Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids
In this note, we prove that intrinsic characteristic classes of Lie algebroids - which in degree one recover the modular class - behave functorially with respect to arbitrary transverse maps, and in particular are weak Morita invariants. In the modular case, this result appeared in [Kosmann-Schwarzb...
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Інститут математики НАН України
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| Цитувати: | Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids / P. Frejlich // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 32 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860269564555689984 |
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| author | Frejlich, P. |
| author_facet | Frejlich, P. |
| citation_txt | Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids / P. Frejlich // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 32 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this note, we prove that intrinsic characteristic classes of Lie algebroids - which in degree one recover the modular class - behave functorially with respect to arbitrary transverse maps, and in particular are weak Morita invariants. In the modular case, this result appeared in [Kosmann-Schwarzbach Y., Laurent-Gengoux C., Weinstein A., Transform. Groups 13 (2008), 727-755], and with a connectivity assumption which we here show to be unnecessary, it appeared in [Crainic M., Comment. Math. Helv. 78 (2003), 681-721] and [Ginzburg V.L., J. Symplectic Geom. 1 (2001), 121-169].
|
| first_indexed | 2025-12-07T19:04:40Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 124, 12 pages
Morita Invariance of Intrinsic Characteristic Classes
of Lie Algebroids
Pedro FREJLICH
UFRGS, Departamento de Matemática Pura e Aplicada, Porto Alegre, Brasil
E-mail: frejlich.math@gmail.com
Received June 18, 2018, in final form November 08, 2018; Published online November 15, 2018
https://doi.org/10.3842/SIGMA.2018.124
Abstract. In this note, we prove that intrinsic characteristic classes of Lie algebroids –
which in degree one recover the modular class – behave functorially with respect to arbitrary
transverse maps, and in particular are weak Morita invariants. In the modular case, this
result appeared in [Kosmann-Schwarzbach Y., Laurent-Gengoux C., Weinstein A., Trans-
form. Groups 13 (2008), 727–755], and with a connectivity assumption which we here show
to be unnecessary, it appeared in [Crainic M., Comment. Math. Helv. 78 (2003), 681–721]
and [Ginzburg V.L., J. Symplectic Geom. 1 (2001), 121–169].
Key words: Lie algebroids; modular class; characteristic classes; Morita equivalence
2010 Mathematics Subject Classification: 53D17; 57R20
1 Introduction
A Lie algebroid A on a manifold M gives rise to intrinsic characteristic classes
char(A) ∈ Hodd(A)
in Lie algebroid cohomology, which obstruct the existence of a metric g on the fibres of Ad(A) :=
A⊕ TM , and a connection ∇ : TM y A, whose induced basic connection ∇bas : Ay Ad(A),
∇bas
a (b, u) =
(
∇%Aba+ [a, b]A, %A∇ua+ [%Aa, u]
)
, a, b ∈ Γ(A), u ∈ X(M),
is g-metric:
Lag(s, s′) = g
(
∇bas
a s, s′
)
+ g
(
s,∇bas
a s′
)
, s, s′ ∈ Γ(Ad(A)).
For example, the familiar statement that there exists a Riemannian (i.e., a torsion-free and
metric) connection associated with a Riemannian metric on M implies that, for tangent bundles
A = TM , these characteristic classes char(A) vanish.
In degree one, char1(A) recovers the modular class of A [10], the obstruction to the existence
of an invariant transverse measure, first discovered in the context of Poisson manifolds [24, 31]
as the ‘Poisson analogue of the modular automorphism group of a von Neumann algebra’. There
is an extensive literature about this important class (see the survey [21]), which is arguably the
only reasonably well-understood among the intrinsic ones. It has been generalized to various
geometric contexts [3, 16, 19, 20, 22, 23, 28, 29, 30], and plays a fundamental role in many
constructions [3, 7, 8, 10, 12, 18, 25, 27, 32].
The purpose of this short note is to show that intrinsic characteristic classes are invariant
under the following version of weak Morita equivalence [13, Section 6.2]: two Lie algebroids B
on N and A on M are weak Morita equivalent if there are submersions N
s← Σ
t→M , and a Lie
algebroid isomorphism (Φ, id) : s!(B) ∼−→ t!(A) between the pullbacks of B and A to Σ. This
establishes a correspondence between cohomology classes in H(B) and H(A), and the claim is
that char(B) and char(A) are related. In fact, we prove slightly more:
mailto:frejlich.math@gmail.com
https://doi.org/10.3842/SIGMA.2018.124
2 P. Frejlich
Theorem. Intrinsic characteristic classes are functorial with respect to transverse maps: if
φ : N →M is transverse to a Lie algebroid A on M , then char(φ!(A)) = φ∗ char(A).
