Lie group extensions associated to projective modules of continuous inverse algebras

Abstract

We call a unital locally convex algebra $A$ a continuous inverse algebra if its unit group $A^\times$ is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group $G$ on a continuous inverse algebra $A$ by automorphisms and any finitely generated projective right $A$-module $E$, we construct a Lie group extension $\hat G$ of $G$ by the group $\GL_A(E)$ of automorphisms of the $A$-module $E$. This Lie group extension is a ``non-commutative'' version of the group $\Aut(\V)$ of automorphism of a vector bundle over a compact manifold $M$, which arises for $G = \Diff(M)$, $A = C^\infty(M,\C)$ and $E = \Gamma\V$. We also identify the Lie algebra $\hat\g$ of $\hat G$ and explain how it is related to connections of the $A$-module $E$.

Full text

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arXiv:0802.2993v1 [math.OA] 21 Feb 2008
Lie group extensions associated to projective
modules of continuous inverse algebras
K.-H. Neeb
February 21, 2008
Abstract
We call a unital locally convex algebra A a continuous inverse
algebra if its unit group A×is open and inversion is a continuous map.
For any smooth action of a, possibly infinite-dimensional, connected
Lie group G on a continuous inverse algebra A by automorphisms and
any finitely generated projective right A-module E, we construct a
Lie group extension?G of G by the group GLA(E) of automorphisms
version of the group Aut(V) of automorphism of a vector bundle over
a compact manifold M, which arises for G = Diff(M), A = C∞(M,C)
and E = ΓV. We also identify the Lie algebra ? g of?G and explain how
AMC Classification: 22E65, 58B34
Keywords: Continuous inverse algebra, infinite dimensional Lie group,
vector bundle, projective module, semilinear automorphism, covariant
derivative, connection
of the A-module E. This Lie group extension is a “non-commutative”
it is related to connections of the A-module E.
Introduction
In [ACM89] it is shown that for a finite-dimensional K-principal bundle P
over a compact manifold M, the group Aut(P) of all bundle automorphisms
carries a natural Lie group structure whose Lie algebra is the Fr´ echet–Lie
algebra of V(P)Kof K-invariant smooth vector fields on M. This applies
in particular to the group Aut(V) of automorphisms of a finite-dimensional
vector bundle with fiber V because this group can be identified with the
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automorphisms group of the corresponding frame bundle P = FrV which is
a GL(V )-principal bundle.
In this paper, we turn to variants of the Lie groups Aut(V) arising in non-
commutative geometry. In view of [Ko76], the group Aut(V) can be identified
with the group of semilinear automorphisms of the C∞(M,R)-module Γ(V)
of smooth sections of V, which, according to Swan’s Theorem, is a finitely
generated projective module. Here the gauge group Gau(V) corresponds to
the group of C∞(M,R)-linear module isomorphisms.
This suggests the following setup: Consider a unital locally convex al-
gebra A and a finitely generated projective right A-module E. When can
we turn groups of semilinear automorphisms of E into Lie groups? First of
all, we have to restrict our attention to a natural class of algebras whose
unit groups A×carry natural Lie group structures, which is the case if A×
is an open subset of A and the inversion map is continuous. Such alge-
bras are called continuous inverse algebras, CIAs, for short. The Fr´ echet
algebra C∞(M,R) is a CIA if and only if M is compact. Then its automor-
phism group Aut(C∞(M,R))∼= Diff(M) carries a natural Lie group structure
with Lie algebra V(M), the Lie algebra of smooth vector fields on M. An-
other important class of CIAs whose automorphism groups are Lie groups
are smooth 2-dimensional quantum tori with generic diophantine properties
(cf. [El86], [BEGJ89]). Unfortunately, in general, automorphism groups of
CIAs do not always carry a natural Lie group structure, so that it is much
more natural to consider triples (A,G,µA), where A is a CIA, G a possibly
infinite-dimensional Lie group, and µA: G → Aut(A) a group homomorphism
defining a smooth action of G on A.
For any such triple (A,G,µA) and any finitely generated projective A-
module E, the subgroup GEof all elements of G for which µA(g) lifts to a
semilinear automorphism of E is an open subgroup. One of our main results
(Theorem 3.3) is that we thus obtain a Lie group extension
1 → GLA(E) = AutA(E) →?GE→ GE→ 1,
where?GEis a Lie group acting smoothly on E by semilinear automorphisms.
