Download Crash Course on Lie groupoid theory Rui Loja Fernandes Departamento de Matemática

Lie Groupoid Theory
Crash Course on Lie groupoid theory
Rui Loja Fernandes
Departamento de Matemática
Instituto Superior Técnico
University of Minnesota 2012
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Groupoids
A groupoid is a small category where every morphism is an
isomorphism.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Groupoids
A groupoid is a small category where every morphism is an
isomorphism.
G ≡ set of arrows
Rui Loja Fernandes
Crash Course on Lie groupoid theory
M ≡ set of objects.
IST
Lie Groupoid Theory
Groupoids
Groupoids
A groupoid is a small category where every morphism is an
isomorphism.
G ≡ set of arrows
M ≡ set of objects.
source and target maps:
u
•
t(g)
g
•
s(g)
t
G
s
//
M
product:
hg
G (2) = {(h, g) ∈ G × G : s(h) = t(g)}
m : G (2) → G
•
t(h)
u
h
Rui Loja Fernandes
Crash Course on Lie groupoid theory
•
s(h)=t(g)
r
g
•
s(g)
IST
Lie Groupoid Theory
Groupoids
Groupoids
A groupoid is a small category where every morphism is an
isomorphism.
G ≡ set of arrows
M ≡ set of objects.
source and target maps:
u
•
t(g)
g
•
s(g)
t
G
s
//
M
product:
hg
G (2) = {(h, g) ∈ G × G : s(h) = t(g)}
m : G (2) → G
•
t(h)
u
h
Rui Loja Fernandes
Crash Course on Lie groupoid theory
•
s(h)=t(g)
r
g
•
s(g)
IST
Lie Groupoid Theory
Groupoids
Groupoids
A groupoid is a small category where every morphism is an
isomorphism.
G ≡ set of arrows
M ≡ set of objects.
identity:
1x
u : M ,→ G
inverse:
ι:G
/
•
x
t
G
g
t(g) •
4
• s(g)
g −1
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Groupoids
A groupoid is a small category where every morphism is an
isomorphism.
G ≡ set of arrows
M ≡ set of objects.
identity:
1x
u : M ,→ G
inverse:
ι:G
/
•
x
t
G
g
t(g) •
4
• s(g)
g −1
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Groupoids
A groupoid is a small category where every morphism is an
isomorphism.
A morphism of groupoids is a functor F : G → H.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Groupoids
A groupoid is a small category where every morphism is an
isomorphism.
A morphism of groupoids is a functor F : G → H.
This means we have a map F : G → H between the sets of arrows,
and a map f : M → N between the sets of objects, such that:
if g : x −→ y is in G, then F(g) : f (x) −→ f (y ) in H.
if g, h ∈ G are composable, then F(gh) = F(g)F(h).
if x ∈ M, then F(1x ) = 1f (x) .
if g : x −→ y , then F(g −1 ) = F(g)−1 .
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Groupoids: basic concepts
right multiplication by g : y ←− x is a bijection between s-fiber
Rg : s−1 (y ) −→ s−1 (x),
h 7→ hg.
left multiplication by g : y ←− x is a bijection between t-fibers:
Lg : t−1 (x) −→ t−1 (y ),
h 7→ gh.
the isotropy group at x:
Gx = s−1 (x) ∩ t−1 (x).
the orbit through x:
Ox := t(s−1 (x)) = {y ∈ M : ∃ g : x −→ y }
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Groupoids: basic concepts
right multiplication by g : y ←− x is a bijection between s-fiber
Rg : s−1 (y ) −→ s−1 (x),
h 7→ hg.
left multiplication by g : y ←− x is a bijection between t-fibers:
Lg : t−1 (x) −→ t−1 (y ),
h 7→ gh.
the isotropy group at x:
Gx = s−1 (x) ∩ t−1 (x).
the orbit through x:
Ox := t(s−1 (x)) = {y ∈ M : ∃ g : x −→ y }
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Groupoids: basic concepts
right multiplication by g : y ←− x is a bijection between s-fiber
Rg : s−1 (y ) −→ s−1 (x),
h 7→ hg.
left multiplication by g : y ←− x is a bijection between t-fibers:
Lg : t−1 (x) −→ t−1 (y ),
h 7→ gh.
the isotropy group at x:
Gx = s−1 (x) ∩ t−1 (x).
the orbit through x:
Ox := t(s−1 (x)) = {y ∈ M : ∃ g : x −→ y }
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Groupoids: basic concepts
right multiplication by g : y ←− x is a bijection between s-fiber
Rg : s−1 (y ) −→ s−1 (x),
h 7→ hg.
left multiplication by g : y ←− x is a bijection between t-fibers:
Lg : t−1 (x) −→ t−1 (y ),
h 7→ gh.
the isotropy group at x:
Gx = s−1 (x) ∩ t−1 (x).
the orbit through x:
Ox := t(s−1 (x)) = {y ∈ M : ∃ g : x −→ y }
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids
Definition
A Lie groupoid is a groupoid G ⇒ M whose spaces of arrows
and objects are both manifolds, the structure maps s, t, u, m, i
are all smooth maps and such that s and t are submersions.
