Crash Course on Lie groupoid theory - math.umn.edu Course on Lie groupoid theory Rui Loja Fernandes...

Lie Groupoid Theory

Crash Course on Lie groupoid theory

Rui Loja Fernandes

Departamento de MatemáticaInstituto Superior Técnico

University of Minnesota 2012

Rui Loja Fernandes IST

Lie Groupoid Theory

Groupoids

A groupoid is a small category where every morphism is anisomorphism.

Lie Groupoid Theory

Groupoids

G ≡ set of arrows M ≡ set of objects.

Lie Groupoid Theory

Groupoids

G ≡ set of arrows M ≡ set of objects.source and target maps:

•t(g)

•s(g)

//t // M

product:

•t(h)

•s(h)=t(g)

•s(g)

G(2) = (h, g) ∈ G × G : s(h) = t(g)

m : G(2) → G

Lie Groupoid Theory

Groupoids

G ≡ set of arrows M ≡ set of objects.source and target maps:

•t(g)

•s(g)

//t // M

product:

•t(h)

•s(h)=t(g)

•s(g)

G(2) = (h, g) ∈ G × G : s(h) = t(g)

m : G(2) → G

Lie Groupoid Theory

Groupoids

G ≡ set of arrows M ≡ set of objects.identity:

u : M → G •x

inverse: ι : G // G t(g)•

44•s(g)

Lie Groupoid Theory

Groupoids

G ≡ set of arrows M ≡ set of objects.identity:

u : M → G •x

inverse: ι : G // G t(g)•

44•s(g)

Lie Groupoid Theory

Groupoids

A morphism of groupoids is a functor F : G → H.

Lie Groupoid Theory

Groupoids

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.

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

Lie Groupoid Theory

Groupoids

Rg : s−1(y) −→ s−1(x), h 7→ hg.

Lg : t−1(x) −→ t−1(y), h 7→ gh.

Gx = s−1(x) ∩ t−1(x).

Ox := t(s−1(x)) = y ∈ M : ∃ g : x −→ y

Lie Groupoid Theory

Groupoids

Rg : s−1(y) −→ s−1(x), h 7→ hg.

Lg : t−1(x) −→ t−1(y), h 7→ gh.

Gx = s−1(x) ∩ t−1(x).

Ox := t(s−1(x)) = y ∈ M : ∃ g : x −→ y

Lie Groupoid Theory

Groupoids

Rg : s−1(y) −→ s−1(x), h 7→ hg.

Lg : t−1(x) −→ t−1(y), h 7→ gh.

Gx = s−1(x) ∩ t−1(x).

Ox := t(s−1(x)) = y ∈ M : ∃ g : x −→ y

Lie Groupoid Theory

Groupoids

Lie groupoids

DefinitionA Lie groupoid is a groupoid G ⇒ M whose spaces of arrowsand objects are both manifolds, the structure maps s, t,u,m, iare 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.

Lie Groupoid Theory

Groupoids

Lie groupoids

DefinitionA Lie groupoid is a groupoid G ⇒ M whose spaces of arrowsand objects are both manifolds, the structure maps s, t,u,m, iare 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.

Lie Groupoid Theory

Groupoids

A picture of a Lie groupoid

t-fibers

s-fibers

t(h) s(h)=t(g)

Lie Groupoid Theory

Groupoids

t-fibers

s-fibers

t(h) s(h)=t(g)

Lie Groupoid Theory

Groupoids

t-fibers

s-fibers

t(h) s(h)=t(g)

Lie Groupoid Theory

Groupoids

Lie groupoids vs (infinite dimensional) Lie groups

DefinitionA bisection of a Lie groupoid G ⇒ M is a smooth mapb : M → G such that s b : M → M and t b : M → M arediffeomorphisms.

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);

One can use bissections to defined the groupoid of jetsJkG ⇒ M of a Lie groupoid G ⇒ M.