Versions of this result have appeared in the literature in various forms; we here quote those
most pertinent to our setting.
In [14, Theorem 4.2] it was shown, building on previous work [15], that the modular class is
a Morita invariant for locally unimodular Poisson manifolds. Shortly afterwards, secondary and
intrinsic characteristic classes were introduced (see [4, 5, 6, 11, 26]), and in [5, Corollary 8] it
was proved that the intrinsic characteristic classes of Poisson manifolds of degree (2q − 1) are
invariant under Morita equivalences whose fibres are at least homologically (2q − 1)-connected;
it was later extended to weak Morita equivalences of Lie algebroids under a similar connectivity
condition [13, Example 6.16].
More recently, it was proved in [22, Theorem 3.10] that the modular class is functorial with
respect to arbitrary transverse maps – thus dropping the connectivity condition – and the authors
pose the question in [22, (iii), p. 729] about the behavior of higher intrinsic characteristic classes
under morphisms. It was this question that piqued our interest, and which our Main Theorem
seeks to answer.
Let us conclude these introductory remarks by pointing out that, in light of the corre-
spondence between 2-term representations up to homotopy and VB-algebroids (see, e.g., [17]),
it would be interesting to revisit the discussion below in the context of VB-algebroids and
groupoids, and to explain the relationship with the results in [9] – which develops a similar line
of inquiry, and through methods that bear great resemblance to the ones employed here.1
The paper is organized as follows: our conventions are discussed in Section 2, where we
summarize the construction of primary, secondary and intrinsic characteristic classes of Lie
algebroids from [4, 6], referring there to proofs. In Section 3 we prove our Main Theorem: as
we explain there, this result is a straightforward consequence of the case of pulling back a Lie
algebroid A on M by a submersion p : Σ → M , and our proof, in that case, reduces to the
construction of appropriate connection and metric on Ad
(
p!(A)
)
, so that the adjoint connection
of p!(A) splits as a direct sum of the pullback of the adjoint connection of A and a metric
subconnection.
2 Characteristic classes
In this section, we give a summary of the main results and constructions needed to contextualize
our discussion, referring to the appropriate references for further details.
1. For vector bundles E and D on M , we denote by Ωp
nl(E;D) the space of nonlinear forms
of degree p on E with values in D – that is, the linear subspace of Hom(∧pΓ(E),Γ(D))
consisting of those elements ω which decrease support, in the sense that ω(e1, . . . , ep)
is identically zero around any point around which some ei ∈ Γ(E) vanishes identically.
When D is the trivial line bundle, we write Ωp
nl(E), and we note that Ωp
nl(E;D) is a module
over Ωp
nl(E). Linear forms ω ∈ Ωp(E,D) = Γ(∧pE∗⊗D) are identified with those elements
of Ωp
nl(E;D) which are C∞(M)-linear in their entries. There are obvious variations when
D is complex or graded; see [1, 4].
2. Let A be a Lie algebroid on M , and let D be the graded, complex vector bundle D0⊕D1,
equipped with an odd endomorphism
∂ =
(
0 ∂1
0
∂0
1 0
)
: D −→ D, ∂2 = 0.
1I thank the anonymous referees for bringing this to my attention.
Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids 3
A nonlinear connection of A on D is a linear map ∇ : Γ(A) → End(Γ(D)), such that, for
all a ∈ Γ(A),
a) ∇ is a local operator;
b) ∇a preserves parity;
c) ∇a commutes with ∂;
d) ∇a satisfies ∇afs = f∇as+ (Laf)s for all f ∈ C∞(M), s ∈ Γ(D).
3. A nonlinear connection of A on D induces:
• a derivation of degree one d∇ : Ωnl(A;D) → Ωnl(A;D), d∇η(a0, . . . , ap) being given
by the usual formula
p∑
i=0
(−1)i∇aiη(a0, . . . , âi, . . . , ap) +
∑
i<j
(−1)i+jη([ai, aj ], a0, . . . , âi, . . . , âj , . . . , ap);
• a dual nonlinear connection ∇∨ of A on D∗, defined by the condition that
La〈θ, s〉 = 〈∇∨a θ, s〉+ 〈θ,∇as〉, a ∈ Γ(A), s ∈ Γ(D), θ ∈ Γ(D∗),
• a nonlinear connection of A on End(D), given by ∇aT = [∇a, T ], whose induced
derivation d∇ : Ωnl(A; End(D)) → Ωnl(A; End(D)) is given by the graded commuta-
tor [∇, ·].
