Quantum symmetry groups of noncommutative spheres

Joseph C. Várilly

Escuela de Matemática, Universidad de Costa Rica, 11501 San José, Costa Rica

Commun. Math. Phys. 221 (2001), 511–523

Abstract

We show that the noncommutative spheres of Connes and Landi are quantum homogeneous
spaces for certain compact quantum groups. We give a general construction of homogeneous
spaces which support noncommutative spin geometries.

1 Introduction
Noncommutative geometry [6] has established itself as a theory which goes beyond the realm of
differentiable manifolds and deals in a unified fashion with many singular geometric spaces, too. A
fundamental feature of NCG is that it fully incorporates all compact, boundaryless spin manifolds
under the heading of “noncommutative spin geometries”: see [7] and [17, Chap. 11].

Outstanding examples of singular geometric spaces are the noncommutative tori [5,9,25], orbit
spaces of discrete group actions, and leaf spaces of foliations. Recently, a new class of examples has
appeared, the “noncommutative spheres” of Connes and Landi [10], from a purely cohomological
construction.

The Moyal-like nature of the twisted products introduced in [10] suggests that the underlying
noncommutative spaces of these spin geometries may be obtained, as 𝐶∗-algebras, by the general
deformation construction of Rieffel [27]. The question arises as to whether these are in fact
noncommutative homogeneous spaces, that is, subalgebras of invariants of certain Hopf algebras
which may be regarded as “quantized symmetry groups”. This question is more delicate than it
might seem, because it must be answered at the 𝐶∗-algebra level: these “symmetry groups” must
be found in the category of “compact quantum groups” in the sense of Woronowicz [37] or perhaps
in the wider category of “locally compact quantum groups” [20]. As it happens, the compact
noncommutative spaces which we discuss below have compact (quantum) symmetry groups, so we
shall restrict ourselves here to Woronowicz’ version.

In Sections 2 and 3 we review the construction of noncommutative spheres and Rieffel’s 𝐶∗-
deformation theory. Section 4 treats compact quantum groups built by such deformations. In
Section 5, we explain how both constructions mesh to yield the desired quantum homogeneous
spaces. In the final section, we briefly discuss noncommutative spin geometries on these homoge-
neous spaces.

1



2 Quantized 4-spheres
The construction of noncommutative spin geometries by Connes and Landi proceeds in two stages.
First, the data (A,H, 𝐷, 𝐶, 𝜒) of an even real spectral triple [6,17] are sought as possible solutions
to a system of equations for the Chern character in cyclic homology:

ch𝑘 (𝑝) ≡ ⟨(𝑝 − 1
2 ) 𝑑𝑝

2𝑘⟩ = 0 for 𝑘 = 0, 1, . . . , 𝑚 − 1, (1a)
𝜋𝐷 (ch𝑚 (𝑝)) = 𝜒, (1b)

where 𝑝 = 𝑝2 = 𝑝∗ is an orthogonal projector in a matrix algebra 𝑀𝑟 (A), ⟨·⟩ denotes the conditional
expectation (or partial trace) ontoA, 𝜒 is the grading operator on theℤ2-graded Hilbert spaceH, and
𝜋𝐷 (𝑎0 𝑑𝑎1 · · · 𝑑𝑎𝑛) := 𝑎0 [𝐷, 𝑎1] · · · [𝐷, 𝑎𝑛] represents elements of the universal graded differential
algebra over A as operators on H.

These equations impose restrictions, first of all, on the algebra A itself. In dimension two,
i.e., when 𝑚 = 1 and 𝑟 = 2, only commutative solutions are found; in fact, Connes showed by an
elementary argument [8] – see also [17, Sect. 11.A] and [22] – that (1a) alone forces A to be a
commutative algebra whose Gelfand spectrum is a closed subset of the 2-sphere 𝕊2. This equation
also makes ch1(𝑝) a Hochschild 2-cycle, whose associated volume form is the standard volume
form on the sphere, so the Gelfand spectrum must be the whole 𝕊2, and thus A ≃ 𝐶∞(𝕊2) on the
basis of (1) alone!

Even in commutative cases such as this, where 𝐷 may be taken as the Dirac operator given by
some metric and spin structure on the spectrum of A, the final condition (1b) does not determine the
metric, but only its volume form; thus the cohomological conditions (1) allow for volume-preserving
variations of the metric, as befits a theory which aspires to incorporate gravity.