Basic Properties For a Lie groupoid G ⇒ M and x ∈ M, one has that:
1
the isotropy groups Gx are Lie groups;
2
the orbits Ox are (regular immersed) submanifolds in M;
3
the unit map u : M → G is an embedding;
4
t : s−1 (x) → Ox is a principal Gx -bundle.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids
Definition
A Lie groupoid is a groupoid G ⇒ M whose spaces of arrows
and objects are both manifolds, the structure maps s, t, u, m, i
are all smooth maps and such that s and t are submersions.
Basic Properties For a Lie groupoid G ⇒ M and x ∈ M, one has that:
1
the isotropy groups Gx are Lie groups;
2
the orbits Ox are (regular immersed) submanifolds in M;
3
the unit map u : M → G is an embedding;
4
t : s−1 (x) → Ox is a principal Gx -bundle.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
A picture of a Lie groupoid
t-fibers
hg
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
A picture of a Lie groupoid
t-fibers
hg
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
A picture of a Lie groupoid
t-fibers
hg
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
t-fibers
G
M
s-fibers
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
t-fibers
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
t-fibers
hg
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
t-fibers
hg
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
t-fibers
hg
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
t-fibers
hg
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
The group of bissections Γ(G) is a Lie group (usually,
infinite dimensional):
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
The group of bissections Γ(G) is a Lie group (usually,
infinite dimensional):
If G = G ⇒ {∗}, then Γ(G) = G;
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
The group of bissections Γ(G) is a Lie group (usually,
infinite dimensional):
If G = G ⇒ {∗}, then Γ(G) = G;
If G = M × M ⇒ M, then Γ(G) = Diff(M);
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Lie groupoids vs (infinite dimensional) Lie groups
Definition
A bisection of a Lie groupoid G ⇒ M is a smooth map
b : M → G such that s ◦ b : M → M and t ◦ b : M → M are
diffeomorphisms.
The group of bissections Γ(G) is a Lie group (usually,
infinite dimensional):
One can use bissections to defined the groupoid of jets
J k G ⇒ M of a Lie groupoid G ⇒ M.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Groupoids
Some classes of groupoids
A Lie groupoid G ⇒ M is called:
source k -connected if the s-fibers s−1 (x) are k -connected for
every x ∈ M. When k = 0 we say that G is a s-connected
groupoid, and when k = 1 we say that G is a s-simply
connected groupoid.
étale if its source map s is a local diffeomorphism.
proper if the map (s, t) : G → M × M is a proper map.
Remark. Proper groupoids are the analogue of compact groups in
Lie groupoid theory.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
From Lie groupoids to Lie algebroids
t-fibers
hg
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
From Lie groupoids to Lie algebroids
t-fibers
hg
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
A=Ker d s
M
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
From Lie groupoids to Lie algebroids
t-fibers
hg
t(h)
G
g
h
s(h)=t(g)
M
s(g)
s-fibers
A=Ker d s
M
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
From Lie groupoids to Lie algebroids
t-fibers
hg
h
t(h)
Rg
s(h)=t(g)
G
g
M
s(g)
s-fibers
A=Ker d s
M
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
From Lie groupoids to Lie algebroids
t-fibers
hg
h
t(h)
Rg
s(h)=t(g)
G
g
M
s(g)
s-fibers
A=Ker d s
M
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
From Lie groupoids to Lie algebroids
t-fibers
hg
h
t(h)
Rg
s(h)=t(g)
G
g
M
s(g)
s-fibers
β
[α,β]= [Xα, X ]
A=Ker d s
M
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
From Lie groupoids to Lie algebroids
t-fibers
hg
h
Rg
G
g
ρ
t(h)
s(h)=t(g)
M
s(g)
s-fibers
M
Rui Loja Fernandes
Crash Course on Lie groupoid theory
β
[α,β]= [Xα, X ]
ρ = dt
A=Ker d s
A
IST
Lie Groupoid Theory
Lie Algebroids
Lie algebroids
Definition
A Lie algebroid is a vector bundle A → M with:
(i) a Lie bracket [ , ]A : Γ(A) × Γ(A) → Γ(A);
(ii) a bundle map ρ : A → TM (the anchor);
such that:
[α, f β]A = f [α, β]A + ρ(α)(f )β,
(f ∈ C ∞ (M), α, β ∈ Γ(A)).
