On the Notion of Noncommutative Submanifold
We review the notion of submanifold algebra, as introduced by T. Masson, and discuss some properties and examples. A submanifold algebra of an associative algebra 𝘈 is a quotient algebra 𝘉 such that all derivations of 𝘉 can be lifted to 𝘈. We will argue that in the case of smooth functions on manifo...
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| description | We review the notion of submanifold algebra, as introduced by T. Masson, and discuss some properties and examples. A submanifold algebra of an associative algebra 𝘈 is a quotient algebra 𝘉 such that all derivations of 𝘉 can be lifted to 𝘈. We will argue that in the case of smooth functions on manifolds, every quotient algebra is a submanifold algebra, derive a topological obstruction when the algebras are deformation quantizations of symplectic manifolds, present some (commutative and noncommutative) examples and counterexamples.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 050, 21 pages
On the Notion of Noncommutative Submanifold
Francesco D’ANDREA
Università di Napoli “Federico II” and I.N.F.N. Sezione di Napoli, Complesso MSA,
Via Cintia, 80126 Napoli, Italy
E-mail: francesco.dandrea@unina.it
URL: http://wpage.unina.it/francesco.dandrea/
Received January 11, 2020, in final form May 30, 2020; Published online June 09, 2020
https://doi.org/10.3842/SIGMA.2020.050
Abstract. We review the notion of submanifold algebra, as introduced by T. Masson, and
discuss some properties and examples. A submanifold algebra of an associative algebra A
is a quotient algebra B such that all derivations of B can be lifted to A. We will argue
that in the case of smooth functions on manifolds every quotient algebra is a submanifold
algebra, derive a topological obstruction when the algebras are deformation quantizations
of symplectic manifolds, present some (commutative and noncommutative) examples and
counterexamples.
Key words: submanifold algebras; tangential star products; coisotropic reduction
2020 Mathematics Subject Classification: 46L87; 53C99; 53D55; 13N15
1 Introduction
Inspired by Gelfand duality, establishing that any commutative C∗-algebra is isomorphic to the
algebra of continuous functions vanishing at infinity on a locally compact Hausdorff space, the
point of view of noncommutative geometry is to regard any associative algebra (possibily with
additional structure, e.g., a Dirac operator [8]) as describing some virtual “noncommutative”
space. It is natural to wonder what is the correct notion of noncommutative space contained
within another noncommutative space.
Motivated by the derivation-based differential calculus of M. Dubois-Violette and P.W. Mi-
chor [14, 17] (reviewed, e.g., in [15, 16]), T. Masson introduced in [32] the notion of submanifold
algebra as a possible candidate for what in the noncommutative realm should be the analogue
of a closed embedded submanifold of a smooth manifold (see [1] for another possible approach).
The starting point is a short exact sequence of associative algebras (over a field K):
0→ J → A
π−→ B → 0. (1)
The sequence (1) induces an exact sequence of complexes
0→ Hom
(
A⊗n, J
)
→ Homπ
(
A⊗n, A
)
→ Hom
(
B⊗n, B
)
, (2)
where the first one is the Hochschild complex of A with coefficients in the A-bimodule J , the
last one is the Hochschild complex of B, and the one in the middle is the sub-complex of the
Hochschild complex of A given by K-linear maps A⊗n → A with image in J if one of the
arguments is in J .
Last arrow in (2) is, in general, not surjective. It is for n = 0, since (2) reduces to (1), and
for n = 1 if restricted to Hochschild coboundaries, i.e., inner derivations. The restriction to
Hochschild 1-cocycles, i.e., derivations, already gives a map
π∗ : Derπ(A)→ Der(B) (3)
that may not be surjective (cf. Examples 10, 17 and 21).
mailto:francesco.dandrea@unina.it
http://wpage.unina.it/francesco.dandrea/
https://doi.org/10.3842/SIGMA.2020.050
2 F. D’Andrea
If M is a smooth manifold and S ⊂ M a closed embedded smooth submanifold, it is well
known that the pullback of the inclusion ı : S →M is surjective, and one has an exact sequence
0→ J → C∞(M)
ı∗−→ C∞(S)→ 0
like in (1), with J the ideal of smooth functions on M that are zero on S. The induced map (3)
on vector fields (derivations) is surjective as well.
With this example in mind, whenever we have a sequence (1) such that (3) is surjective, we
will call B a submanifold algebra of A [32].
Submanifold algebras have been recently studied from the point of view of Drinfel’d twists
in [20, 21, 39].
The aim of this paper is to understand the meaning of submanifold algebras from the point
of view of noncommutative geometry. The plan is the following. In Section 2 we fix the nota-
tions and provide some algebraic background. In Section 3 we investigate the case of smooth
manifolds (and closed subsets), exhibit examples of commutative submanifold algebras that are
not algebras of functions on smooth manifolds, examples of quotient algebras that are not sub-
manifold algebras, and argue that when both A and B in (1) are algebras of smooth functions
the condition on π∗ is in fact redundant (Theorem 11). In Section 4 we discuss some examples of
submanifold algebras of noncommutative algebras, including “almost commutative” spaces. In
Section 5 we study deformation quantizations of Poisson manifolds and discuss some topological
obstructions for a quotient algebra to be a submanifold algebra (cf. Lemma 31).
2 Preliminaries and notations
2.1 Notations
Throughout the following, and unless stated otherwise, A will denote an associative algebra
over a field K, M a smooth manifolds (without boundary), C∞(M) the algebra of real -valued
smooth function on M , C0(M) the C∗-algebra of complex -valued continuous functions vanishing
at infinity. For S ⊂M we define
I(S) :=
{
f ∈ C0(M) : f |S = 0
}
and call both I(S) and I(S)∩C∞(M) the vanishing ideal of S (it will be clear from the context
which one we are referring to). Recall that a derivation D of an algebra A is a K-linear map
A→ A satisfying the Leibniz rule
D(ab) = D(a)b+ aD(b), a, b ∈ A. (4)
The set of all derivations of A will be denoted by Der(A). A derivation is inner if it is of the
form a 7→ [x, a] for some fixed x ∈ A. The set of all inner derivations will be denoted by Inn(A).
2.2 Submanifold algebras
Let π : A→ B be a surjective homorphism of associative algebras over a field K and J := kerπ.
Let
Derπ(A) :=
{
D ∈ Der(A) : D(a) ∈ J ∀ a ∈ J
}
.
Lemma 1. For every D ∈ Derπ(A) there is a unique derivation D̃ ∈ Der(B) such that D̃ ◦ π =
π ◦D.
On the Notion of Noncommutative Submanifold 3
Proof. Let a, b ∈ A with a − b ∈ J . Then D(a) − D(b) ∈ J and π(D(a)) ∈ B depends only
on the class x := π(a) of a. We define D̃(x) := π(D(a)). Now if x = π(a) and y = π(b), then
xy = π(ab), and
D̃(xy) = π(D(ab)) = π(D(a)b+ aD(b)) = D̃(x)y + xD̃(y).
Similarly one proves that D̃ is linear. �
The assignment D 7→ D̃ gives a Lie algebra map:
π∗ : Derπ(A)→ Der(B).
Definition 2. If π∗ is surjective, we will call π : A → B a coembedding1 and B a submanifold
algebra of A.
Surjectivity of (3) guarantees, among other things, that there is a surjective homomorphism
ΩDer(A)→ ΩDer(B) of “minimal” derivation based differential calculi [32, Proposition IV.1].
It is straightforward to check that, since J is a two-sided ideal, Inn(A) is a Lie subalgebra of
Derπ(A) and π∗ maps Inn(A) surjectively to Inn(B).
Remark 3. If all derivations of B are inner, then B is a submanifold algebra of A.