In dimension four, with 𝑚 = 2 and 𝑟 = 4, there is also a commutative solution given in [8],
namely the smooth function algebra 𝐶∞(𝕊4). Later, Connes and Landi [10] found a family of
noncommutative solutions, parametrized by a complex number of modulus one _ = 𝑒2𝜋𝑖\: these
are the algebras 𝐶∞(𝕊4

\
) (together with their corresponding Dirac operators), which may be called

“smooth function algebras for noncommutative 4-spheres 𝕊4
\
”, in the standard parlance of quantum

group theorists. Their representations are uniformly bounded and in each case a𝐶∗-norm is quickly
found, allowing to complete them to “continuous function algebras”, denoted 𝐶 (𝕊4

\
).

This procedure extends directly to higher dimensions, yielding noncommutative spheres in any
even dimension greater than 2 from the corresponding “instanton algebras” (so called because the
finite projective modules 𝑝A𝑟 may be regarded as vector bundles over A). Starting from the odd
Chern character in cyclic homology, one can also search for odd-dimensional noncommutative
spaces with this method (in the odd case, H is ungraded and 𝜒 in (1b) is replaced by 1).

A striking feature of this construction is that these noncommutative manifolds are parametrized
by numbers of modulus one, in contrast to the real numbers 𝑞 ≠ ±1 which label the well-known
2-spheres 𝕊2

𝑞𝑐 of Podleś [24], which were originally constructed as homogeneous spaces of the
compact quantum groups SU𝑞 (2). By combining features of both constructions, Da̧browski, Landi
and Masuda [12] built a family of quantized 4-spheres 𝕊4

𝑞; on computing the Chern characters of
the instantons, they found that (1a) is violated, inasmuch as ch1(𝑝) = (1− 𝑞2) times a nonvanishing
term.

In any case, it is clear that the Connes–Landi spheres 𝕊4
\

lie outside the realm of 𝑞-spheres of the
Podleś type. Indeed, several other variants on the 𝕊4

𝑞 spheres have since appeared [2, 3, 31], which,
however, do not incorporate the 𝕊4

\
family [11]. Of particular note is the construction by Hong and

2



Szymański [18] of a large family of quantized 𝑛-spheres 𝕊𝑛𝑞, for 𝑛 ⩾ 2 and 𝑞 > 0, by deforming
𝐶 (𝕊𝑛) to Cuntz–Krieger 𝐶∗-algebras based on certain directed graphs; but again, the 𝕊4

\
family is

not included. Therefore, it behooves us to ask whether that family may be realized as “quantum
homogeneous spaces”.

3 Deformations of homogeneous spaces
The second stage of the Connes–Landi construction is the provision of spin geometries on the spheres
𝕊4
\
. This is accomplished by a deformation of the commutative spectral triple (𝐶∞(𝕊4),H, /𝐷),

where /𝐷 denotes a Dirac operator on the Hilbert space H of square-integrable spinors over 𝕊4. In
the deformation, /𝐷 is kept fixed, so that all spectral data, including the classical dimension (four!)
of the geometry are unchanged: only the algebra and its representation on H are modified.

One declares a kind of Moyal product on 𝐶∞(𝕊4) by the following recipe: first, note that there
is an isometric action of the 2-torus 𝕋 2 on 𝕊4, allowing us to decompose any smooth function on
𝕊4 as a series 𝑓 =

∑
𝑟 𝑓𝑟 indexed by 𝑟 ∈ ℤ2, where 𝑓𝑟 lies in the 𝑟 th spectral subspace:

(𝑒2𝜋𝑖𝜙1 , 𝑒2𝜋𝑖𝜙2) · 𝑓𝑟 = 𝑒2𝜋𝑖(𝑟1𝜙1+𝑟2𝜙2) 𝑓𝑟 .

The series converges rapidly in the Fréchet topology of 𝐶∞(𝕊4). By introducing the following
star-product of homogeneous elements:

𝑓𝑟 × 𝑔𝑠 := 𝑒2𝜋𝑖\𝑟1𝑠2 𝑓𝑟𝑔𝑠, (2)

Connes and Landi constructed a representation of 𝐶 (𝕊4
\
) on the spinor space H (having bounded

commutators with /𝐷); in essence, the representation is explicit only on the smooth subalgebra, which
is just the vector space 𝐶∞(𝕊4) with the commutative product replaced by the star-product (2).

