, ˜γ)) |Q = T LQ ((X, α), (Y, β), (Z, γ))
⇓
Theorem: i) L a Dirac structure on P , Q a
clean submanifold with respect to L ⇒ L Q
a Dirac structure on Q
ii) The characteristic foliation associated
with L Q is
L Q ∩ T Q ∼ = L Q ∩ (T Q ⊕ T Q 0 )
L |Q ∩ T Q 0
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Poisson reduction via Dirac structures
π a Poisson structure on P
⇓
L π = {(# π (α), α) ∈ T P ⊕ T ∗ P/α ∈ T ∗ P }
a Dirac structure on P
• G a Lie group
φ : G × P → P a Poisson action
(⇔ φ g : P → P a Poisson automorphism, ∀g ∈ G)
•
J : P → g ∗ an equivariant momentum map
i) ξ ∈ g ⇒ ξ p = X π Ĵ(ξ)
Ĵ(ξ) : P → R,
x ∈ P → J(x)(ξ) ∈ R
ii)
P
J
✲ g
∗
✫
φ g
❄
P
J
✲
g ∗
Ad ∗ g
❄
Ad ∗ g ◦ J = J ◦ φ g , ∀g ∈ G
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µ a regular value of J
⇓
✩
Q = J −1 (µ) a regular submanifold of P
x ∈ Q
i) ξ p (x) ∈ T x Q ⇔ ξ ∈ g µ = {ξ ∈ g/ξ g ∗(µ) = 0}
ii) S x = {ξ ∈
g/d(Ĵ(ξ))(x) = 0}
g x = {ξ ∈ g/ξ p (x) = 0}
⇓
L π (x) ∩ (T x Q) 0 ∼ = g x /S x
Q is a clean submanifold with respect to L π
⇕
dim g x /S x = c, ∀x ∈ Q
Q is clean with respect to L π
⇕
(L π ) Q a Dirac structure on Q
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φ µ : G µ × Q → Q
action of G µ = {g ∈ G/Ad ∗ gµ = µ} on Q
i) (L π ) Q is invariant under the action φ µ
ii) (L π ) Q (x) ∩ T x Q ∼ = g µ /g x , ∀x ∈ Q
⇓
The leaves of the characteristic foliation of (Lπ) Q are
orbits of the action φ µ
Assumption: The space of the orbits of the
action φ µ , ˆQµ = Q/G µ , is a smooth manifold
and the canonical projection Q → ˆQ µ is a
submersion
i)⇒ (L π ) Q induces a Dirac structure ( ̂L π ) Q on ˆQ µ
ii) ⇒
The characteristic distribution of ( ̂Lπ ) Q is zero
⇓
ˆQ µ is a Poisson manifold
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