4. A Hermitian metric h on D, regarded as a complex-antilinear map D → D∗, conjugates
a nonlinear connection ∇ of A on D to an h-dual nonlinear connection ∇h of A on D,
given by ∇ha := h−1◦∇∨a ◦h. If ∇ = ∇h, we say that h is invariant under ∇, or that ∇ is h-
metric. Note that every Hermitian metric h is invariant under some nonlinear connection;
e.g., ∇m := 1
2
(
∇+∇h
)
.
5. A nonlinear subconnection of a nonlinear connection ∇ of A on D is the restriction ∇′ =
∇|D′ of ∇ to an invariant subbundle D′, i.e., one for which ∇aΓ(D′) ⊂ Γ(D′) for all
a ∈ Γ(A). If that is the case, there is an induced quotient nonlinear connection ∇/D′
of A on D/D′, (∇/D′)a[s] = [∇as]. When D = D′ ⊕ D′′ where ∇′′ := ∇|D′′ is another
subconnection, we say that ∇ splits as a direct sum, and write ∇ = ∇′ ⊕∇′′.
6. For a nonlinear connection ∇ of A on D, d2
∇ = R∇∧, where R∇ denotes the curvature
of ∇,
R∇ ∈ Ω2
nl(A; End(D)), R∇(a, b) = [∇a,∇b]−∇[a,b], a, b ∈ Γ(A),
and it is always the case that d∇R∇ = 0. If R∇ = 0, we call ∇ a nonlinear representation.
Because the supertrace Trs(T ) = Tr(T00) − Tr(T11) induces a linear map intertwining
derivations,
Trs : Ωnl(A; End(D)) −→ Ωnl(A;C), dA Trs = Trs d∇,
it follows in general that Trs
(
Rq∇
)
∈ Ω2q
nl (A;C) are dA-closed for every integer q; see [4].
7. If (Φ, φ) : B → A is a morphism of Lie algebroids, and ∇ is a nonlinear connection of A
on D, there is an induced pullback nonlinear connection (Φ, φ)!∇ of B on φ∗(D),
(Φ, φ)!∇aφ†(s) := φ†(∇Φ(a)s), a ∈ Γ(φ!(A)), s ∈ Γ(D),
4 P. Frejlich
in which case (Φ, φ) : (Φ, φ)!∇ → ∇ defines a pullback morphism of nonlinear connections,
in the sense that the induced linear map
(Φ, φ)∗ : Ωnl(A;D)→ Ωnl(B;φ∗(D))
intertwines the derivations d∇ and d(Φ,φ)!∇.2 If φ : N →M is a smooth map transverse to
a Lie algebroid A on M , i.e.,
φ∗(TxN) + %A(Aφ(x)) = Tφ(x)M, x ∈ N,
then there is a pullback Lie algebroid φ!(A) := TN ×TM A on N , and an induced pullback
morphism of Lie algebroids
(
φ̃, φ
)
: φ!(A) → A. In this case, we will write simply φ!∇
and φ∗ instead of
(
φ̃, φ
)!∇ and
(
φ̃, φ
)∗
.
8. A nonlinear connection is a connection tout court if
∇fas = f∇as, f ∈ C∞(M), a ∈ Γ(A), s ∈ Γ(D),
that is, if it is C∞(M)-linear in the Γ(A)-entry, in which case we write ∇ : A y D. Two
nonlinear connections∇0, ∇1 are equivalent provided that there exists θ ∈ Ω1
nl(A; End(D)),
such that
∇1
a −∇0
a = [θ(a), ∂], a ∈ Γ(A),
in which case Trs
(
Rq∇0
)
= Trs
(
Rq∇1
)
for all q (see [6]). A nonlinear connection ∇ of A
on D will be called a connection up to homotopy if it is equivalent to a connection; in
this case, we will write ∇ : A y D. Both connections and connections up to homotopy
are preserved by all operations on nonlinear connections described in items 3–7. Note
that, for a connection up to homotopy ∇, Trs
(
Rq∇
)
are linear forms, Trs
(
Rq∇
)
∈ Ω2q(A).
A representation up to homotopy3 is a connection up to homotopy for which R∇ vanishes
identically, in which case d∇ turns Ω•nl(A;D) into a cochain complex.