Im ρ ⊂ TM is integrable ⇒ characteristic foliation of M;
For each x ∈ M, Ker ρx is a finite dim Lie algebra (isotropy
Lie algebra).
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
Lie algebroids
Definition
A Lie algebroid is a vector bundle A → M with:
(i) a Lie bracket [ , ]A : Γ(A) × Γ(A) → Γ(A);
(ii) a bundle map ρ : A → TM (the anchor);
such that:
[α, f β]A = f [α, β]A + ρ(α)(f )β,
(f ∈ C ∞ (M), α, β ∈ Γ(A)).
Im ρ ⊂ TM is integrable ⇒ characteristic foliation of M;
For each x ∈ M, Ker ρx is a finite dim Lie algebra (isotropy
Lie algebra).
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
Lie algebroids vs (infinite dimensional) Lie algebras
The space of sections Γ(A) is a Lie algebra (usually infinite
dimensional):
If A = g → {∗}, then Γ(A) = g;
If A = TM, then Γ(A) = X(M);
Rmk. If G ⇒ M is a Lie groupoid with Lie algebroid A → M one
can define the exponential map exp : Γ(A) → Γ(G).
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
From Lie algebroids to Lie groupoids
Theorem (Lie I)
Let G be a Lie groupoid with Lie algebroid A. There exists a unique
(up to isomorphism) source 1-connected Lie groupoid Ge with Lie
algebroid A.
Theorem (Lie II)
Let G and H be Lie groupoids with Lie algebroids A and B, where G is
source 1-connected. Given a Lie algebroid homomorphism
φ : A → B, there exists a unique Lie groupoid homomorphism
Φ : G → H with (Φ)∗ = φ.
. . . but Lie III does not hold!
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
From Lie algebroids to Lie groupoids
Theorem (Lie I)
Let G be a Lie groupoid with Lie algebroid A. There exists a unique
(up to isomorphism) source 1-connected Lie groupoid Ge with Lie
algebroid A.
Theorem (Lie II)
Let G and H be Lie groupoids with Lie algebroids A and B, where G is
source 1-connected. Given a Lie algebroid homomorphism
φ : A → B, there exists a unique Lie groupoid homomorphism
Φ : G → H with (Φ)∗ = φ.
. . . but Lie III does not hold!
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
A non-integrable Lie algebroid
Fix ω ∈ Ω2 (M), closed, and take the associated Lie
algebroid A = TM ⊕ R.
Theorem
The Lie algebroid A integrates to a Lie groupoid G iff the group
of spherical periods of ω:
Z
Nx := { ω | γ ∈ π2 (M, x)} ⊂ R
γ
is discrete.
Example
If M = S2 × S2 and ω = dA ⊕ λdA, then Nx is discrete iff λ ∈ Q.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
A non-integrable Lie algebroid
Fix ω ∈ Ω2 (M), closed, and take the associated Lie
algebroid A = TM ⊕ R.
Theorem
The Lie algebroid A integrates to a Lie groupoid G iff the group
of spherical periods of ω:
Z
Nx := { ω | γ ∈ π2 (M, x)} ⊂ R
γ
is discrete.
Example
If M = S2 × S2 and ω = dA ⊕ λdA, then Nx is discrete iff λ ∈ Q.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
A non-integrable Lie algebroid
Fix ω ∈ Ω2 (M), closed, and take the associated Lie
algebroid A = TM ⊕ R.
Theorem
The Lie algebroid A integrates to a Lie groupoid G iff the group
of spherical periods of ω:
Z
Nx := { ω | γ ∈ π2 (M, x)} ⊂ R
γ
is discrete.
Example
If M = S2 × S2 and ω = dA ⊕ λdA, then Nx is discrete iff λ ∈ Q.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
Obstructions to integrability
Theorem (Crainic & RLF, 2003)
For a Lie algebroid A, there exist monodromy groups Nx ⊂ Ax such
that A is integrable iff the groups Nx are uniformly discrete for x ∈ M.
Each Nx is the image of a monodromy map:
^
∂ : π2 (L, x) → G(g
x)
with L the leaf through x and gx := Ker ρx the isotropy Lie algebra.
Corollary
A Lie algebroid A is integrable provided either of the following hold:
(i) All leaves have finite π2 ;
(ii) The isotropy Lie algebras have trivial center.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
Obstructions to integrability
Theorem (Crainic & RLF, 2003)
For a Lie algebroid A, there exist monodromy groups Nx ⊂ Ax such
that A is integrable iff the groups Nx are uniformly discrete for x ∈ M.