More generally, if 𝑀 is a compact Riemannian manifold admitting a Lie group of isometries of
rank 𝑙 ⩾ 2, so that 𝑀 carries an isometric action of the torus 𝕋 𝑙 , one can decompose 𝐶∞(𝑀) into
spectral subspaces indexed by ℤ𝑙 . The Moyal product of two homogeneous functions 𝑓𝑟 and 𝑔𝑠 is
then given by

𝑓𝑟 × 𝑔𝑠 := 𝜌(𝑟, 𝑠) 𝑓𝑟𝑔𝑠, (3)
where 𝜌 : ℤ𝑙 × ℤ𝑙 → 𝕋 is a 2-cocycle on the additive group ℤ𝑙 . The cocycle relation

𝜌(𝑟, 𝑠 + 𝑡)𝜌(𝑠, 𝑡) = 𝜌(𝑟, 𝑠)𝜌(𝑟 + 𝑠, 𝑡)

guarantees associativity of the new product. For instance [17, 34], one may take

𝜌(𝑟, 𝑠) := exp
{
−2𝜋𝑖

∑
𝑗<𝑘 𝑟 𝑗\ 𝑗 𝑘 𝑠𝑘

}
,

where \ = [\ 𝑗 𝑘 ] is a real 𝑙 × 𝑙 matrix. Complex conjugation of functions remains an involution for
the new product provided that the matrix \ is skewsymmetric.

The relation (3) is easily recognized as the product rule for the twisted group 𝐶∗-algebra
𝐶∗(ℤ𝑙 , 𝜌). We may replace 𝜌 by its skewsymmetrized version

𝜎(𝑟, 𝑠) := exp
{
−𝜋𝑖∑𝑙

𝑗 ,𝑘=1 𝑟 𝑗\ 𝑗 𝑘 𝑠𝑘
}
, (4)

because 𝜌 and 𝜎 are cohomologous [26], and we obtain 𝐶∗(ℤ𝑙 , 𝜎) = 𝐶 (𝕋 𝑙
\
), which is precisely the

𝐶∗-algebra of the noncommutative 𝑙-torus with parameter matrix \.

3



The relation (3) is clearly, then, a discretized version of the usual Moyal product, due to the
periodicity of the 𝕋 𝑙-action. Recall that the standard Moyal product on the phase space ℝ2𝑚 may be
expressed either by the familiar series in powers of ℏ whose first nontrivial term gives the Poisson
bracket, or alternatively in the integral form [16]:

( 𝑓 ×𝐽 𝑔) (𝑥) := (2𝜋ℏ)−𝑛
∬

𝑓 (𝑥 + 𝑠)𝑔(𝑥 + 𝑡) 𝑒𝑖𝑠·𝐽𝑡/ℏ 𝑑𝑠 𝑑𝑡,

where 𝐽 is the skewsymmetric matrix giving the standard symplectic structure on ℝ2𝑚 (and the dot
is the usual scalar product on ℝ2𝑚). This may be interpreted as an oscillatory integral for suitable
classes of functions and distributions on ℝ2𝑚, and yields the familiar series as an asymptotic
expansion in powers of ℏ [14, 35]. It is, therefore, a better starting point than that series for a 𝐶∗-
algebraic theory of deformations. Indeed, this was the form of the Moyal product used by Rieffel in
his general deformation theory [27]. He found, in fact, an improvement over the board by rewriting
it as

( 𝑓 ×𝐽 𝑔) (𝑥) :=
∬

𝑓 (𝑥 + 𝐽𝑠)𝑔(𝑥 + 𝑡) 𝑒2𝜋𝑖𝑠·𝑡 𝑑𝑠 𝑑𝑡.

He then generalized this to

𝑎 ×𝐽 𝑏 :=
∬
𝑉×𝑉

𝛼𝐽𝑠 (𝑎)𝛼𝑡 (𝑏) 𝑒2𝜋𝑖𝑠·𝑡 𝑑𝑠 𝑑𝑡, (5)

where 𝑎, 𝑏 belong to a 𝐶∗-algebra 𝐴, 𝛼 : 𝑉 → Aut(𝐴) is a (strongly continuous) action of a vector
group 𝑉 ≃ ℝ𝑙 on 𝐴, and 𝐽 is a skewsymmetric real 𝑙 × 𝑙 matrix. The oscillatory integral (5) makes
sense, a priori, only for elements 𝑎, 𝑏 of the smooth subalgebra 𝐴∞ of 𝐴 (under the action 𝛼), which
is a Fréchet pre-𝐶∗-algebra.