In Section 3 we will study the case of commutative algebras, where no non-zero derivation is
inner.
2.3 Compositions of coembeddings
Given a sequence A
f−→ B
g−→ C of coembeddings, in general g∗ ◦ f∗ and (g ◦ f)∗ are not equal
(they don’t even have the same domain). It is then not obvious that the composition g ◦f is still
a coembedding. In order to discuss compositions of coembeddings, it is convenient to rephrase
properties of derivations in terms of homomorphisms of associative algebras. We are going to
need the next lemma, whose proof is straightforward.
Lemma 4. Let A be an associative algebra. A map D : A→ A is a derivation if and only if the
map
A→M2(A), a 7→
(
a D(a)
0 a
)
(5)
is a homomorphism (of associative algebras).
A homomorphism A → M2(A) will be called admissible if it is of the form (5) for some
(unique) D ∈ Der(A).
Given a map f : A → B, we will denote by F : M2(A) → M2(B) the map obtained by app-
lying f to each matrix element. The definition of coembedding can then be restated in the
following form.
Lemma 5. A surjective homomorphism f : A → B is a coembedding if and only if for every
admissible homomorphism ξ : B → M2(B) there exists an admissible homomorphism η : A →
M2(A) making the following diagram commute
1Since in the motivating example of smooth manifolds it is somehow dual to a (closed) embedding.
4 F. D’Andrea
A B
M2(A) M2(B)
f
F
η ξ
It is now easy to prove the following:
Proposition 6. Consider a sequence A
f−→ B
g−→ C of maps between associative algebras. If f
and g are coembeddings, then g ◦ f is a coembedding.
Proof. The composition g◦f of two surjective homomorphisms is a surjective homomorphisms.
Consider the diagram:
A B
M2(A) M2(B)
C
M2(C)
f g
F G
η ξ φ
If φ is admissible, since g is a coembedding there exists an admissible ξ making the right square
commute. But f is a coembedding as well, so there exists an admissible η making the left
square – and then the outer rectangle – commute. Since for every φ there exists η making the
outer rectangle commute, g ◦ f is a coembedding. �
3 Function algebras
3.1 Submanifold algebras of commutative C∗-algebras
Let us start with topological spaces. Suppose that (1) is an exact sequence of commutative
complex C∗-algebras. It is well known that, by Gelfand duality, A ' C0(M) and B ' C0(N)
where M and N are locally compact Hausdorff spaces. Every ∗-homomorphism π : C0(M) →
C0(N) is the pullback of a proper continuous map F : N → M , and it is surjective only if F is
injective: it is then a topological embedding of N as a closed subset of M . Up to isomorphisms,
every exact sequence (1) of commutative C∗-algebras is then of the form
0→ I(S)→ C0(M)
π−→ C0(S)→ 0
with S ⊂ M a closed topological subspace and π(f) := f |S ∀ f ∈ C0(M) (and every inclusion
S ↪→ M of a closed subset gives rise to such a short exact sequence). We refer to Chapter 1
of [23] for the details.
Since commutative C∗-algebras have no non-zero derivations,2 the associated map π∗ is sur-
jective as well.
3.2 Submanifold algebras of C∞-algebras
The crucial point in the discussion in previous section is that the functor C0 is an equivalence
between the category of locally compact Hausdorff topological spaces, with proper continuous
functions, and the (opposite) category of commutative C∗-algebras with ∗-homomorphisms (see,
e.g., [4, Section II.2.2.7]).
2In fact, a stronger statement is due to Kadison [27, Theorem 2]: each derivation of a (possibly non-
commutative) C∗-algebra annihilates its center.
On the Notion of Noncommutative Submanifold 5
Things with smooth manifolds become more involved. We can associate to every smooth
manifold (without boundary) M the algebra C∞(M), and to any smooth map F : M → N
the homomorphism F ∗ : C∞(N) → C∞(M). This gives a functor from the category of smooth
manifolds to the category of commutative (real unital associative) algebras, which obviously fails
to be surjective on objects.3 Nevertheless it is a full and faithful functor (see [28, Corollary 35.10]
or [34, Theorem 2.8]), and this is enough for our purposes.
Let us consider then a short exact sequence
0→ J → C∞(M)
π−→ B → 0, (6)
where we know a priori that the one in the middle is the algebra of smooth functions on some
smooth manifold M . We will assume that dim(M) ≥ 1 and use the standard identification of
Der(C∞(M)) with the set X(M) of smooth global vector fields on M .
There are several natural questions we may ask:
(i) Is there a commutative example where π is surjective but π∗ is not? (Are these two condi-
tions independent from each other or is every quotient algebra of C∞(M) a submanifold
algebra?)
(ii) Is there an example where π and π∗ are both surjective (B is a submanifold algebra
of C∞(M)), but B 6' C∞(N) for any smooth manifold N?
(iii) If we known that J is the vanishing ideal of a subset S ⊂ M and B ' C∞(M)/J a sub-
manifold algebra, can we conclude that S is a submanifold of M?
(iv) What can we conclude under the assumption that B = C∞(N)?
The first two examples give a positive answer to question (ii).
Example 7 (dual numbers). Let M = R and B be the subalgebra of M2(R) spanned by the
identity matrix and the matrix
ε :=
(
0 1
0 0
)
.
Up to an isomorphism, B ' R[ε]/
(
ε2
)
is the algebra of dual numbers. Let π : C∞(R) → B be
the homomorphism (the identity matrix is omitted):
π(f) := f(0) + f ′(0)ε ∀ f ∈ C∞(R).
Clearly π is surjective, and it is not difficult to check that π∗ is surjective as well. One has
Der(C∞(R)) =
{
f 7→ φf ′ : φ ∈ C∞(R)
}
,
Der(B) =
{
λε d
dε : λ ∈ R
}
.
Here J = kerπ is given by those smooth functions on R vanishing at 0 together with their first
derivative. Given a derivation D : f 7→ φf ′, from (φf ′)′(0) = φ(0)f ′′(0) + φ′(0)f ′(0) it follows
that the derivation maps J to J if and only if φ(0) = 0. Hence:
Derπ(C∞(R)) =
{
f 7→ φf ′ : φ ∈ C∞(R), φ(0) = 0
}
.
If Df := φf ′ with φ(0) = 0, then π∗(D) = φ′(0)ε d
dε . Since φ′(0) can be any real number, the
map π∗ is surjective.
3We could, however, consider the category of C∞-rings, for which there are characterizations of those objects
that are isomorphic to C∞(M) for some M . See, e.g., [33].
6 F. D’Andrea
We saw in previous example that: the algebra of dual numbers is a submanifold algebra
of C∞(R); it is not isomorphic to any algebra of smooth functions on a manifold, since it has
a non-zero nilpotent element ε; the kernel of π is not the vanishing ideal of any subset of R,
even if Derπ(C∞(R)) is the set of vector fields on R vanishing on the subset S = {0}.
More generally, for arbitrary M , given a point p ∈M and a non-zero tangent vector v ∈ TpM ,
we can construct a surjective homomorphism
π : C∞(M)→ R[ε]/
(
ε2
)
, π(f) := f(p) + v(f)ε,
and prove (using local coordinates) that π∗ is surjective as well.