Γ(V) for a smooth vector bundle V, and G = Diff(M), the Lie group?G is
but our construction contains a variety of other interesting settings. From a
different perspective, the Lie group structure on?G also tells us about possible
2
For the special case where M is a compact manifold, A = C∞(M,R), E =
isomorphic to the group Aut(V) of automorphisms of the vector bundle V,

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smooth actions of Lie groups on finitely generated projective A-modules by
semilinear maps which are compatible with a smooth action on the algebra
A.
A starting point of our construction is the observation that the connected
components of the set Idem(A) of idempotents of a CIA coincide with the
orbits of the identity component A×
Using the natural manifold structure on Idem(A) (cf. [Gram84]), the action
of A×on Idem(A) even is a smooth Lie group action.
On the Lie algebra side, the semilinear automorphisms of E correspond to
the Lie algebra DEnd(E) of derivative endomorphisms, i.e., those endomor-
phisms ϕ ∈ EndK(E) for which there is a continuous derivation D ∈ der(A)
with ϕ(s.a) = ϕ(s).a + s.(D.a) for s ∈ E and a ∈ A. The set?
all pairs (ϕ,D) ∈ EndK(E)×der(A) satisfying this condition is a Lie algebra
and we obtain a Lie algebra extension
0of A×under the conjugation action.
DEnd(E) of
0 → EndA(E) = glA(E) ֒→?
DEnd(E) → → der(A) → 0.
Pulling this extension back via the Lie algebra homomorphism g → der(A)
induced by the action of G on A yields the Lie algebra ? g of the group?G from
In Section 5 we briefly discuss the relation between linear splittings of
the Lie algebra extension ? g and covariant derivatives in the context of non-
Thanks: We thank Hendrik Grundling for reading erlier versions of this
paper and for numerous remarks which lead to several improvements of the
presentation.
above (Proposition 4.7).
commutative geometry (cf. [Co94], [MMM95], [DKM90]).
Preliminaries and notation
Throughout this paper we write I := [0,1] for the unit interval in R and
K either denotes R or C. A locally convex space E is said to be Mackey
complete if each smooth curve γ: I → E has a (weak) integral in E. For a
more detailed discussion of Mackey completeness and equivalent conditions,
we refer to [KM97, Th. 2.14].
A Lie group G is a group equipped with a smooth manifold structure
modeled on a locally convex space for which the group multiplication and
the inversion are smooth maps. We write 1 ∈ G for the identity element
and λg(x) = gx, resp., ρg(x) = xg for the left, resp., right multiplication on
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G. Then each x ∈ T1(G) corresponds to a unique left invariant vector field
xl with xl(g) := dλg(1).x,g ∈ G. The space of left invariant vector fields
is closed under the Lie bracket of vector fields, hence inherits a Lie algebra
structure. In this sense we obtain on g := T1(G) a continuous Lie bracket
which is uniquely determined by [x,y]l= [xl,yl] for x,y ∈ g. We shall also
use the functorial notation ? L(G) for the Lie algebra of G and, accordingly,
? L(ϕ) = T1(ϕ): ? L(G1) → ? L(G2) for the Lie algebra morphism associated to a
morphism ϕ: G1→ G2of Lie groups.
A Lie group G is called regular if for each ξ ∈ C∞(I,g), the initial value
problem
γ(0) = 1,γ′(t) = γ(t).ξ(t) = T(λγ(t))ξ(t)
has a solution γξ∈ C∞(I,G), and the evolution map
evolG: C∞(I,g) → G,ξ ?→ γξ(1)
is smooth (cf. [Mil84]). For a locally convex space E, the regularity of the
Lie group (E,+) is equivalent to the Mackey completeness of E ([Ne06,
Prop. II.5.6]). We also recall that for each regular Lie group G its Lie algebra
g is Mackey complete and that all Banach–Lie groups are regular (cf. [Ne06,
Rem. II.5.3] and [GN08]).
A smooth map expG: ? L(G) → G is called an exponential function if each
curve γx(t) := expG(tx) is a one-parameter group with γ′
group G is said to be locally exponential if it has an exponential function for
which there is an open 0-neighborhood U in ? L(G) mapped diffeomorphically
by expGonto an open subset of G.