In the remainder of this section, we recall the discussion in [6], referring there to proofs and
further details.
Lemma. There is a rule cs which assigns to all non-negative integers p, q > 0 and connections
up to homotopy ∇0, . . . ,∇p : A y D, a cochain
csq(∇0, . . . ,∇p) ∈ Ω2q−p(A;C)
with the property that, for every permutation σ and Hermitian metric h on D:
CS1) csq(∇) = Trs
(
Rq∇
)
,
CS2) csq(∇σ(0), . . . ,∇σ(p)) = (−1)σcsq(∇0, . . . ,∇p),
CS3) dAcsq(∇0, . . . ,∇p) =
p∑
i=0
(−1)icsq
(
∇0, . . . , ∇̂i, . . . ,∇p
)
,
CS4) csq
(
∇h0 , . . . ,∇hp
)
= (−1)qcsq(∇0, . . . ,∇p).
2As explained in [22], it is best to think that a connection ∇ : A y D induces the derivation d∇∨ , simply
because the map of modules induced by a pair of vector bundle maps Φ: B → A and Ψ: DB → DA covering the
same smooth map φ : N →M is (Φ,Ψ, φ)∗ : Ω(A;D∗A)→ Ω(B;D∗B). A morphism from a connection∇B : B y DB
to a connection ∇A : Ay DA is then a such triple (Φ,Ψ, φ) for which (Φ,Ψ, φ)∗ intertwines the derivations d∇∨
A
and d∇∨
B
. When Ψ is fibrewise an isomorphism – as in the case of a pullback morphism – we may dualize the
construction above to a map of modules Ω(A;DA)→ Ω(B;DB) intertwining the derivations d∇A and d∇B .
3For the convenience of the reader, we chose to maintain the term representation up to homotopy as it appears
in [4, 6], in spite of the fact that terminology has come to mean something else [1].
Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids 5
Such cochains are given explicitly by
csq(∇0, . . . ,∇p) :=
{
Trs
(
Rq∇0
)
if p = 0,
(−1)b
p+1
2
c −
∫
∆p Trs
(
Rq∇aff
)
if p > 0,
where btc the greatest integer no greater than t and:
• −
∫
∆p : Ω•
(
pr!(A);C
)
→ Ω•−p(A;C) denotes the linear map of fibre integration4 associated
to the canonical projection from the product of M with the standard p-simplex pr : M ×
∆p→M ;
• ∇aff : pr!(A) y pr∗(D) denotes the connection up to homotopy ∇aff =
p∑
i=0
ti pr!(∇i).
Given a connection up to homotopy ∇ : A y D, define
cs(∇) = Trs(exp(iR∇)) =
∑ iq
q!
csq(∇) ∈ Ω(A;C).
Proposition 1 (primary characteristic classes).
a) For a connection up to homotopy, we have dAcs(∇) = 0 and cs(∇0⊕∇1) = cs(∇0)+cs(∇1);
b) for all Lie algebroid morphisms (Φ, φ) : B → A and connection up to homotopy ∇ : Ay D,
we have cs((Φ, φ)!(∇)) = (Φ, φ)∗cs(∇);
c) the cohomology class [cs(∇)] does not depend on the choice of connection up to homotopy ∇;
d) [cs(∇)]∈H2•(A) is a real cohomology class lying in the image of the map (%A, id)∗ : H(M)→
H(A) induced by the anchor of A;
e) [cs(∇)] ∈ H4•(A) if ∇ is a real5 connection up to homotopy.
We call the Chern character of D the element ch(D) ∈ H(A) represented by cs(∇), for
some connection up to homotopy ∇ : A y D. We regard it as a primary characteristic class,
obstructing the existence of a representation up to homotopy of A on D. The vanishing of
cs(∇) allows one to define secondary characteristic classes u(∇) ∈ Hodd(A), which obstruct the
existence of an invariant metric. For a connection up to homotopy ∇ : A y D and a Hermitian
metric h on D, define
u(∇, h) :=
∑
iq+1csq
(
∇,∇h
)
∈ Ωodd(A;C).