Each Nx is the image of a monodromy map:
^
∂ : π2 (L, x) → G(g
x)
with L the leaf through x and gx := Ker ρx the isotropy Lie algebra.
Corollary
A Lie algebroid A is integrable provided either of the following hold:
(i) All leaves have finite π2 ;
(ii) The isotropy Lie algebras have trivial center.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
Obstructions to integrability
Theorem (Crainic & RLF, 2003)
For a Lie algebroid A, there exist monodromy groups Nx ⊂ Ax such
that A is integrable iff the groups Nx are uniformly discrete for x ∈ M.
Each Nx is the image of a monodromy map:
^
∂ : π2 (L, x) → G(g
x)
with L the leaf through x and gx := Ker ρx the isotropy Lie algebra.
Corollary
A Lie algebroid A is integrable provided either of the following hold:
(i) All leaves have finite π2 ;
(ii) The isotropy Lie algebras have trivial center.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
The Maurer-Cartan Form on a Lie Groupoid
For a Lie group G the Maurer-Cartan form is the right-invariant
g-valued 1-form:
ωMC (ξ) = (dg Rg −1 )(ξ) ∈ g.
It satisfies the Maurer-Cartan equation:
dωMC +
1
[ωMC , ωMC ] = 0.
2
For a Lie groupoid G, right translation by g is a diffeomorphism
Rg : s−1 (t(g)) → s−1 (s(g)),
so the Maurer-Cartan form is a s-foliated 1-form.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
The Maurer-Cartan Form on a Lie Groupoid
For a Lie group G the Maurer-Cartan form is the right-invariant
g-valued 1-form:
ωMC (ξ) = (dg Rg −1 )(ξ) ∈ g.
It satisfies the Maurer-Cartan equation:
dωMC +
1
[ωMC , ωMC ] = 0.
2
For a Lie groupoid G, right translation by g is a diffeomorphism
Rg : s−1 (t(g)) → s−1 (s(g)),
so the Maurer-Cartan form is a s-foliated 1-form.
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
The Maurer-Cartan Form on on a Lie Groupoid
Definition
The Maurer-Cartan form on G is the s-foliated A-valued 1-form
T sG
G
defined by
ωMC
t
/A
/M
ωMC (ξ) = (dg Rg −1 )(ξ) ∈ At(g) ,
ξ ∈ Tgs G
The Maurer-Cartan form satisfies the Maurer-Cartan equation:
1
[ωMC , ωMC ]∇ = 0.
2
for an auxiliar connection ∇.
d∇ ωMC +
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
The Maurer-Cartan Form on on a Lie Groupoid
Definition
The Maurer-Cartan form on G is the s-foliated A-valued 1-form
T sG
G
defined by
ωMC
t
/A
/M
ωMC (ξ) = (dg Rg −1 )(ξ) ∈ At(g) ,
ξ ∈ Tgs G
The Maurer-Cartan form satisfies the Maurer-Cartan equation:
1
[ωMC , ωMC ]∇ = 0.
2
for an auxiliar connection ∇.
d∇ ωMC +
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
Overlook
There is a huge body of results on Lie groupoid theory, developed in
the last 15 years, which include, e.g.,:
Normal form results and slice theorems for proper Lie groupoids.
Van Est type theorems for Lie groupoid cohomology and
vanishing theorems for cohomology of proper Lie groupoids.
Equivariant cohomology and classifying spaces for Lie
groupoids.
Deformation cohomology, stability and rigidity results for Lie
algebroids.
Stratification structure for the orbit space of a proper Lie
groupoid.
[. . . ]
Rui Loja Fernandes
Crash Course on Lie groupoid theory
IST
Lie Groupoid Theory
Lie Algebroids
Local form of a Lie algebroid
Choose local coordinates (x 1 , . . . , x d ) on X and local basis of
sections {e1 , . . . , en } of A → M. Then:
[ei , ej ] = Cijk (x)ek ,
ρ(ei ) = Bia
∂
.
∂x a
Then:
The Jacobi identity for [ , ]A gives:
Bja
∂Cki ,l
∂x a
+ Bka
i
∂Cl,j
∂x a
+ Bla
i
∂Cj,k
∂x a
i
i
m
i
m
= Cm,j
Ckm,l + Cmk
Cl,j
+ Cm,l
Cj,k
The fact that ρ : Γ(A) → X(M) preserves Lie brackets gives:
Bib
Rui Loja Fernandes
Crash Course on Lie groupoid theory
∂Bja
∂x b
− Bjb
∂Bia
l
= Ci,j
Bla .
∂x b
IST