If A is an associative algebra with unit, we write 1 for the identity ele-
ment, A×for its group of units, Idem(A) = {p ∈ A: p2= p} for its set of
idempotents and ηA(a) = a−1for the inversion map A×→ A. A homomor-
phism ρ: A → B is unital algebras is called isospectral if ρ−1(B×) = A×. We
write GLn(A) := Mn(A)×for the unit group of the unital algebra Mn(A) of
n × n-matrices with entries in A.
Throughout, G denotes a (possibly infinite-dimensional) Lie group , A
a Mackey complete unital continuous inverse algebra (CIA for short) and
G × A → A,(g,a) ?→ g.a = µA(g)(a) is a smooth action of G on A.
x(0) = x. The Lie
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1Idempotents and finitely generated projec-
tive modules
The set Idem(A) of idempotents of a CIA A plays a central role in (topolog-
ical) K-theory. In [Gram84, Satz 2.13], Gramsch shows that this set always
carries a natural manifold structure, which implies in particular that its con-
nected components are open subsets. Since we shall need it in the following,
we briefly recall some basic facts on Idem(A) (cf. [Gram84]; see also [Bl98,
Sect. 4]).
Proposition 1.1 For each p ∈ Idem(A), the set
Up:= {q ∈ Idem(A): pq + (1 − p)(1 − q) ∈ A×}
is an open neighborhood of p in Idem(A) and, for each q ∈ Up, the element
sq:= pq + (1 − p)(1 − q) ∈ A×
satisfiessqqs−1
q
= p.
The connected component of p in Idem(A) coincides with the orbit of the
identity component A×
Idem(A),(g,p) ?→ cg(p) := g.p := gpg−1.
0of A×under the conjugation action A××Idem(A) →
Proof.
p to 1 and since A×is open, Upis an open neighborhood of p. Hence, for
each q ∈ Up, the element sqis invertible, and a trivial calculation shows that
sqq = psq.
If q is sufficiently close to p, then sq∈ A×
open neighborhood of 1 in A (recall that A is locally convex and A×is open).
This implies that q = s−1
Conversely, since the orbit map A×→ Idem(A),g ?→ cg(p) is continuous,
it maps the identity component A×
Idem(A).
Since the map q ?→ pq + (1 − p)(1 − q) is continuous, it maps
0because sp= 1 and A×
0is an
qpsqlies in the orbit {cg(p): g ∈ A×
0} of p under A×
0.
0into the connected component of p in
Lemma 1.2 Let A be a CIA, n ∈ N, and p = p2∈ Mn(A) an idempotent.
Then the following assertions hold:
(1) The subalgebra pMn(A)p is a CIA with identity element p.
(2) The unit group (pMn(A)p)×is a Lie group.
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Proof.
complete if A is so, it suffices to prove the assertion for n = 1.
(1) From the decomposition of the identity 1 as a sum 1 = p+(1−p) of
two orthogonal idempotents, we obtain the direct sum decomposition
Since Mn(A) also is a CIA ([Swa62]; [Gl02]) which is Mackey
A = pAp ⊕ pA(1 − p) ⊕ (1 − p)Ap ⊕ (1 − p)A(1 − p).
We claim that an element a ∈ pAp is invertible in this algebra if and
only if the element a + (1 − p) is invertible in A. In fact, if b ∈ pAp satisfies
ab = ba = p, then
(a + (1 − p))(b + (1 − p)) = ab + (1 − p) = 1 = (b + (1 − p))(a + (1 − p)).
If, conversely, c ∈ A is an inverse of a+(1−p) in A, then ca+c(1−p) = 1 =
ac + (1 − p)c leads after multiplication with p to ca = p = ac, which implies
(pcp)a = p = a(pcp), so that pcp is an inverse of a in pAp. The preceding
argument implies in particular that (pAp)×= pAp ∩ (A×− (1 − p)) is an
open subset of pAp, and that the inversion map
ηpAp: (pAp)×→ pAp,a ?→ a−1= ηA(a + 1 − p) − (1 − p)
is continuous.
(2) is an immediate consequence of (1) (cf. [Gl02], [Ne06], Ex. II.1.4,
Th. IV.1.11).