Example 8 (germs of smooth functions). Fix p ∈ M , let B := C∞p (M) be the algebra of all
germs of smooth functions at p, and let
π : C∞(M)→ C∞p (M)
be the map sending a function to its germ at p. Such a map is surjective (every germ can be
represented by a global smooth function). Its kernel J is the set of all functions that are zero
near p; since derivations are local, X(J) ⊂ J for every X ∈ Der(C∞(M)) and Derπ(C∞(M)) ≡
Der(C∞(M)). The map
π∗ : X(M)→ TpM
is just the evaluation of vector fields at p. It is well known that such a map is surjective [30,
Proposition 8.7]. Stalks of germs of smooth functions on M are then examples of submanifold
algebras of C∞(M) not isomorphic to C∞(N) for any smooth manifold N .4
We want now to answer to question (iii). If S ⊂M is a closed subset of a smooth manifold M ,
we denote by C∞(S)◦ the algebra of real-valued functions on S that admit a smooth prolongation
to an open neighbourhood of S in M (and then to the whole M , cf. [30, Lemma 2.26]). This
is the standard way of defining smooth functions on a general subset of a manifold, when such
a subset has no intrinsic smooth structure. If S is a closed embedded submanifold, then C∞(S)◦
coincides with the set of functions that are smooth with respect to the smooth structure of S.
(However, it should be noted that if S is an immersed submanifold this is not true.)
Theorem 9 (closed subsets). C∞(S)◦ is a submanifold algebra of C∞(M).
Proof. By construction, elements of C∞(S)◦ are functions on S that are in the image of the
restriction map π : C∞(M)→ C∞(S)◦, π(f) := f |S . We need to prove surjectivity of π∗.
Any derivation of C∞(S)◦ composed with the evaluation at a point p ∈ S gives a derivation
of C∞(M) at p, i.e., a vector Xp ∈ TpM . We need to prove that the map X : S → TM , p 7→ Xp,
is the restriction to S of a global smooth vector field on M .
Let p ∈ S, let
(
U,ϕ =
(
x1, . . . , xn
))
be a chart on M centered at p and B a coordinate ball
with p ∈ B and B ⊂ U . For all q ∈ U ∩ S we can write
Xq =
n∑
i=1
vi(q)
∂
∂xi
∣∣∣
q
for some (unique) vi(q) ∈ R. Choose smooth functions x̃i ∈ C∞(M) such that x̃i coincide with xi
on B [30, Lemma 2.26]. For every i = 1, . . . , n, since X(x̃i|S) ∈ C∞(S)◦ and π is surjective, we
can choose f i ∈ C∞(M) such that
X
(
x̃i|S
)
= π
(
f i
)
.
4By contraddiction, assume that C∞p (M) ' C∞(N) for some smooth manifold N . Since X(N) ' TpM is finite-
dimensional, N must be 0-dimensional. But then X(N) must be {0}, that implies dim(M) = dim(TpM) = 0.
On the Notion of Noncommutative Submanifold 7
For all q ∈ B ∩ S, one has f i(q) = Xq
(
x̃i
)
= Xq(x
i) = vi(q). As a consequence, the smooth
vector field Y ∈ X(U) given by
Y := f i
∂
∂xi
satisfies Y |B = X|B. The local vector field X is locally the restriction of a local smooth vector
field on M (it is a smooth vector field along S according to the terminology of [30]). It follows
from [30, Lemma 8.6] that X is the restriction to S of a global smooth vector field on M . �
Thus, for any closed subset S of a smooth manifold M , C∞(S)◦ is a submanifold algebra
of C∞(M) (with π the pullback of the inclusion S ↪→ M and J the vanishing ideal of S).
This notion of noncommutative submanifold, even in the commutative case, covers then more
examples than the standard geometric notion of submanifold. This is in some sense analogous
to what happens in diffeology, where on any subset of a diffeological space there is a canonical
“subset diffeology” induced by the ambient space [26].
We can now answer to question (i) and give a commutative example of quotient algebra that
is not a submanifold algebra.
Example 10 (the cross). Let S :=
{
(x, y) ∈ R2 : xy = 0
}
, let ı : R→ S be the map x 7→ (x, 0),
and π = ı∗ : C∞(S)◦ → C∞(R) its pullback. It follows from Theorem 9 that a derivation X of
C∞(S)◦ is a smooth vector field along S of the form
Xp =
2∑
i=1
vi(p)
∂
∂xi
∣∣∣
p
,
where x1, x2 are the standard coordinates on R2. Let f(x, y) := xy. The condition that
Xp(f) = 0 ∀ p ∈ S implies that v1(0, y) = 0 ∀ y 6= 0 and v2(x, 0) = 0 ∀x 6= 0. By continuity,
Xp=0 = 0. The image π∗(X) is a vector field on R vanishing at 0. The vector field ∂/∂x1 is not
in the image of π∗, which is then not surjective.
Notice that Theorem 9 doesn’t apply in this case, since S is not a manifold (it is not locally
Euclidean at the origin). So while it is true that C∞(S)◦ is a submanifold algebra of C∞
(
R2
)
(by Theorem 9), C∞(R) is not a submanifold algebra of C∞(S)◦.
Consider now an exact sequence (6), with B = C∞(N) the algebra of smooth function on
a second smooth manifold N (question (iv)). The following theorem holds:
Theorem 11. Let M and N be smooth manifolds and π : C∞(M) → C∞(N) a surjective
homomorphism. Then π = F ∗ is the pullback of a proper embedding F : N →M and S := F (N)
is a closed embedded submanifold of M .
Proof. By [28, Corollary 35.10], every homomorphism π : C∞(M) → C∞(N) is the pullback
of a smooth map F : N →M . Assuming that π = F ∗ is surjective, we will prove first that F is
injective, then that it is an immersion, and finally that is a proper embedding.
F is injective: by contraddiction, suppose that F is not injective: i.e., F (p) = F (q) for
some p 6= q in N . Then, every function f ∈ C∞(N) in the image of F ∗ has f(p) = f(q). Since
(for every p 6= q) there exists a smooth function with f(p) 6= f(q) (closed disjoint subsets of
a smooth manifold can be separated by a smooth function), F ∗ is not surjective.
F is an immersion: under the usual identification TpR ' R and Tpf = df |p for a scalar
function, for all f ∈ C∞(N) and all p ∈ N one has
df |p = dg|F (p) ◦ TpF, (7)
8 F. D’Andrea
where g ∈ C∞(M) is any function satisfying F ∗(g) = f . It follows from (7) that the covector
df |p ∈ T ∗pN vanishes on kerTpF for all f , which implies that kerTpF = {0} and F is an
immersion at p (covectors df |p span T ∗pN , as one can prove by taking the differential of smooth
prolongations of local coordinates).
F is a proper embedding: we proved that S := F (N) is an immersed submanifold
of M . It is well known that the restriction map C∞(M) → C∞(S), f 7→ f |S , to an immersed
submanifold is surjective if and only if S is properly embedded (see, e.g., [30, Example 5-18(b)]),
and in particular closed in M . �
An immediate corollary of previous theorem is that, for smooth functions on manifolds, the
condition of surjectivity of π∗ is redundant: if C∞(N) is a quotient algebra of C∞(M), then it
is also a submanifold algebra.
Example 10 allows us to illustrate a fundamental difference between embeddings and coem-
beddings. Embeddings of smooth manifolds satisfy the following property:
Proposition 12. Consider a sequence N
α−→ S
β−→M of smooth maps between smooth manifolds.
If β ◦ α is a closed embedding, then so is α.
The proof is straightforward if we rephrase it in terms of algebra morphisms. Recall that every
morphism C∞(M)→ C∞(S) is the pullback of a smooth map α : S →M [28, Corollary 35.10],
and it is surjective if and only if α is a closed embedding (Theorem 11). Proposition 12 is then
equivalent to the next Proposition 13, whose proof is straightforward:
Proposition 13. Consider a sequence C∞(M)
f−→ C∞(S)
g−→ C∞(N) of algebra morphisms. If
g ◦ f is surjective, then so is g.
Proof. The image of g ◦ f is a subset of the image of g. �
The noncommutative analogue of Proposition 13 does not hold. Namely, there are sequences
A
f−→ B
g−→ C of surjective algebra maps such that f and g ◦ f are coembeddings but g is not.