Lie Groupoid Theory

Groupoids

t-fibers

s-fibers

t(h) s(h)=t(g)

Lie Groupoid Theory

Groupoids

t-fibers

s-fibers

t(h) s(h)=t(g)

Lie Groupoid Theory

Groupoids

t-fibers

s-fibers

t(h) s(h)=t(g)

Lie Groupoid Theory

Groupoids

t-fibers

s-fibers

t(h) s(h)=t(g)

Lie Groupoid Theory

Groupoids

t-fibers

s-fibers

t(h) s(h)=t(g)

Lie Groupoid Theory

Groupoids

t-fibers

s-fibers

t(h) s(h)=t(g)

Lie Groupoid Theory

Groupoids

Lie Groupoid Theory

Groupoids

If G = G ⇒ ∗, then Γ(G) = G;

If G = M ×M ⇒ M, then Γ(G) = Diff(M);

Lie Groupoid Theory

Groupoids

Lie Groupoid Theory

Groupoids

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 forevery x ∈ M. When k = 0 we say that G is a s-connectedgroupoid, and when k = 1 we say that G is a s-simplyconnected 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 inLie groupoid theory.

Lie Groupoid Theory

Lie Algebroids

From Lie groupoids to Lie algebroids

s-fibers

Ms(g)s(h)=t(g)t(h)

t-fibers

A=Ker d sM

[X , X ]β

[α,β]=α

Lie Groupoid Theory

Lie Algebroids

s-fibers

Ms(g)s(h)=t(g)t(h)

t-fibers

A=Ker d sM

[X , X ]β

[α,β]=α

Lie Groupoid Theory

Lie Algebroids

s-fibers

Ms(g)s(h)=t(g)t(h)

t-fibers

A=Ker d sM

[X , X ]β

[α,β]=α

Lie Groupoid Theory

Lie Algebroids

s-fibers

Ms(g)s(h)=t(g)t(h)

t-fibers

A=Ker d sM

[X , X ]β

[α,β]=α

Lie Groupoid Theory

Lie Algebroids

s-fibers

Ms(g)s(h)=t(g)t(h)

t-fibers

A=Ker d sM

[X , X ]β

[α,β]=α

Lie Groupoid Theory

Lie Algebroids

s-fibers

Ms(g)s(h)=t(g)t(h)

t-fibers

A=Ker d sM

[X , X ]β

[α,β]=α

Lie Groupoid Theory

Lie Algebroids

s-fibers

Ms(g)s(h)=t(g)t(h)

t-fibers

A=Ker d sM

[X , X ]β

[α,β]=α

Lie Groupoid Theory

Lie Algebroids

Lie algebroids

DefinitionA 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, fβ]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 (isotropyLie algebra).

Lie Groupoid Theory

Lie Algebroids

Lie algebroids

DefinitionA 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, fβ]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 (isotropyLie algebra).

Lie Groupoid Theory

Lie Algebroids

Lie algebroids vs (infinite dimensional) Lie algebras

The space of sections Γ(A) is a Lie algebra (usually infinitedimensional):

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 onecan define the exponential map exp : Γ(A)→ Γ(G).

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 G with Liealgebroid A.

Theorem (Lie II)

Let G and H be Lie groupoids with Lie algebroids A and B, where G issource 1-connected. Given a Lie algebroid homomorphismφ : A→ B, there exists a unique Lie groupoid homomorphismΦ : G → H with (Φ)∗ = φ.

. . . but Lie III does not hold!

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 G with Liealgebroid A.

Theorem (Lie II)

Let G and H be Lie groupoids with Lie algebroids A and B, where G issource 1-connected. Given a Lie algebroid homomorphismφ : A→ B, there exists a unique Lie groupoid homomorphismΦ : G → H with (Φ)∗ = φ.

. . . but Lie III does not hold!

Lie Groupoid Theory

Lie Algebroids

A non-integrable Lie algebroid

Fix ω ∈ Ω2(M), closed, and take the associated Liealgebroid A = TM ⊕ R.