Let E be a finitely generated projective right A-module. Then there is
some n ∈ N and an idempotent p = p2∈ Mn(A) with E∼= pAn, where
A acts by multiplication on the right.
p ∈ Idem(Mn(A)), the right submodule pAnof Anis finitely generated (as
a quotient of An) and projective because it is a direct summand of the free
module An ∼= pAn⊕ (1 − p)An. The following lemma provides some infor-
mation on A-linear maps between such modules.
Conversely, for each idempotent
Lemma 1.3 Let p,q ∈ Idem(Mn(A)) be two idempotents. Then the follow-
ing assertions hold:
(1) The map x ?→ λx|pAn (left multiplication) yields a bijection
αp,q: qMn(A)p = {x ∈ Mn(A): qx = x,xp = x} → HomA(pAn,qAn).
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(2) pAn ∼= qAnif and only if there are x,y ∈ Mn(A) with xy = q and yx = p.
If, in particular, q = gpg−1for some g ∈ GLn(A), then x := gp and
y := g−1satisfy xy = q and yx = p.
(3) pAn ∼= qAnif and only if there are x ∈ qMn(A)p and y ∈ pMn(A)q with
xy = q and yx = p.
(4) If pAn ∼= qAn, then there exists an element g ∈ GL2n(A) with g˜ pg−1= ˜ q,
where
?p 0
0 0
˜ p =
?
and˜ q =
?q 0
0 0
?
.
Proof.
matrix, so that Mn(A)∼= EndA(An). Since pAnand qAnare direct summands
of An, each element of Hom(pAn,qAn) can be realized by left multiplication
with a matrix, and we have the direct sum decomposition
(1) Each element of EndA(An) is given by left multiplication with a
EndA(An)∼= Hom(pAn,qAn) ⊕ Hom(pAn,(1 − q)An)
⊕ Hom((1 − p)An,qAn) ⊕ Hom((1 − p)An,(1 − q)An),
which corresponds to the direct sum decomposition
Mn(A)∼= qMn(A)p⊕(1−q)Mn(A)p⊕qMn(A)(1−p)⊕(1−q)Mn(A)(1−p).
Now the assertion follows from qMn(A)p = {x ∈ Mn(A): qx = x,xp = x}.
(2), (3) If pAnand qAnare isomorphic, there is some x ∈ qMn(A)p∼=
Hom(pAn,qAn) for which λx: pAn→ qAn,s ?→ xs is an isomorphism. Writ-
ing λ−1
x
as λyfor some y ∈ Hom(qAn,pAn)∼= pMn(A)q, we get
p = λy◦ λx(p) = yxp = yxandq = λx◦ λy(q) = xyq = xy.
If, conversely, p = yx and q = xy hold for some x,y ∈ Mn(A), then p2= p
implies p = yx = yxyx = yqx and likewise q = xy = xpy, which leads to
(pyq)(qxp) = pyqxp = p3= p and(qxp)(pyq) = qxpyq = q3= q.
Therefore, we also have p = y′x′and q = x′y′with x′:= qxp ∈ qMn(A)p
and y′:= pyq ∈ pMn(A)q. Then λx′: pAn→ qAnand λy′: qAn→ pAnare
module homomorphisms with λx′ ◦ λy′ = λx′y′ = λq= idqAn and λy′ ◦ λx′ =
λy′x′ = λp= idpAn.
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(4) (cf. [Bl98, Prop. 4.3.1]) Pick x,y as in (3). Let
?1 − q
y1 − p
α :=
x
?
andβ :=
?1 − pp
p1 − p
?
∈ M2n(A).
Then a direct calculation yields α2= 1 = β2. Therefore z := βα ∈ GL2n(A).
Moreover, we have
?0 0
0 p
α˜ qα−1=
?
andβ
?0 0
0 p
?
β−1= ˜ p,
so that z˜ qz−1= ˜ p.
Proposition 1.4 For each finitely generated projective right A-module E,
we pick some idempotent p ∈ Mn(A) with E∼= pAn. Then we topologize
EndA(E) by declaring
αp,p: pMn(A)p → EndA(E),x ?→ λx|pAn
to be a topological isomorphism. Then the algebra EndA(E) is a CIA and
GLA(E) is a Lie group. This topology does not depend on the choice of p and
if A is Mackey complete, then GLA(E) is locally exponential.
Proof.
obtain a CIA structure on EndA(E), so that GLA(E) is a Lie group which is
locally exponential if A is Mackey complete ([Gl02]).