Proposition 2 (secondary characteristic classes).
a) The cochains u(∇, h) are real;
b) for all Lie algebroid morphism (Φ, φ) : B → A, connection up to homotopy ∇ : A y D and
Hermitian metric h, we have u((Φ, φ)!(∇), φ∗(h)) = (Φ, φ)∗u(∇, h);
4To construct −
∫
∆p , fix a splitting σ : pr∗(A) → pr!(A) to p̃r, and denote by q: Ω
(
pr! A
)
→ Ω(V ) the ho-
momorphism induced by the inclusion of V = ker pr∗ ⊂ T (M × ∆p). Then for ω ∈ Ωp+q
(
pr!(A)
)
and sections
a1, . . . , aq ∈ Γ(A), define −
∫
∆p ω so that the identity below is satisfied:
ιaq · · · ιa1 −
∫
∆p
ω :=
∫
∆p
q(ισ(aq) · · · ισ(a1)ω).
5We consider real vector bundles D as complex ones via complexification D ⊗R C, and we observe that a real
nonlinear connection ∇ of A on D induces a complex nonlinear connection ∇C of A on D⊗RC, and that a metric g
on D induces a Hermitian metric gC on the complexification D ⊗R C, in such a way that (∇g)C = (∇C)gC .
6 P. Frejlich
c) If cs(∇) = 0, then dAu(∇, h) = 0, in which case:
i) u(∇) := [u(∇, h)] ∈ Hodd(A) is independent of h;
ii) u(∇) ∈ H4•+1(A) if ∇ is a real connection up to homotopy.
Main Example. Let the adjoint bundle Ad(A) of a Lie algebroid A on M be A in even parity,
and TM in odd parity, equipped with ∂(a, u) = (0, %A(a)). Then ∇ad given by
∇ad
a (b, u) := ([a, b]A, [%Aa, u])
defines a representation up to homotopy. This can be seen as follows: every linear connection
∇ : TM y A induces a linear basic connection ∇bas : Ay Ad(A),
∇bas
a (b, u) :=
(
∇%Aba+ [a, b]A, %A∇ua+ [%Aa, u]
)
;
and the nonlinear representation ∇ad is equivalent to ∇bas
∇ad = ∇bas + [θ∇, ∂], θ∇(a)(b, u) := (∇ua, 0).
∇ad can be alternatively defined as the unique representation up to homotopy which under the
canonical Lie algebroid map (pr, id) : J1(A)→ A pulls back to the canonical representation
∇j1 : J1(A) y Ad(A), ∇j1j1a(b, u) = ∇ad
a (b, u).
Definition. The intrinsic characteristic classes charq(A) ∈ H2q−1(A) of the Lie algebroid A are
the secondary characteristic classes uq(∇ad) of the adjoint representation up to homotopy ∇ad.
Note that it follows from the discussion in the Main Example, and item b) of Proposition 2,
that char(A) can be alternatively defined as the unique element Hodd(A) which pulls back under
the Lie algebroid map (pr, id) : J1(A) → A to the secondary characteristic class u(∇j1) of the
canonical representation of J1(A) on Ad(A).
Example. The modular class mod(A) of A coincides with 2π char1(A) ∈ H1(A).
Remark. Intrinsic characteristic classes are not a complete obstruction to the existence of
a metric which is invariant under a basic connection. This is in contrast to the case of the
modular class, whose vanishing implies the existence of an invariant measure. As an example,
let g be the 3-dimensional Lie algebra given by
[e1, e2] = 0, [e1, e3] = ae1 + be2, [e2, e3] = ce1 + de2, Q =
(
a b
c d
)
∈ GL2(R),
which we regard as a Lie algebroid over a point. By dimensional reasons, we have
char(g) = 0 ⇐⇒ mod(g) = 0 ⇐⇒ TrQ = 0.
On the other hand, the only basic connection is ∇bas
x y = [x, y], and Ad(g) admits a (positive-
definite) ad-invariant metric iff g is abelian.
3 Proof of the Main Theorem
While primary and secondary characteristic classes are functorial with respect to pullbacks
essentially by inspection of the construction, for intrinsic characteristic classes the situation is
slightly more intricate because the adjoint representation up to homotopy of a pullback is not
itself a pullback representation up to homotopy. The following special case will turn out to be
key:
Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids 7
Proposition 3. Intrinsic characteristic classes are functorial with respect to surjective submer-
sions.
The proof of the Main Theorem requires the following direct consequence of Proposition 3:
Proposition 4. Intrinsic characteristic classes are functorial with respect to transverse, closed
embeddings.