Consider for example the sequence C∞
(
R2
) f−→ C∞(S)◦
g−→ C∞(R), where S is the cross, f is
the pullback of the inclusion S ↪→ R2 and g the map in Example 10. By Theorem 9, the map f
is a coembedding, and g ◦ f is the pullback of the inclusion R → R2 as horizontal axis, so it is
a coembedding as well. However g is not a coembedding, cf. Example 10.
3.3 Polynomial algebras
For commutative algebras an equivalent formulation of the condition of submanifold algebra is
via Kähler differentials. Let A be a commutative algebra over a field K, M an A-module and
(ΩA/K , d) the module of Kähler differentials. Then, the map
HomA(ΩA/K ,M) ∼−→ DerK(A,M),
f 7→ f ◦ d
sending A-linear maps ΩA/K → M to M -valued derivations of A (K-linear maps A → M
satisfying (4)) is a bijection [18, Chapter 16]. The universal derivation will be denoted always
by d, whatever is the algebra considered.
If π : A→ B is a homomorphism of commutative algebras, since d◦π : A→ ΩB/K is a deriva-
tion, by the universal property of Kähler differentials there exists π∗∗ ∈ HomA(ΩA/K ,ΩB/K)
(where we think of B-modules as A-modules via π) that makes the following diagram commute:
On the Notion of Noncommutative Submanifold 9
ΩA/K
ΩB/K
A
B
π
d
d
π∗∗
Explicitly π∗∗(adb) = π(a)dπ(b) ∀ a, b ∈ A. Note that if π is surjective, π∗∗ is surjective as well.
We can now rephrase the definition of submanifold algebra in terms of Kähler differential.
Proposition 14. Let π : A → B be a surjective homomorphism of commutative algebras.
Then B is a submanifold algebra of A if and only if for every f ∈ HomB(ΩB/K , B) there exists
f̃ ∈ HomA(ΩA/K , A) that makes the following diagram commute:
ΩA/K
ΩB/K
A
B
π
f̃
f
π∗∗
Proof. The situation is illustrated in the following diagram:
A
B
ΩA/K
ΩB/K
A
B
π
d
d
π∗∗
f̃
f
D̃
D
π
Suppose every D ∈ Der(B) admits a lift D̃ making the outer diagram commutative (automati-
cally D̃(J) ⊂ J and D̃ ∈ Derπ(A)). Then for every f ∈ HomB(ΩB/K , B) we can lift the deriva-
tion D = f ◦ d to a derivation D̃ of A. By universality D̃ = f̃ ◦ d for some f̃ ∈ HomA(ΩA/K , A),
and f̃ is the lift of f we are looking for.
Vice versa, if every f admits a lift f̃ , given any D ∈ Der(B) and given f such that f ◦d = D,
we can lift f to f̃ and form the derivation D̃ = f̃ ◦ d. Automatically π∗(D̃) = D, so that π∗ is
surjective. �
Proposition 15. If A is commutative and ΩA/K is a free A-module, then every quotient algebra
of A is a submanifold algebra.
Proof. It follows from Proposition 14. Let S be a free generating set for ΩA/K . Then, every
Kähler differential can be written in a unique way as a finite sum
∑
ξ∈S
aξ · ξ with aξ ∈ A. Let
f ∈ HomB(ΩB/K , B). For every ξ ∈ S there exists (by surjectivity of π) an element ξ̃ ∈ A such
that π
(
ξ̃
)
= f ◦ π∗∗(ξ). Let f̃ be defined by
f̃
(∑
ξ∈S
aξ · ξ
)
:=
∑
ξ∈S
aξ · ξ̃.
By construction f̃ ∈ HomA(ΩA/K , A) and π ◦ f̃ = f ◦ π∗∗ as requested. �
10 F. D’Andrea
Example 16 (affine sets). Let A := K[x1, . . . , xn], S ⊂ Kn an affine algebraic set, J the radical
ideal of S, B := A/J its coordinate ring and π : A→ B the quotient map. Since ΩA/K is a free
A-module [18, Proposition 16.1], B is a submanifold algebra of A.
Similarly to the case of smooth functions, the coordinate algebra B of S in previous example
is a submanifold algebra even when S is not a smooth manifold. In the case of affine varieties
there is however a simple criteria to know if S is smooth by looking at derivations: if K is
algebraically closed and S is irreducible, then S is a smooth submanifold of Kn if and only if
the Lie algebra Der(B) is simple (see, e.g., [3]).
We close this section with an example of quotient of a polynomial algebra that is not a sub-
manifold algebra (the polynomial version of Example 10).
Example 17 (the ordinary double point). Let A := K[x, y]/ (xy), B := K[x] and π the surjec-
tive homomorphism defined by π(y) = 0. A simple computation shows that: Der(A) is generated
by the two derivations x ∂
∂x and y ∂
∂y ; Derπ(A) = Der(A); the image of π∗ is freely generated (as
a B-module) by x d
dx ; Der(B) is freely generated by d
dx . The derivation d
dx is not in the image
of π∗, that is then not surjective.
4 Noncommutative submanifolds: examples
In parallel with previous section, let’s start with (separable complex) C∗-algebras. In this case,
it is well known that any derivation of a quotient algebra can be lifted [35]. Thus:
Example 18 (C∗-algebras). If π : A→ B is a surjective morphism between separable complex
C∗-algebras, then π∗ is surjective.
In Section 3 we studied examples with no non-zero inner derivations. On the other side
of the spectrum, any quotient algebra with only inner derivations is a submanifold algebra
(Remark 3). This includes the universal enveloping algebra of a semisimple finite-dimensional
Lie algebra (over a field K of characteristic 0), any central simple algebra (as a corollary of
Skolem–Noether theorem, cf. [13, p. 80, Example 4c]), Weyl algebras [12, Lemma 4.6.8], all
von Neumann algebras (Sakai–Kadison theorem [27, 36]) and in particular finite-dimensional
complex C∗-algebras (since they coincide with their weak closure). It is not difficult to extend
the latter result to the real case. Let us record the result for future use.
Lemma 19. Let B be a finite-dimensional (real or complex) C∗-algebra. Then all derivations
of B are inner.
Proof. We need to prove the statement in the real case. Assume then that B is real, so that
BC = B + iB is a complex finite-dimensional C∗-algebra. Every R-linear endomorphism φ of
the vector space underlying B can be extended to a C-linear endomorphism φC of BC by
φC(a+ ib) = φ(a) + iφ(b), ∀ a, b ∈ B.
It is straightforward to check that if φ ∈ Der(B) then φC ∈ Der(BC). Since BC has only inner
derivations, φC = [x+ iy, · ] for some fixed x, y ∈ B. But then for all a ∈ B:
φ(a) = φC(a) = [x, a] + i[y, a].
Since φ has image in B, y must be central, and φ(a) = [x, a] for all a ∈ B. �
As a corollary:
On the Notion of Noncommutative Submanifold 11
Example 20. Let π : A→ B be a surjective homomorphism with B a finite-dimensional real or
complex C∗-algebra (and A any real or complex algebra). Then B is a submanifold algebra of A.
In this list of sporadic examples, the next is a noncommutative example of quotient algebra
that is not a submanifold algebra.
Example 21. Let A := U(sb(2,R)) be the universal enveloping algebra of the Lie algebra
spanned by two elements H and E with relation [H,E] = E. Let B := R[x] be the algebra
of polynomials in an indeterminate x. By Peter–Weyl theorem A has basis
{
HjEk
}
j,k≥0
. The
map π : A→ B defined by
π
(
HjEk
)
= xjδk,0
is a surjective homomorphism. The kernel J has basis
{
HjEk
}
j≥0,k≥1
and is generated by E.