To verify the independence of the topology on EndA(E) from p, we first
note that for any matrix
?p 0
0 0
We simply combine Lemma 1.2 with Lemma 1.3(1) to see that we
˜ p =
?
∈ MN(A), N > n,
we have a natural isomorphism ˜ pMN(A)˜ p∼= pMn(A)p, because all non-zero
entries of matrices of the form ˜ pX˜ p, X ∈ MN(A), lie in the upper left (p×p)-
submatrix and depend only on the corresponding entries of X.
If q ∈ Idem(Mℓ(A)) is another idempotent with qAℓ ∼= E, then the preced-
ing argument shows that, after passing to max(n,ℓ), we may w.l.o.g. assume
that ℓ = n. Then Lemma 1.3 yields a g ∈ GL2n(A) with gpg−1= q, and then
conjugation with g induces a topological isomorphism
pMn(A)p∼= pM2n(A)p
cg
− − − − − →qM2n(A)q∼= qMn(A)q.
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Example 1.5 (a) Let M be a smooth paracompact finite-dimensional man-
ifold. We endow A := C∞(M,K) with the smooth compact open topology,
i.e., the topology inherited by the natural embedding
C∞(M,K) ֒→
∞
?
k=0
C(TkM,TkK),f ?→ (Tk(f))k∈N0,
where all spaces C(TkM,TkK) carry the compact open topology which co-
incides with the topology of uniform convergence on compact subsets.
If E is the space of smooth sections of a smooth vector bundle
q: V → M, then E is a finitely generated projective A-module ([Swa62]).
The algebra EndA(E) is the space of smooth sections of the vector bundle
End(V) and its unit group GLA(E)∼= Gau(V) is the corresponding gauge
group. We shall return to this class of examples below.
(b) We obtain a similar picture if A is the Banach algebra C(X,K), where
X is a compact space and E is the space of continuous sections of a finite-
dimensional topological vector bundle over X. Then EndA(E) is a Banach
algebra, so that its unit group GLA(E) is a Banach–Lie group.
2Semilinear automorphisms of finitely gen-
erated projective modules
In this section we take a closer look at the group ΓL(E) of semilinear au-
tomorphism of a right A-module E. One of our main results, proved in
Section 3 below, asserts that if E is a finitely generated projective module of
a CIA A, certain pull-backs of this group by a smooth action of a Lie group
G on A lead to a Lie group extension?G of G by the Lie group GLA(E) (cf.
The discussion in this section is very much inspired by Y. Kosmann’s
paper [Ko76].
Proposition 1.4) acting smoothly on E.
Definition 2.1 Let E be a topological right module of the CIA A, i.e., we
assume that the module structure E × A → E,(s,a) ?→ s.a =: ρE(a)s is a
continuous bilinear map. We write EndA(E) for the algebra of continuous
module endomorphisms of E and GLA(E) := EndA(E)×for its group of units,
the module automorphism group of E. For A = K we have in particular
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GL(E) = GLK(E). The group GLA(E) is contained in the group
ΓL(E)
:= {ϕ ∈ GLK(E): (∃ϕA∈ Aut(A))(∀s ∈ E)(∀a ∈ A) ϕ(s.a) = ϕ(s).ϕA(a)}
= {ϕ ∈ GLK(E): (∃ϕA∈ Aut(A))(∀a ∈ A) ϕ ◦ ρE(a) = ρE(ϕA(a)) ◦ ϕ}.
of semilinear automorphisms of E, where we write Aut(A) for the group of
topological automorphisms of A. We put
?
where the multiplication is componentwise multiplication in the product
group. In [Har76], the elements of?
E are constructed for this group with values in differential forms over A. If
the representation of A on E is faithful, then ϕAis uniquely determined by
ϕ, so that?
1 → GLA(E) →?
where Aut(A)Edenotes the image of the group?
Remark 2.2 (a) For each ψ ∈ Aut(A), we define the corresponding twisted
module Eψby endowing the vector space E with the new A-module structure
defined by s∗ψa := s.ψ(a), i.e., ρEψ = ρE◦ψ. Then a continuous linear map
ϕ: E → Eψis a morphism of A-modules if and only if ϕ(s.a) = ϕ(s).ψ(a)
holds for all s ∈ E and a ∈ A, i.e.,
ΓL(E) := {(ϕ,ϕA) ∈ GLK(E)×Aut(A): (∀a ∈ A) ϕ◦ρE(a) = ρE(ϕA(a))◦ϕ},
ΓL(E) are called semilinear automor-
phisms and, for A commutative, certain characteristic cohomology classes of
ΓL(E)∼= ΓL(E).