Proof. Let i : X ↪→M be a closed embedding transverse to A, and p : NX := TM |X/TX → X
the normal bundle toX. By the normal form theorem in [2], we can find an open subset U ⊂ NX,
and an isomorphism of Lie algebroids (Φ, φ) : p!i!(A) ∼−→ A|φ(U), such that the following triangle
of morphisms of Lie algebroids commutes
p!i!(A)
(Φ,φ)
'
// A|φ(U)
i!(A),
(z̃,z)
cc
(̃i,i)
;;
where
(̃
i, i
)
is the pullback morphism of Lie algebroids induced by the inclusion i : X ↪→M , and
(z̃, z) is the pullback morphism of Lie induced by the zero section z : X ↪→ U . Because (Φ, φ) is
an isomorphism, we have that char
(
p!i!(A)
)
= (Φ, φ)∗ char(A), and this implies
i∗ char(A) = z∗ char
(
p!i!(A)
)
= (pz)∗ char
(
i!(A)
)
= char
(
i!(A)
)
,
where in the middle equality we used Proposition 3. �
Proof of the Main Theorem. Let φ : N →M be a smooth map, and A a Lie algebroid on M .
Factor φ as pr2 i, where
N
pr1←− N ×M pr2−→M
denote the canonical projections, and where
i : N −→ N ×M, i(x) := (x, φ(x))
is the embedding of N as the graph of φ. Because pr2 is a surjective submersion, φ is transverse
to A exactly when i is transverse to pr!
2(A). Hence
φ∗ char(A) = i∗ pr∗2 char(A) = i∗ char
(
pr!
2A
)
= char
(
i! pr!
2A
)
= char
(
φ!A
)
,
where in the second equality we used Proposition 3, and in the third, Proposition 4. �
So everything boils down to
Proof of Proposition 3. Let p : Σ → M be a surjective submersion, and A a Lie algebroid
on M . Our goal is to show that
char
(
p!(A)
)
= p∗(char(A)),
and by item b) of Proposition 2, it suffices to show that
char
(
p!(A)
)
= u
(
p!
(
∇bas
))
,
where ∇bas is the basic connection associated (in the sense of the Main Example) with some
linear connection ∇ : TM y A.
8 P. Frejlich
To do so, it is enough to give a recipe which to a connection ∇ : TM y A and metrics gA
on A and gM on TM , assigns a connection ∇ : TΣ y p!(A), and metrics gp!(A) on p!(A) and gΣ
on TΣ, such that
cs
(
∇bas
,∇bas,g)
= p∗cs
(
∇bas,∇bas,g
)
, (3.1)
where g = (gA, gM ) and g = (gp!(A), gΣ). Our recipe for
(
∇, gp!(A), gΣ
)
will depend on choices of
a metric gV on the vertical bundle V = ker p∗, and an Ehresmann connection H ⊂ TΣ for p,
all of which we fix once and for all. Denote by h : p∗(TM) → TΣ the horizontal lift associated
with H and by V the subbundle V ⊕ V ⊂ Ad
(
p!(A)
)
.
Consider the exact sequence of vector bundles over Σ:
0 −→ V −→ Ad
(
p!(A)
) p̃−→ p∗Ad(A) −→ 0
and define
hor : p∗(A) ∼−→ C ⊂ p!(A), hor(a) := (h(%Aa), a) ∈ TΣ×TM A.
This induces a linear splitting (hor, h) : p∗Ad(A) → Ad
(
p!(A)
)
to the exact sequence above,
and we define metrics gΣ on TΣ and gp!A on p!(A) so that
(V, gV )⊕
(
p∗(TM), p∗gM
)
−→ (TΣ, gΣ),
(
v, p†(u)
)
7→ v + h(u),
(V, gV )⊕
(
p∗(A), p∗gA
)
−→
(
p!A, gp!A
)
,
(
v, p†(a)
)
7→ v + hor(a)
be isometries.
The metric g = (gp!(A), gΣ) on Ad
(
p!(A)
)
is the one in the output of our recipe. The
construction of ∇ which satisfies (3.1), on the other hand, is subtler, and proceeds in steps.
Step one. First consider the Riemannian connection ∇R : TΣ y TΣ of gΣ, which satisfies
∇R,bas = ∇R = ∇R,gΣ .
Step two. Let the horizontal and vertical projections corresponding to gΣ be denoted by
PH ,PV : TΣ→ TΣ, and define a new connection
∇Σ : TΣ y TΣ, ∇Σ
u v := PH∇R
uPH(v) + PV∇R
uPV (v).