From [
H, 1
kH
jEk
]
= HjEk ∀ j ≥ 0, k ≥ 1
we deduce that J is contained in the commutator ideal, and since B ' A/J is commutative
J = [A,A] must be exactly the commutator ideal.
Let us prove that the map π∗ is not surjective. Let D̃ ∈ Der(B) be the derivation D̃ = x d
dx
and suppose D̃ = π∗(D) for some D. Set
D(H) =
∑
j,k≥0
aj,kH
jEk and D(E) =
∑
j,k≥0
bj,kH
jEk
with aj,k, bj,k ∈ R. From D̃ = π∗(D) we deduce that a1,0 = 1 and aj,0 = 0 for all j 6= 1. From
D(E) ∈ J we deduce that bj,0 = 0 ∀ j ≥ 0. From the Leibniz rule we get
0 = D([H,E]− E) = [D(H), E] + [H,D(E)]−D(E)
=
[
H +
∑
j≥0, k≥1
aj,kH
jEk, E
]
+
[
H,
∑
j≥0, k≥1
bj,kH
jEk
]
−
∑
j≥0,k≥1
bj,kH
jEk
= E +
∑
j≥0, k≥1
aj,k
[
Hj , E
]
Ek +
∑
j≥0, k≥2
bj,k(k − 1)HjEk.
Since
[
Hj , E
]
∈ J , we get 0 = E + (· · ·)E2, which is not zero whatever is the element that
multiplies E2. We arrived at a contraddiction, proving that D̃ /∈ Im(π∗).
Example 21 has a “folkloristic” interpretation: (the complexification of) the algebra A can
be interpreted as the coordinate algebra of κ-Minkowski space [31] in 1 + 1 dimensions, that
is generated by a time operator t and position operator q subject to the relation [t, q] = iκ−1q
(with κ 6= 0 a constant). The algebra B can be interpreted as the coordinate algebra on the
time “axis”. Such an axis is not a submanifold of κ-Minkowski in the sense of Definition 2.
4.1 Free algebras
Any homomorphism π : A → B maps the center Z(A) of A into the center Z(B) of B. If π is
surjective, π∗ : Derπ(A)→ Der(B) is a module map that covers the map π|Z(A) : Z(A)→ Z(B),
meaning that
π∗(zD) = π(z)π∗(D) ∀D ∈ Derπ(A) and z ∈ Z(A).
In Example 21 the map π|Z(A) : Z(A)→ Z(B) is not surjective (since Z(A) = K is trivial while
Z(B) = B is not). One may think that this condition has something to do with non-surjectivity
of π∗, but next example shows that this is not the case.
12 F. D’Andrea
Example 22 (free algebras). Let A = K〈x, y〉 be the free algebra with two generators x and y,
B = K[x, y], J = [A,A] the commutator ideal. Note that every derivation D sends the commu-
tator ideal into itself, since
D([a, b]) = [D(a), b] + [a,D(b)] ∀ a, b ∈ A.
Thus Derπ(A) = Der(A). Since Z(A) = K is the set of constant polynomials and Z(B) = B,
clearly π|Z(A) : Z(A)→ Z(B) is not surjective.
If we pick up two elements a, b ∈ A, there exists a unique derivation D such that D(x) = a
and D(y) = b (by universality, extended linearly and via the Leibniz rule). Thus Der(A) ' A2
as a vector space. For such a D, π∗(D) = π(a) ∂
∂x + π(b) ∂∂y , hence π∗ is surjective.
In fact, any quotient algebra of a free algebra is a submanifold algebra. The tensor algebra
example in [32] is a special case of Proposition 23.
Proposition 23. Let S be a set, A := K〈S〉 the free algebra generated by S and π : A → B
a surjective homomorphism to a second associative algebra B. Then B is a submanifold algebra
of A.
Proof. By the universal property, for every map f : S → C from S to an associative algebra C
there exists a unique homomorphism f̃ : A → C such that f̃ |S = f . Let D ∈ Der(B). For all
x ∈ S there exists δx ∈ A such that π(δx) = D(π(x)) (by surjectivity of π). Let C ⊂M2(A) be
the subalgebra of matrices of the form(
a b
0 a
)
, a, b ∈ A.
Let f : S → C be the map given by
f(x) :=
(
x δx
0 x
)
and f̃ : A → C the corresponding algebra morphism. For i = 1, 2 denote by pri : C → A the
projection on the matrix element in position (i, i). Since pri ◦f̃ : A → A is a homomorphism
given on S by pri ◦f = IdS , by unicity of the lift it must be pri ◦f̃ = IdA. Therefore
f̃(a) =
(
a D̃(a)
0 a
)
for some map D̃ : A → A. Since f̃ is a homomorphism, from Lemma 4 we deduce that D̃ is
a derivation.
Extend π to M2(A) in the obvious way. By construction
π(f̃(x)) =
(
π(x) π(δx)
0 π(x)
)
=
(
π(x) D(π(x))
0 π(x)
)
= f(π(x))
for all x ∈ S. Since π ◦ f̃ and f ◦ π are homomorphism that coincide on generators, they must
be equal, which means π ◦ D̃ = D ◦ π. The latter automatically implies that D̃ ∈ Derπ(A). �
4.2 Almost commutative spaces
One of the main applications of vector bundles in physics is to Yang–Mills theories: here L2
sections of a complex smooth Hermitian vector bundle π : E → M describe the physical state
of a particle (or several particles) “living” in the manifold M ; vectors in a fiber describe the
On the Notion of Noncommutative Submanifold 13
“internal” degrees of freedom of such a particle. In the celebrated Dirac equation, for example,
M is a 4-dimensional Riemannian spin manifold, π : E → M the spinor bundle and one looks
for solutions of the equation – among smooth sections of such a bundle – describing the state of
a couple particle-antiparticle with spin 1/2.
Inspired by Kaluza–Klein theories, where one derives a 4-dimensional Yang–Mills theory
coupled with gravity from a purely gravitational theory on some auxiliary 5-dimensional mani-
fold, A. Connes suggested to replace the unobserved extra dimension of Kaluza–Klein by a 0-
dimensional noncommutative space. The starting point is the tensor product C∞(M) ⊗ F of
smooth functions on a manifold and a finite-dimensional real algebra F , describing some kind of
virtual 0-dimensional noncommutative space. Starting from a purely geometric theory on such
a product, one is able to derive the complicated Lagrangian of the standard model of particle
physics coupled with gravity5 (see, e.g., [9] or [38] and references therein).
Following the point of view of Connes, an “almost commutative space” is something de-
scribed by the tensor product of the algebra of smooth functions on a manifold and some
finite-dimensional algebra or, more generally, by an algebra bundle over a manifold.
Let K = R or C and F be a finite-dimensional K-algebra. A smooth algebra bundle over
a (real) smooth manifold M , with typical fiber F , is a smooth vector bundle π : E →M whose
fibers are K-algebras and whose local trivializations give maps Ep → {p} × F (∀ p ∈ M) that
are not only isomorphisms of K-vector spaces, but of K-algebras as well [24, p. 377].
Given any smooth vector bundle π : E →M , it is well known that M is a submanifold of E
(via the zero section); moreover if S ⊂ M is a submanifold, then π−1(S) ⊂ E is a subman-
ifold (inverse image of a submanifold by means of a submersion) and in particular all fibers
Ep = π−1(p) are submanifolds of E. Having at our disposal an algebraic notion of submanifold,
we wonder if analogous properties hold for algebra bundles.
If π : E →M is an algebra bundle, the module Γ∞(π) of global smooth sections is a K-algebra
with pointwise product. For ξ, η ∈ Γ∞(π) we define
(ξ · η)(p) := ξ(p)η(p) ∀ p ∈M,
where the one on the right is the product in the fiber Ep. By construction for any p ∈ M , the
evaluation at p gives a homomorphism evp : Γ∞(π)→ Ep.