The map (ϕ,ϕA) ?→ ϕAdefines a short exact sequence of groups
ΓL(E) → Aut(A)E→ 1,
ΓL(E) in Aut(A).
ϕ ◦ ρE(a) = ρEψ(a) ◦ ϕ fora ∈ A.
Therefore (ϕ,ψ) ∈?
ΓL(E) is equivalent to ϕ: E → Eψbeing a module iso-
morphism. This shows that
Aut(A)E= {ψ ∈ Aut(A): Eψ∼= E}.
(b) Let ψ ∈ Diff(M) and q: V → M be a smooth vector bundle over M.
We consider the pull-back vector bundle
Vψ:= ψ∗V := {(x,v) ∈ M × V: ψ(x) = qV(v)}
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with the bundle projection qψ
If s: M → Vψis a smooth section, then s(x) = (x,s′(ψ(x))), where
s′: M → V is a smooth section, and this leads to an identification of the
spaces of smooth sections of V and Vψ. For a smooth function f : M → K
and s ∈ Γ(Vψ), we have (s.f)(x) = f(x)s(x) = (x,f(x)s′(ψ(x))), so that the
corresponding right module structure on E = ΓV is given by s′∗f = s′.(ψ.f),
where ψ.f = f◦ψ−1. This shows that Eψ= (ΓV)ψ ∼= Γ(Vψ), i.e., the sections
of the pull-back bundle form a twisted module.
(c) Let E be a finitely generated projective right A-module and p ∈
Idem(Mn(A)) with E∼= pAn. For ψ ∈ Aut(A) we write ψ(n)for the cor-
responding automorphisms of An, resp., Mn(ψ) for the corresponding auto-
morphism of Mn(A). Then the map ψn: Mn(ψ)−1(p)An→ pAninduces a
module isomorphism Mn(ψ)−1(p)An ∼= (pAn)ψ.
(d) Let ρE: A → End(E) denote the representation of A on E defining the
right module structure. Then, for each a ∈ A×, we have (ρE(a),c−1
because ρE(a)(s.b) = s.ba = (s.a)(a−1ba).
V: Vψ→ M,(x,v) ?→ x.
a) ∈?
ΓL(E)
Definition 2.3 Let G be a group acting by automorphism on the group N
via α: G → Aut(N). We call a map f : G → N a crossed homomorphism if
f(g1g2) = f(g1)α(g1)(f(g2)) forg1,g2∈ G.
Note that f is a crossed homomorphism if and only if (f,idG): G → N ⋊αG
is a group homomorphism. The set of all crossed homomorphisms G → N is
denoted by Z1(G,N). The group N acts naturally on Z1(G,N) by
(n ∗ f)(g) := nf(g)α(g)(n)−1
and the set of N-orbits in Z1(G,N) is denoted H1(G,N). If N is not abelian,
Z1(G,N) and H1(G,N) do not carry a group structure; only the constant
map 1 is a distinguished element of Z1(G,N), and its class [1] is distinguished
in H1(G,N). The crossed homomorphisms in the class [1] are called trivial.
They are of the form f(g) = nα(g)(n)−1for some n ∈ N.
Proposition 2.4 Let ρE: A → EndK(E) denote the action of A on the right
A-module E and consider the action of the unit group A×on the group ΓL(E)
by ˜ ρE(a)(ϕ) := ρE(a)−1ϕρE(a). To each ψ ∈ Aut(A) we associate the func-
tion
C(ψ): A×→ ΓL(E),a ?→ ρE(ψ(a)a−1)−1= ρE(ψ(a))−1ρE(a).
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Then C(ψ) ∈ Z1(A×,ΓL(E)), and we thus get an exact sequence of pointed
sets
1 → GLA(E) →?
characterizing the subgroup Aut(A)Eas C
ΓL(E) → Aut(A)
C
− − →H1(A×,ΓL(E)),
−1([1]).