Note that V,H ⊂ TΣ are subconnections by construction. We claim that ∇Σ is gΣ-metric,
∇Σ = ∇Σ,gΣ . Indeed, note that by definition of ∇Σ, we have
gΣ
(
∇Σ
u v, w
)
= gΣ
(
∇R
uPHv,PHw
)
+ gΣ
(
∇R
uPV v,PV w
)
and because gΣ(V,H) = 0 and ∇R is gΣ-metric,
gΣ
(
∇Σ
u v, w
)
+ gΣ
(
v,∇Σ
uw
)
= LugΣ(PHv,PHw) + LugΣ(PV v,PV w) = LugΣ(v, w).
Step three. There exist unique C∞(Σ)-linear maps
D : Γ(H)→ End(Γ(V )), E : Γ(H) −→ End(Γ(C)),
satisfying the Leibniz rule
Dw(fv) = fDw(v) + (Lwf)v, Ew(fα) = fEw(α) + (Lwf)α,
Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids 9
for all f ∈ C∞(Σ), v ∈ Γ(V ), w ∈ Γ(H) and α ∈ Γ(C), and such that
Dh(u)v = [h(u), v], Eh(u)hor(a) = hor(∇ua),
for all u ∈ X(M) and a ∈ Γ(A). Concretely, we identify Γ(H) with C∞(Σ) ⊗C∞(M) X(M)
and Γ(C) with C∞(Σ)⊗C∞(M) Γ(A). Then D is just obtained by extension of scalars Dλh(u) :=
λDh(u). In turn, for each fixed u ∈ X(M), the linear map
Eh(u) : Γ(A) −→ Γ(C), Eh(u)(a) = hor(∇ua)
extends to an endomorphism of Γ(C) via Eh(u)(µ ⊗ a) := µEh(u)(a) + (Lh(u)µ)hor(a), and E is
obtained by extension of scalars: Eλh(u) := λEh(u).
Step four. Let now ∇ : TΣ y p!(A) be the connection which satisfies
a) ∇vv′ := ∇Σ,bas
v v′, b) ∇wv′ := Dw(v′),
c) ∇vα := ∇Σ,bas
v %p!(A)α, d) ∇wα := Ewα+ c(w, %p!(A)α)
for all v, v′ ∈ Γ(V ), w ∈ Γ(H) and α ∈ Γ(C), and where c denotes the extension of
X(M)× X(M) −→ Γ(V ), (u, u′) 7→ [h(u), h(u′)]− h[u, u′]
to a form c ∈ Γ(∧2p∗(T ∗M)⊗ V ). This concludes our recipe
(∇, gA, gM ) 7→
(
∇, gp!(A), gΣ
)
and all there is left to do is to check that (3.1) is satisfied.
We begin by computing the basic connection ∇bas
: p!(A) y Ad
(
p!(A)
)
:
a) ∇bas
v v′ = ∇Σ
v v
′, b) ∇bas
hor(a)v
′ = ∇Σ
h(%Aa)v
′,
c) ∇bas
v hor(b) = 0, d) ∇bas
v h(u) = 0,
e) ∇bas
hor(a)hor(b) = hor
(
∇bas
a b
)
, f) ∇bas
hor(a)h(u) = h
(
∇bas
a u
)
,
where v, v′ ∈ Γ(V ), u ∈ X(M) and a, b ∈ Γ(A). In particular, it follows from a) and b) that
∇bas
α Γ(V) ⊂ Γ(V), α ∈ Γ
(
p!(A)
)
, (3.2)
whereas from c)–f) it follows that
∇bas
α Γ(p∗Ad(A)) ⊂ Γ(p∗Ad(A)), α ∈ Γ
(
p!(A)
)
. (3.3)
Because V and p∗Ad(A) are g-orthogonal, it follows from (3.2), (3.3) and the definition of g-dual
connection that
∇bas,g
α Γ(V) ⊂ Γ(V), ∇bas,g
α Γ(p∗Ad(A)) ⊂ Γ(p∗Ad(A)), α ∈ Γ
(
p!(A)
)
. (3.4)
The explicit description a)–f) of ∇bas
also implies that ∇bas
restricts to a subconnection ∇V :=
∇bas|V : p!(A) y V, which is g-metric:
∇V = ∇bas|V = ∇bas,g|V, (3.5)
and that for α, β ∈ Γ
(
p!(A)
)
, a, b ∈ Γ(A), w ∈ X(Σ) and u ∈ X(M),
α ∼p a, β ∼p b, w ∼p u =⇒ ∇bas
α (β,w) ∼p ∇bas
a (b, u). (3.6)
10 P. Frejlich
We conclude from equations (3.2), (3.3), (3.5) and (3.6) that
∇bas
= ∇V ⊕ p!