If E = M ×F and π is the projection on the first factor, then Γ∞(π) ' C∞(M)⊗F similarly
to the standard model example.
Proposition 24. Let π : E → M be an algebra bundle with typical fiber a finite-dimensional
real or complex C∗-algebra.6 Then, for every p ∈M , the map
evp : Γ∞(π)→ Ep
is a coembedding.
Proof. It is true for every vector bundle that any vector in a fiber can be extended to a global
smooth section. The map evp is then surjective. Since every derivation of Ep is inner (Lem-
ma 19), the induced map on derivations is surjective as well (Remark 3). �
In the example of the standard model, previous proposition can be interpreted by saying that
the 0-dimensional noncommutative space encoding the internal degrees of freedom of particles
is a “noncommutative submanifold” of the product space. One may wonder if M is a “non-
commutative submanifold” as well: it is difficult to answer such a question in general, since
5This is of course an oversimplification: the full story is beyond the scope of this paper. The interested reader
can consult the books [9, 38].
6For example F = C⊕H⊕M3(C) in the case of the standard model, where H is the (real) algebra of quaternions.
14 F. D’Andrea
a homomorphism Γ∞(π) → C∞(M) may not even exist. We will investigate this question for
trivial algebra bundles, i.e., tensor products of algebras, cf. Example 25.
Another example covered by Proposition 24 is the rational noncommutative torus. Let θ =
p/q ∈ Q be a rational number, with p and q coprime. The algebra of “complex-valued smooth
functions” on the noncommutative torus Tθ is isomorphic to Γ∞(π) with π : E → T2 a suitable
algebra bundle over the (ordinary) 2-torus [23]. The typical fiber is the algebra Mq(C) of all
q × q complex matrices. The spectral triple of the rational noncommutative torus was recently
studied from the point of view of algebra bundles in [6].
Example 25 (tensor products). Let K be any field and A := A1 ⊗A2 a tensor product of two
associative K-algebras. Suppose ε : A1 → K is a non-zero augmentation. Then π := ε⊗Id : A→
A2 is a surjective homomorphism. For every D ∈ Der(A2), the formula
D̃(a1 ⊗ a2) := a1 ⊗D(a2), ∀ a1 ∈ A1, a2 ∈ A2,
defines a derivation D̃ ∈ Derπ(A) satisfying by construction π∗(D̃) = D. Thus, A2 is a subman-
ifold algebra of A.
5 Formal deformations
A rich source of examples of “noncommutative spaces” is from deformation quantizations of
Poisson manifolds. Only in this section, C∞(M) will denote complex -valued smooth functions
on a (real) smooth manifold M .
Definition 26. A star product on a Poisson manifold (M, {, }) is a C[[}]]-bilinear associative
binary operation ? on C∞(M)[[}]] of the form:
f ? g =
∞∑
k=0
}kCk(f, g), ∀ f, g ∈ C∞(M), (8)
where each Ck : C∞(M)× C∞(M)→ C∞(M) is a bi-differential operator and for all f, g ∈
C∞(M):
C0(f, g) = fg is the pointwise multiplication,
C1(f, g)− C1(g, f) = 2i {f, g} is the Poisson bracket,
Ck(1, f) = 0 ∀ k ≥ 1 (1 is a neutral element for ?).
From now on we will always assume that C1 is antisymmetric,7 so that
C1(f, g) = i {f, g} .
If we stop the sum (8) at order r ≥ 1 and work over the ring C[}]/
(
}r+1
)
we get the notion of
order r deformation of a Poisson manifold.
In the framework of deformation quantization, we can consider the problem of star products
that are tangential to submanifolds, and investigate under what conditions every derivation
of the star product on the submanifold admits a prolongation (cf. Section 5.2). For 1st order
deformations this becomes a problem of prolongation of Poisson vector fields, i.e., vector fields
X ∈ X(M) satisfying
X
(
{f, g}
)
=
{
X(f), g
}
+
{
f,X(g)
}
, ∀ f, g ∈ C∞(M).
In Section 5.3 we will consider short exact sequences of formal deformations coming from
coisotropic reduction of a Poisson manifold.
7This can be done without loss of generality: any star product is equivalent to one with C1 antisymmetric [25,
Proposition 2.23].
On the Notion of Noncommutative Submanifold 15
5.1 }-linear derivations
Let A :=
(
C∞(M)[[}]], ?
)
be a deformation quantization of a Poisson manifold M . When
dealing with star products, we will only consider derivations of star products that are }-linear.
An element of Der(A) will be then a formal power series
D =
∞∑
k=0
}kDk
of differential operators on M satisfying the Leibniz rule:
D(f ? g) = (Df) ? g + f ? (Dg), ∀ f, g ∈ C∞(M).
At order 0 in } this means that D0 is a derivation of C∞(M), i.e., a vector field. At order 1 we
get
D1(fg)− (D1f)g − f(D1g) = C1(D0f, g) + C1(f,D0g)−D0C1(f, g).
Since in previous equality the left hand side is symmetric and the right hand side antisymmetric,
we deduce that they must both vanish. Thus, D1 must be a vector field as well, and D0
must be a Poisson vector field. If we are interested in first order deformations, this completely
characterizes derivations.
Lemma 27. An ε-linear derivation of
(
C∞(M)[ε]/
(
ε2
)
, ?
)
is a sum
D0 + εD1
of a Poisson vector field D0 and an arbitrary vector field D1 on M .
On a symplectic manifold, the correspondence between derivations of a star product and
Poisson (in this case symplectic) vector fields holds at any order in }. Suppose M is a symplectic
manifold and denote by Xsym(M)[[}]] the space of formal symplectic vector fields on M . Elements
of Xsym(M)[[}]] are formal power series
X =
∞∑
k=0
}kXk
with Xk a symplectic vector field on M for all k ≥ 0. Every symplectic vector field on a con-
tractible open set is Hamiltonian. We can then cover M by contractible open subsets, and on
each U of this cover find functions fUk ∈ C∞(U) such that
Xk(g)|U =
{
fUk , g
}
for all g ∈ C∞(M) and all k ≥ 0. These functions are determined by X up to an additive
constant (U is connected). We can then define a new function Dg ∈ C∞(M) given, on each
set U of this cover, by
Dg|U =
1
}
∑
k≥0
}k
(
fUk ? g − g ? fUk
)
.
For all g we get a well-defined formal power series of global smooth functions Dg, and a well
define derivation D of the star product. Such a derivation depends only on X. A simple
argument by induction shows that every derivation of the star product is in fact of this form:
Theorem 28 ([25, Proposition 3.5]). Every }-linear derivation of a star product on a symplectic
manifold M corresponds to a formal symplectic vector field via the construction above.
If H1(M,R) = 0 every symplectic vector field is Hamiltonian, and every derivation of the
star product is essentially inner,8 given by 1
} [f ?,] for some f ∈ C∞(M)[[}]].
8Inner except for the factor 1
} in front.
16 F. D’Andrea
5.2 Deformations of Poisson submanifolds
Suppose
π :
(
C∞(M)[[}]], ?
)
→
(
C∞(N)[[}]], ?
)
(9)
is a homomorphisms between deformation quantizations of two Poisson manifolds M and N .
We will assume that π is }-linear, i.e., of the form
π =
∞∑
k=0
}kπk (10)
where each πk maps C∞(M) to C∞(N) and is extended to formal power series by C[[}]]-linearity.