This problem of good definition is overcome [27] by introducing a suitable 𝐶∗-norm on 𝐴∞ for
which the ×𝐽 product is continuous, and then completing it in this norm to obtain the deformed
𝐶∗-algebra 𝐴𝐽 . The construction is functorial in that morphisms of 𝐴 restrict to 𝐴∞ and then
extend uniquely to morphisms of 𝐴𝐽 . In more detail: if (𝐴, 𝛼(𝑉)) and (𝐵, 𝛽(𝑉)) are two 𝐶∗-
algebras carrying actions of 𝑉 , and if 𝜙 : 𝐴 → 𝐵 is a ∗-homomorphism intertwining the actions 𝛼
and 𝛽, then 𝜙(𝐴∞) ⊆ 𝐵∞ and the restriction of 𝜙 to 𝐴∞ extends uniquely to a ∗-homomorphism
𝜙𝐽 : 𝐴𝐽 → 𝐵𝐽 . Moreover, if the original map 𝜙 is injective, then 𝜙𝐽 is injective, too; and 𝜙𝐽 is
surjective whenever 𝜙 is surjective.

In particular, when 𝐵 = 𝐴 and 𝛽 = 𝛼, each 𝛼𝑥 intertwines 𝛼 with itself since 𝑉 is an abelian
group, and this gives an action 𝛼𝐽 : 𝑉 → Aut(𝐴𝐽) whose restriction to 𝐴∞ coincides with 𝛼. Then
(𝐴𝐽 , 𝛼𝐽) can be deformed in turn, using a new skewsymmetric matrix 𝐾 , say; and the result turns
out to be isomorphic to 𝐴𝐽+𝐾 . By taking 𝐾 = −𝐽, we see that the change 𝐴 ↦→ 𝐴𝐽 is reversible. It
is therefore unsurprising, but still a deep and important result, that the smooth subalgebra remains
unchanged during this mutation: (𝐴𝐽)∞ = 𝐴∞ as vector spaces, although they have different
multiplications [27, Thm. 7.1].

The case of particular interest to us occurs when the action 𝛼 of𝑉 is periodic, so that 𝛼𝑥 = id𝐴 for
𝑥 ∈ 𝐿, a cocompact lattice in𝑉 ; in which case, 𝛼 is effectively an action of the compact abelian group
𝐻 = 𝑉/𝐿. Then 𝐴∞ decomposes into spectral subspaces labelled by elements of 𝐿 (or characters
of 𝐻) and one can check [27, Prop. 2.21] that if 𝛼𝑠 (𝑎𝑝) = 𝑒2𝜋𝑖𝑝·𝑠𝑎𝑝 and 𝛼𝑡 (𝑏𝑞) = 𝑒2𝜋𝑖𝑞·𝑡𝑏𝑞 with
𝑝, 𝑞 ∈ 𝐿, then

𝑎𝑝 ×𝐽 𝑏𝑞 = 𝑒−2𝜋𝑖𝑝·𝐽𝑞𝑎𝑝𝑏𝑞 .

4



On comparing this with (3) (with the cocycle 𝜌 replaced there by 𝜎), we see that it suffices to take
𝐴 := 𝐶 (𝕋 𝑙) and 𝐽 := 1

2\ in order to obtain any noncommutative torus𝐶 (𝕋 𝑙
\
) ≃ 𝐴𝐽 by this algorithm.

In fine, the isospectral deformation procedure of [10], based on the star-product (2), is, as far
as the algebra is concerned, a special case of Rieffel’s 𝐶∗-deformation theory. (This same point is
made by Sitarz in a recent announcement [32].)

Moreover, if the isometric action of 𝕋 𝑙 on 𝑀 alluded to above is free on some orbit, so that
𝑀 contains an embedded 𝑙-torus, then the restriction map 𝜋 : 𝐶 (𝑀) → 𝐶 (𝕋 𝑙) induces a surjective
∗-homomorphism 𝜋2𝐽 : 𝐶 (𝑀)2𝐽 → 𝐶 (𝕋 𝑙

\
), so that the noncommutative torus appears as a quotient

of the deformed 𝐶 (𝑀). In particular, if \ is irrational, then the noncommutative sphere 𝐶 (𝕊4
\
) is

not a type I 𝐶∗-algebra.

4 Compact quantum groups from deformations
It stands to reason, then, that this 𝐶∗-deformation process should yield compact quantum groups
when applied to the 𝐶∗-algebra 𝐶 (𝐺) of continuous functions on a compact Lie group. This proves
to be the case, by a further construction of Rieffel. There are two issues to address here: first,
which vector group actions on 𝐶 (𝐺) are admissible and useful, and second, how to deal with the
coproduct, counit and antipode which define the Hopf algebra structure of 𝐶 (𝐺) (or rather, of its
dense subalgebra of representative functions).