(
∇bas
)
. (3.7)
Because p∗g(b, b′) = g(hor(b), hor(b′)) and ∇bas
v hor(b) = 0, it follows that
∇bas,g
v hor(b) = 0, v ∈ Γ(V ), b ∈ Γ(A),
and because ∇bas
hor(a)hor(b) = hor
(
∇bas
a b
)
, it follows that
∇bas,g
hor(a)hor(b) = hor
(
∇bas,g
a b
)
,
whence
α ∼p a, β ∼p b, w ∼p u =⇒ ∇bas,g
α (β,w) ∼p ∇bas,g
a (b, u). (3.8)
Equations (3.4), (3.5) and (3.8) hence imply that
∇bas,g
= ∇V ⊕ p!
(
∇bas,g
)
. (3.9)
Now form the affine connections
∇aff : A× T∆1 y Ad(A)×∆1, ∇aff = t0∇bas + t1∇bas,g,
∇aff
: p!(A)× T∆1 y Ad
(
p!(A)
)
×∆1, ∇aff
= t0∇
bas
+ t1∇
bas,g
used respectively to compute cs
(
∇bas,∇bas,g
)
and cs
(
∇bas
,∇bas,g)
. Then equations (3.7) and (3.9)
imply that
∇aff
= pr!
(
∇V)⊕ (p, id∆1)!
(
∇aff
)
,
whence
Trs
(
Rq
∇aff
)
= pr∗Trs
(
Rq∇V
)
+
(
p, id∆1
)∗
Trs
(
Rq∇aff
)
and so
csq
(
∇bas
,∇bas,g)
= − −
∫
∆1
Trs
(
Rq
∇aff
)
= − −
∫
∆1
(
p, id∆1
)∗
Trs
(
Rq∇aff
)
= −p∗ −
∫
∆1
Trs
(
Rq∇aff
)
= p∗cs
(
∇bas,∇bas,g
)
.
This shows that (3.1) holds true, and concludes the proof that char
(
p!(A)
)
= p∗ char(A). �
Acknowledgements
Work partially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek
(Vrije Competitie grant “Flexibility and Rigidity of Geometric Structures” 612.001.101) and by
IMPA (CAPES-FORTAL project). I would like to thank Ioan Mărcuţ, Ori Yudilevich, Rui Loja
Fernandes, Olivier Brahic and David Mart́ınez-Torres. I am also grateful to the anonymous
referees for their many useful comments.
Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids 11
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1 Introduction
2 Characteristic classes
3 Proof of the Main Theorem
References
|
| id | nasplib_isofts_kiev_ua-123456789-209880 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:04:40Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Frejlich, P. 2025-11-28T09:40:50Z 2018 Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids / P. Frejlich // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D17; 57R20 arXiv: 1805.00542 https://nasplib.isofts.kiev.ua/handle/123456789/209880 https://doi.org/10.3842/SIGMA.2018.124 In this note, we prove that intrinsic characteristic classes of Lie algebroids - which in degree one recover the modular class - behave functorially with respect to arbitrary transverse maps, and in particular are weak Morita invariants. In the modular case, this result appeared in [Kosmann-Schwarzbach Y., Laurent-Gengoux C., Weinstein A., Transform. Groups 13 (2008), 727-755], and with a connectivity assumption which we here show to be unnecessary, it appeared in [Crainic M., Comment. Math. Helv. 78 (2003), 681-721] and [Ginzburg V.L., J. Symplectic Geom. 1 (2001), 121-169]. Work partially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Vrije Competitie grant "Flexibility and Rigidity of Geometric Structures" 612.001.101) and by IMPA (CAPES-FORTAL project). I would like to thank Ioan Mărcut¸, Ori Yudilevich, Rui Loja Fernandes, Olivier Brahic, and David Martínez-Torres. I am also grateful to the anonymous referees for their many useful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids Article published earlier |
| spellingShingle | Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids Frejlich, P. |
| title | Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids |
| title_full | Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids |
| title_fullStr | Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids |
| title_full_unstemmed | Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids |
| title_short | Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids |
| title_sort | morita invariance of intrinsic characteristic classes of lie algebroids |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209880 |
| work_keys_str_mv | AT frejlichp moritainvarianceofintrinsiccharacteristicclassesofliealgebroids |