If we look at the condition π(f ? g) = π(f) ? π(g) at order 0 we get that π0 must be a ho-
momorphism between the commutative algebras C∞(M) and C∞(N), hence the pullback of
a smooth map ϕ0 : N →M [28, Corollary 35.10]. At order 1 we get:
ϕ∗0
(
{f, g}
)
− {ϕ∗0f, ϕ∗0g} = π1(f)π1(g)− π1(fg).
Since the symmetric and antisymmetric part must both vanish, we deduce that ϕ0 is a Poisson
map and π1 a homomorphism, hence the pullback of a smooth map ϕ1 : N →M .
Lemma 29. The map (10) is surjective if and only if π0 : C∞(M)→ C∞(N) is surjective.
Proof. “⇒” If π is surjective, for all g =
∞∑
k=0
}kgk, with gk ∈ C∞(N), there exists f =
∞∑
k=0
}kfk,
with fk ∈ C∞(M), such that
π(f) = π0(f0) +O(}) = g = g0 +O(}).
Thus π0(f0) = g0 and π0 is surjective.
“⇐” Suppose π0 is surjective and let g =
∞∑
k=0
}kgk, with gk ∈ C∞(N). It follows from
surjectivity of π0 that the recursive equation
π0(fk) = gk −
k∑
j=1
πj(fk−j)
admits a solution f =
∞∑
k=0
}kfk, with fk ∈ C∞(M). Such a formal power series satisfies by
construction π(f) = g, hence π is surjective. �
If π0 is surjective, it follows from Theorem 11 that ϕ0(N) is a closed embedded Poisson
submanifold of M .
Conversely, suppose S is a closed embedded submanifold of a Poisson manifold M and J is
the vanishing ideal of S. If J is a Poisson ideal, i.e., {J, f} ⊂ J for all f ∈ C∞(M), the Poisson
structure of M induces one on S and S becomes a Poisson submanifold of M .9
A star product (8) on M is tangential to S if Ck(J, f) ⊂ J for all k ≥ 1 and for all f ∈ C∞(M).
If such a condition is satisfied, we get by restriction a star product on S. In such a situation, the
pullback of the inclusion ı : S →M extends by C[[}]]-linearity to a surjective homomorphism
ı∗ : A :=
(
C∞(M)[[}]], ?
)
→ B :=
(
C∞(S)[[}]], ?
)
. (11)
This is an instance of surjective homomorphism (9) where π = π0 = ı∗ has no higher order
terms. Among the examples in this class we find regular coadjoint orbits of compact Lie groups.
9Note that J being a Poisson ideal means that all Hamiltonian vector fields Xf := {f, · } belong to
Derı∗(C∞(M)).
On the Notion of Noncommutative Submanifold 17
Example 30 (regular coadjoint orbits). Let G be a compact connected Lie group, ı : O ↪→ g∗ a
regular coadjoint orbit, and equip C∞(g∗)[[}]] with BCH star product ?BCH. Then there exists
a star product ? on O and a series of g-invariant differential operators T = Id +
∞∑
k=1
}kTk on g∗
such that
ı∗ ◦ T :
(
C∞(g∗)[[}]], ?BCH
)
→
(
C∞(O)[[}]], ?
)
is a homomorphism [2, Theorem 5.2]. If we define a new star product ? on g∗ (equivariantly
equivalent to ?BCH) by
f ? g := T
(
T−1f ?BCH T
−1g
)
,
then the new star product is tangential to O, and
π := ı∗ :
(
C∞(g∗)[[}]], ?
)
→
(
C∞(O)[[}]], ?
)
is a surjective homomorphism. The coadjoint orbits of a compact connected Lie group are simply
connected [19, Theorem 2.3.7], thus H1(O,R) = 0, derivations of (C∞(O)[[}]], ?) are essentially
inner and π∗ is surjective.
One could conjecture that (11) is always a coembedding if S is symplectic, since at least
locally the derivations are essentially inner (Theorem 28). Unfortunately this is not the case, as
shown by the next lemma.
Lemma 31 (obstructions). Consider a surjective homomorphism like in (11).
(a) If there is a Poisson vector field on S that cannot be extended to a Poisson vector field
on M , then (11) is not a coembedding.
Assume that both M and S are symplectic. Then:
(b) the morphism (11) induces a linear map:
H1(M,R)→ H1(S,R). (12)
If (11) is a coembedding, the map (12) is surjective.
(c) If H1(M,R) = 0, then (11) is a coembedding if and only if H1(S,R) = 0.
Proof. (a) Let π = ı∗ be the pullback of the inclusion as in (11), and assume that π∗ is surjective
on derivations (of the star products). For any Poisson vector field D̃0 on S there is a derivation
D̃ of the star product on S such that D̃ = D̃0 +O(}) [37, Proposition 3.1]. By hypothesis, there
exists a derivation D = D0 +O(}) of the star product on M such that
π∗(D) = D̃.
This in particular means that D̃0 is the restriction to S of the vector field D0 on M . Being the
zeroth order part of a derivation (of a star product), D0 is a Poisson vector field on M . Thus,
if (11) is a coembedding, every Poisson vector field D̃0 on S can be extended to a Poisson vector
field D0 on M .
(b) Recall that on a symplectic manifold M the 1st de Rham cohomology is isomorphic to
the quotient of symplectic by Hamiltonian vector fields:
H1(M,R) ' Xsym(M)/Xham(M).
18 F. D’Andrea
The map π∗ sends symplectic vector fields into symplectic vector fields and Hamiltonian into
Hamiltonian, and it is surjective on Hamiltonian vector fields.10 If π is a coembedding, π∗ is
surjective on symplectic vector fields as well (as we proved at point (a)). The only thing we have
to prove is that every symplectic vector field Y is in the domain of π∗, i.e., satisfies Y (J) ⊂ J
where J is the vanishing ideal of S. In fact, we are going to prove that this is true for any
vector field.
Let ω be the symplectic form on M and Y ∈ X(M). Any 1-form on M can be written as
a finite sum
∑
finite
fidgi for some fi, gi ∈ C∞(M). Thus
ω(Y, · ) =
∑
finite
fidgi =
∑
finite
fiω(Xgi , · ) =
∑
finite
ω(fiXgi , · ),
for some fi, gi and where Xgi denotes the Hamiltonian vector field of gi. By non-degeneracy of ω:
Y =
∑
finite
fiXgi .
In other words, Xham(M) generates X(M) as a C∞(M)-module.
Now, Xgi(J) ⊂ J since J is a Lie ideal, and fiXgi(J) ⊂ J since J is an associative ideal,
hence the thesis: Y (J) ⊂ J .
(c) is a simple corollary of (b). If H1(S,R) = 0, then up to a factor 1/} every derivation
of B is inner and π∗ is surjective on inner derivations. If H1(S,R) 6= 0, then (12) cannot be
surjective and (11) cannot be a coembedding. �
Examples of pairs of symplectic manifolds M,S with S (closed embedded) symplectic sub-
manifold of M , H1(M,R) = 0 and H1(S,R) 6= 0 can be constructed as follows. Take M = CPn
a complex projective space with standard symplectic structure and S any complex submanifold:
CPn has vanishing cohomology in odd degree, and complex submanifolds are symplectic sub-
manifolds. Now it is not difficult to find examples where S is not simply connected: take any
Riemann surface with genus ≥ 1.
An easier and explicit example of quotient map (11) that is not a coembedding is the following.
Example 32. On M := R2 with Cartesian coordinates (x, y) consider the commuting vector
fields
X :=
∂
∂x
, Y := y
∂
∂y
,
the Poisson structure given by the bivector field X ∧ Y , and the associated Weyl-type star
product
f ? g := µ ◦ ei}(X⊗Y−Y⊗X)(f ⊗ g), ∀ f, g ∈ C∞
(
R2
)
, (13)
where µ is the pointwise multiplication map.