Proof.That C(ψ) is a crossed homomorphism follows from
C(ψ)(ab) = ρE(ψ(ab))−1ρE(ab) = ρE(ψ(a))−1ρE(ψ(b))−1ρE(b)ρE(a)
= C(ψ)(a)ρE(a)−1C(ψ)(b)ρE(a) = C(ψ)(a)˜ ρE(a)?C(ψ)(b)?.
That the crossed homomorphism C(ψ) is trivial means that there is a ϕ ∈
ΓL(E) with
C(ψ)(a) = ρE(ψ(a))−1ρE(a) = ϕρE(a)−1ϕ−1ρE(a),
which means that ρE(ψ(a))ϕ = ϕρE(a) for a ∈ A×. Since each CIA A is
generated by it unit group, which is an open subset, the latter relation is
equivalent to (ϕ,ψ) ∈?
Example 2.5 Let qV: V → M be a smooth K-vector bundle on the com-
pact manifold M and Aut(V) the group of smooth bundle isomorphisms.
Then each element ϕ of this group permutes the fibers of V, hence induces a
diffeomorphism ϕMof M. We thus obtain an exact sequence of groups
ΓL(E). We conclude that C
−1([1]) = Aut(A)E.
1 → Gau(V) → Aut(V) → Diff(M)[V]→ 1,
where Gau(V) = {ϕ ∈ Aut(V): ϕM= idM} is the gauge group of V and
Diff(M)V= {ψ ∈ Diff(M): ψ∗V∼= V}
is the set of all diffeomorphisms ψ of M lifting to automorphisms of V (cf.
Remark 2.2(b)). The group Diff(M) carries a natural Fr´ echet–Lie group
structure for which Diff(M)Vis an open subgroup, hence also a Lie group.
Furthermore, it is shown in [ACM89] that Aut(V) and Gau(V) carry natural
Lie group structures for which Aut(V) is a Lie group extension of Diff(M)V
by Gau(V).
Consider the CIA A := C∞(M,K) and recall from Example 1.5(a) that
the space E := C∞(M,V) of smooth sections of V is a finitely generated
projective A-module. The action of Aut(V) on V induces an action on E by
ϕE(s)(x) := ϕ.s(ϕ−1
M(x)).
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For any smooth function f : M → K, we now have ϕE(fs)(x) := f(ϕ−1
ϕE(s)(ϕ−1
on E by semilinear automorphisms of E and that we obtain a commutative
diagram
Gau(V)→ Aut(V) → Diff(M)[V]
?
Next, we recall that
M(x))·
M(x)), i.e., ϕE(fs) = (ϕM.f)·ϕE(s). We conclude that Aut(V) acts
??
GLA(E) →ΓL(E)→ Aut(A)E.
Aut(A) = Aut(C∞(M,K))∼= Diff(M) (1)
(cf. [Ne06, Thm. IX.2.1], [Bko65], [Gra06], [Mr05]). Applying [Ko76, Prop. 4]
to the Lie group G = Z, it follows that the map Aut(V) → ΓL(E) is a
bijective group homomorphism. Let us recall the basic idea of the argument.
First we observe that the vector bundle V can be reconstructed from
the A-module E as follows. For each m ∈ M, we consider the maximal
closed ideal Im:= {f ∈ A: f(m) = 0} and associate the vector space Em:=
E/ImE. Using the local triviality of the vector bundle V, it is easy to see
that Em∼= Vm. We may thus recover V from E as the disjoint union
?
m∈M
V =Em.
Any ϕ ∈ ΓL(E) defines an automorphism ϕAof A, which we identify with
a diffeomorphism ϕM of M via ϕA(f) := f ◦ ϕ−1
implies that ϕ induces an isomorphism of vector bundles
M. Then ϕA(Im) = IϕM(m)
V → V,s + ImE ?→ ϕ(s + ImE) = ϕ(s) + IϕM(m)E.
Its smoothness follows easily by applying it to a set of sections s1,...,sn
which are linearly independent in m. This implies that each element ϕ ∈
ΓL(E) corresponds to an element of Aut(V), so that the vertical arrows in
the diagram above are in fact isomorphisms of groups.