TheoremThe Lie algebroid A integrates to a Lie groupoid G iff the groupof spherical periods of ω:

Nx := ∫γω | γ ∈ π2(M, x) ⊂ R

is discrete.

Example

If M = S2 × S2 and ω = dA⊕ λdA, then Nx is discrete iff λ ∈ Q.Rui Loja Fernandes IST

Lie Groupoid Theory

Lie Algebroids

Nx := ∫γω | γ ∈ π2(M, x) ⊂ R

is discrete.

Example

Lie Groupoid Theory

Lie Algebroids

Nx := ∫γω | γ ∈ π2(M, x) ⊂ R

is discrete.

Example

Lie Groupoid Theory

Lie Algebroids

Obstructions to integrability

Theorem (Crainic & RLF, 2003)

For a Lie algebroid A, there exist monodromy groups Nx ⊂ Ax suchthat 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(gx )

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.

Lie Groupoid Theory

Lie Algebroids

∂ : π2(L, x)→ G(gx )

Corollary

Lie Groupoid Theory

Lie Algebroids

∂ : π2(L, x)→ G(gx )

Corollary

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-invariantg-valued 1-form:

ωMC(ξ) = (dgRg−1 )(ξ) ∈ g.

It satisfies the Maurer-Cartan equation:

dωMC +12

[ωMC, ωMC] = 0.

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.

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-invariantg-valued 1-form:

ωMC(ξ) = (dgRg−1 )(ξ) ∈ g.

It satisfies the Maurer-Cartan equation:

dωMC +12

[ωMC, ωMC] = 0.

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.

Lie Groupoid Theory

Lie Algebroids

The Maurer-Cartan Form on on a Lie Groupoid

DefinitionThe Maurer-Cartan form on G is the s-foliated A-valued 1-form

T sGωMC //

t// Mdefined by

ωMC(ξ) = (dgRg−1 )(ξ) ∈ At(g), ξ ∈ T sg G

The Maurer-Cartan form satisfies the Maurer-Cartan equation:

d∇ωMC +12

[ωMC, ωMC]∇ = 0.

for an auxiliar connection ∇.Rui Loja Fernandes IST

Lie Groupoid Theory

Lie Algebroids

The Maurer-Cartan Form on on a Lie Groupoid

DefinitionThe Maurer-Cartan form on G is the s-foliated A-valued 1-form

T sGωMC //

t// Mdefined by

ωMC(ξ) = (dgRg−1 )(ξ) ∈ At(g), ξ ∈ T sg G

The Maurer-Cartan form satisfies the Maurer-Cartan equation:

d∇ωMC +12

[ωMC, ωMC]∇ = 0.

for an auxiliar connection ∇.Rui Loja Fernandes IST

Lie Groupoid Theory

Lie Algebroids

OverlookThere is a huge body of results on Lie groupoid theory, developed inthe 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 andvanishing theorems for cohomology of proper Lie groupoids.

Equivariant cohomology and classifying spaces for Liegroupoids.

Deformation cohomology, stability and rigidity results for Liealgebroids.

Stratification structure for the orbit space of a proper Liegroupoid.

[. . . ]

Lie Groupoid Theory

Lie Algebroids

Local form of a Lie algebroidChoose local coordinates (x1, . . . , xd ) on X and local basis ofsections e1, . . . ,en of A→ M. Then:

[ei ,ej ] = Ckij (x)ek , ρ(ei ) = Ba

∂xa .

The Jacobi identity for [ , ]A gives:

Baj∂C i

∂xa + Bak

∂C il,j

∂xa + Bal

∂C ij,k

∂xa =(C i

m,jCmk,l + C i

mk Cml,j + C i

m,lCmj,k)

The fact that ρ : Γ(A)→ X(M) preserves Lie brackets gives:

∂Baj

∂xb − Bbj∂Ba

i∂xb = C l

i,j Bal .

Crash Course on Lie groupoid theory - math.umn.edu Course on Lie groupoid theory Rui Loja Fernandes...

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