Embed S := R in R2 as horizontal axis. Then ? is tangential to S and we have a surjective
homomorphism
π : A :=
(
C∞
(
R2
)
[[}]], ?
)
→ B := C∞(R)[[}]] (14)
10This is true even when π is not the pullback of the inclusion and in full generality follows from the fact that
Hamiltonian vector fields are the order zero part of essentially inner derivations, which can always be lifted. For
every f ∈ C∞(S), the essentially inner derivation 1
} [f ?, · ] = {f, · }+O(}) of the star product on S and can be lifted
to a derivation 1
}
[
f̃ ?, ·
]
= {f̃ , · }+O(}) of the star product on M by choosing an extension f̃ ∈ C∞(M) of f . The
map π∗ sends the Hamiltonian vector field {f̃ , · } to {f, · }, and it is then surjective on Hamiltonian vector fields.
On the Notion of Noncommutative Submanifold 19
where the product on the right is the C[[}]]-linear extension of the pointwise product. A vector
field
D0 = a
∂
∂x
+ b
∂
∂y
, a, b ∈ C∞
(
R2
)
,
is a Poisson vector field on R2 if and only if
X(a) + Y (b) = 0.
This implies ∂a
∂x |y=0 = 0, so a(x, 0) = c is constant on the horizontal axis and
π∗(D) = c
∂
∂x
+O(}).
The (Poisson) vector field x ∂
∂x on R is not the restriction of any Poisson vector field on R2.
Using Lemma 31(c) we conclude that the map (14) is not a coembedding.
Remark 33. Note that, similarly to Example 21, in Example 32 one has
x ? y − y ? x = 2i} y.
Replacing the formal parameter by a non-zero real number, C[x, y] with star product (13)
becomes an algebra isomorphic to the complexification of U(sb(2,R)).
5.3 Coisotropic reduction
A more general class of examples of surjective homomorphisms of formal deformations comes
from coisotropic reduction. Let us review the classical notion of phase space reduction from an
algebraic point of view. Suppose M is a Poisson manifold, that we interpret as a phase space of
a physical system, and imagine that the system is constrained to move in a (closed embedded)
submanifold S of M . To obtain a phase space which represents in some sense the true degrees of
freedom of the system, we can then perform phase space reduction. Recall that S is a coisotropic
submanifold of M if the vanishing ideal
J := I(S) =
{
f ∈ C∞(M) : f |S = 0
}
is closed under the Poisson bracket. Let
A :=
{
f ∈ C∞(M) : {f, J} ⊂ J
}
be the normalizer of J . Since J is closed under Poisson bracket, J ⊂ A. Since f 7→ −Xf :=
−{f, · } is a Lie algebra morphism, A is closed under Poisson bracket; sinceXfg = fXg+gXf and
J is an ideal, A is closed under pointwise product: it is a Poisson subalgebra of
(
C∞(M), { , }
)
.
Finally, by construction J is a Poisson ideal in A.11 We thus have an exact sequence (1) of
Poisson algebra maps, where the quotient algebra B is interpreted as algebra of smooth functions
on what we would call geometrically the reduced phase space Mred.12
“Good” star products on M induce formal deformations of the constrained ideal, normalizer
and reduced phase space, fitting an exact sequence:
0→ (J [[}]], ?)→ (A[[}]], ?)
π−→
(
C∞(Mred)[[}]], ?
)
→ 0. (15)
11The normalizer is indeed the largest Poisson subalgebra of
(
C∞(M), { , }
)
containing J as a Poisson ideal.
12Note that if S is a symplectic submanifold, then A = C∞(M) and Mred ' S.
20 F. D’Andrea
A procedure that always works when S has codimension 1 in M is in [22]. For a general discussion
of the problem one can see the review [5] and references therein. A more recent “categorical”
approach is in [11].
Since symplectic submanifolds are special examples of this construction, we cannot ex-
pect (15) to be always a coembedding.
As a concrete example one can take M := Cn+1 \ {0} with standard symplectic structure.
The submanifold S := S2n+1 is then coisotropic. Let ∂θ be the vector field on M (tangent to S)
generating the obvious U(1) action given by multiplication of all complex coordinates by the
same phase. The normalizer J is given by those functions f ∈ C∞(M) such that ∂θf vanishes
on S. The quotient algebra is isomorphic to the algebra of smooth functions on S2n+1 that
are U(1)-invariant, that we identify with smooth functions on Mred = CPn. It is shown in [22]
that Wick star product on M can be reduced to CPn and one has a sequence (15) of formal
deformations. In this example π∗ is surjective again for trivial reasons: Mred is simply connected
and all derivations are essentially inner.
One could enlarge the class of examples by looking at reductions coming from an action
of a Poisson–Lie group on a Poisson manifold [10], or at coisotropic submanifolds of Jacobi
manifolds (rather than Poisson) [29]. It is also worth mentioning the paper [7], where the authors
prove a relative version of Kontsevich’s formality theorem, involving formal deformations of pairs
of a manifold and a submanifold (covering in particular the case of Poisson manifold together
with a coisotropic submanifold). It could be interesting to see if there is any connection with
submanifold algebras. This investigation is postponed to a future work.
Acknowledgements
I would like to thank Alessandro De Paris for suggesting Example 17 and Chiara Esposito for
her comments on a preliminary version of the paper. A special thanks goes to the anonymous
referees for carefully reading the paper and suggesting some interesting future research lines.
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1 Introduction
2 Preliminaries and notations
2.1 Notations
2.2 Submanifold algebras
2.3 Compositions of coembeddings
3 Function algebras
3.1 Submanifold algebras of commutative C*-algebras
3.2 Submanifold algebras of C-algebras
3.3 Polynomial algebras
4 Noncommutative submanifolds: examples
4.1 Free algebras
4.2 Almost commutative spaces
5 Formal deformations
5.1 -linear derivations
5.2 Deformations of Poisson submanifolds
5.3 Coisotropic reduction
References
|
| id | nasplib_isofts_kiev_ua-123456789-210700 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:03:41Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | D'Andrea, Francesco 2025-12-15T15:24:18Z 2020 On the Notion of Noncommutative Submanifold. Francesco D'Andrea. SIGMA 16 (2020), 050, 21 pages 1815-0659 2020 Mathematics Subject Classification: 46L87; 53C99; 53D55; 13N15 arXiv:1912.01225 https://nasplib.isofts.kiev.ua/handle/123456789/210700 https://doi.org/10.3842/SIGMA.2020.050 We review the notion of submanifold algebra, as introduced by T. Masson, and discuss some properties and examples. A submanifold algebra of an associative algebra 𝘈 is a quotient algebra 𝘉 such that all derivations of 𝘉 can be lifted to 𝘈. We will argue that in the case of smooth functions on manifolds, every quotient algebra is a submanifold algebra, derive a topological obstruction when the algebras are deformation quantizations of symplectic manifolds, present some (commutative and noncommutative) examples and counterexamples. I would like to thank Alessandro De Paris for suggesting Example 17 and Chiara Esposito for her comments on a preliminary version of the paper. A special thanks goes to the anonymous referees for carefully reading the paper and suggesting some interesting future research lines. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Notion of Noncommutative Submanifold Article published earlier |
| spellingShingle | On the Notion of Noncommutative Submanifold D'Andrea, Francesco |
| title | On the Notion of Noncommutative Submanifold |
| title_full | On the Notion of Noncommutative Submanifold |
| title_fullStr | On the Notion of Noncommutative Submanifold |
| title_full_unstemmed | On the Notion of Noncommutative Submanifold |
| title_short | On the Notion of Noncommutative Submanifold |
| title_sort | on the notion of noncommutative submanifold |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210700 |
| work_keys_str_mv | AT dandreafrancesco onthenotionofnoncommutativesubmanifold |