Finally, we take a look at the Lie structures on these groups. A priori, the
automorphism group Aut(A) of a CIA carries no natural Lie group structure,
but the group isomorphism Diff(M) → Aut(A) from (1) defines a smooth
action of the Lie group Diff(M) on A. Indeed, this can be derived quite
directly from the smoothness of the map
Diff(M) × C∞(M,K) × M → K,(ϕ,f,m) ?→ f(ϕ−1(m))
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which is smooth because it is a composition of the smooth action map
Diff(M) × M → M and the smooth evaluation map A × M → M (cf.
[NeWa08, Lemma A.2]).
Since the vector bundle V can be embedded into a trivial bundle
M × Kn, we obtain a topological embedding of Gau(V) as a closed sub-
group of Gau(M × Kn)∼= C∞(M,GLn(K))∼= GLn(A). Accordingly, we
obtain an embedding E ֒→ Anand the preceding discussion yields an iden-
tification of GLA(E) with Gau(V) as the same closed subgroups of GLn(A)
(cf. Proposition 1.4). Since both groups are locally exponential Lie groups,
the homeomorphism Gau(V) → GLA(E) is an isomorphism of Lie groups (cf.
[Ne06, Thm. IV.1.18] and [GN08] for more details on locally exponential Lie
groups).
3 Lie group extensions associated to projec-
tive modules
In this section we consider a Lie group G, acting smoothly by automorphisms
on the CIA A. We write µA: G → Aut(A) for the corresponding homomor-
phism. For each right A-module E, we then consider the subgroup
GE:= {g ∈ G: Eg∼= E} = µ−1
A(Aut(A)E),
where we write Eg:= EµA(g)for the corresponding twisted module (cf. Re-
mark 2.2(a)). The main result of this section is Theorem 3.3 which asserts
that for G = GE, the pull-back of the group extension?
ΓL(E) of Aut(A)Eby
GLA(E) yields a Lie group extension?G of G by GLA(E).
Proposition 3.1 If E is a finitely generated projective right A-module, then
the subgroup GE of G is open. In particular, we have µA(G) ⊆ Aut(A)E if
G is connected.
Proof.
A-module of the form pAnfor some idempotent p ∈ Mn(A). We recall from
Remark 2.2(c) that for any automorphism ψ ∈ Aut(A) and γ ∈ GLn(A) with
Mn(ψ)−1(p) = γ−1pγ, the maps
Since E is finitely generated and projective, it is isomorphic to an
Mn(ψ)−1(p)An→ (pAn)ψ,x ?→ ψ(n)(x),Mn(ψ)−1(p)An→ pAn,s ?→ γ · s
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are isomorphisms of A-modules. According to Proposition 1.1, all orbits of
the group GLn(A)0 in Idem(Mn(A)) are connected open subsets of
Idem(Mn(A)), hence coincide with its connected components. Therefore the
subset {g ∈ G: g.p ∈ GLn(A)0.p} of G is open. In view of Lemma 1.3(2),
this open subset is contained in the subgroup GE, hence GEis open.
From now on we assume that G = GE. Then we obtain a group extension
1 → GLA(E) →?G
q
− − →G → 1,
where q(ϕ,g) = g, and
?G := {(ϕ,g) ∈ ΓL(E) × G: (ϕ,µA(g)) ∈?
acts on E via π(ϕ,g).s := ϕ(s) by semilinear automorphisms. The main
result of the present section is that?G carries a natural Lie group structure
precise:
ΓL(E)}∼= µ∗
A?
ΓL(E) (2)
and that it is a Lie group extension of G by GLA(E). Let us make this more
Definition 3.2 An extension of Lie groups is a short exact sequence
1 → N
ι
− − →?G
q
− − →G → 1
of Lie group morphisms, for which?G is a smooth (locally trivial) principal
In the following, we identify N with the subgroup ι(N) ??G.
Theorem 3.3 If A is a CIA, G is a Lie group acting smoothly on A by
µA: G → Aut(A), and E is a finitely generated projective right A-module with
µA(G) ⊆ Aut(A)E, then GLA(E) and?G carry natural Lie group structures
1 → GLA(E) →?G
defines a Lie group extension of G by GLA(E).
N-bundle over G with respect to the right action of N given by (? g,n) ?→ ? gn.
such that the short exact sequence
q
− − →G → 1
Proof.
GLA(E) is a Lie group. The assumption µA(G) ⊆ Aut(A)E implies that
G = GE, so that the group?G is indeed a group extension of G by GLA(E).
15
In view of Proposition 1.4, EndA(E) is a CIA and its unit group