The solution to the second problem could not be simpler: the coalgebra structure and antipode
can be left completely untouched, and only the algebra structure need be deformed! The matter
is not quite trivial, as one must ensure that the coproduct is still an algebra homomorphism for
the new product. This possibility was pointed out by Dubois-Violette [13], who noticed that
Woronowicz’ matrix corepresentations for 𝐶 (SU𝑞 (𝑁)) and similar bialgebras could be seen as
different star-products on the same coalgebra.

Now suppose that 𝐻 is a closed connected abelian subgroup of 𝐺 (usually we may take 𝐻 to
be a maximal torus, but it is not really necessary that it be maximal); following Rieffel [28], we
consider the action of 𝐻 × 𝐻 on 𝐺 given by (ℎ, 𝑘) · 𝑥 := ℎ𝑥𝑘−1, and the corresponding action on
𝐶 (𝐺):

[(ℎ, 𝑘) · 𝑓 ] (𝑥) := 𝑓 (ℎ−1𝑥𝑘). (6)

We may regard this as a periodic action of the Lie algebra h⊕h, with the following notation. Choose
and fix a basis for the vector space h ≃ ℝ𝑙 , so that the exponential mapping from h onto 𝐻 may
be expressed as a homomorphism 𝑒 : ℝ𝑙 → 𝐻 whose kernel is the integer lattice ℤ𝑙 ; by taking
_ := 𝑒(1, 1, . . . , 1) we may write _𝑠 := 𝑒(𝑠) for 𝑠 ∈ ℝ𝑙 with a multiindex notation; the action of
𝑉 := h ⊕ h on 𝐶 (𝐺) is then written as

[𝛼(𝑠, 𝑡) 𝑓 ] (𝑥) := 𝑓 (_−𝑠𝑥_𝑡). (7)

(In the sequel, we shall refer to this as an action of h ⊕ h or of 𝐻 × 𝐻, interchangeably.) If 𝐽 is now
any skewsymmetric matrix in 𝑀2𝑙 (ℝ), then (5) now defines a Moyal product on 𝐶∞(𝐺), and the
procedure of Section 3 extends this to a 𝐶∗-algebra 𝐶 (𝐺)𝐽 , which yields a quantization of 𝐺 as a
noncommutative space.

The remaining difficulty is that an arbitrary choice of 𝐽 will not mesh well with the coalgebra
structure of 𝐶∞(𝐺), so we shall not always get a quantization of the group structure of 𝐺. For that,
one may follow the approach of Drinfeld by first equipping 𝐺 with a compatible Poisson bracket

5



(i.e., the product map𝐺 ×𝐺 → 𝐺 must be a Poisson map). By a well-known procedure [4] this can
be done at the infinitesimal level by equipping its Lie algebra g with a cocycle 𝜙 : g → Λ2g whose
dual defines a Lie bracket on g∗. For instance, one may take 𝜙(𝑋) = ad𝑋 (𝑟), where 𝑟 ∈ g ⊗ g is a
skewsymmetric solution of the classical Yang–Baxter equation [𝑟12, 𝑟13] + [𝑟12, 𝑟23] + [𝑟13, 𝑟23] = 0.
If 𝑟 =

∑
𝑘 𝑋𝑘 ⊗ 𝑌𝑘 , then since [𝑟12, 𝑟13] =

∑
𝑗 𝑘 [𝑋 𝑗 , 𝑋𝑘 ] ⊗ 𝑌 𝑗 ⊗ 𝑌𝑘 and similarly for the other terms,

this equation is satisfied when 𝑟 ∈ h ⊗ h for an abelian Lie subalgebra h of g. Although there
are other solutions (see [21] for an exhaustive treatment of Poisson Lie group structures on simple
compact Lie groups and the several algebraic quantizations of the Hopf algebra of representative
functions), we shall focus on the case 𝑟 ∈ h ⊗ h. On using our previous identification of h with ℝ𝑙 ,
we can write 𝑟 as a skewsymmetric 𝑙 × 𝑙 matrix 𝑄. The corresponding Poisson structure on 𝐺 is
given by the bivector field 𝑊 , where 𝑊𝑥 := _𝑥 (𝑟) − 𝜌𝑥 (𝑟) is the difference of the left and right
translates of 𝑟 from Λ2g to Λ2𝑇𝑥𝐺. Therefore [29], at the infinitesimal level we should take

𝐽 :=
(
𝑄 0
0 −𝑄

)
as the 2𝑙 × 2𝑙 matrix of deformation parameters for the action of h ⊕ h.

We can now write the twisted product on 𝐶∞(𝐺) as

( 𝑓 ×𝐽 𝑔) (𝑥) :=
∫
h4
𝑓 (_−𝑄𝑠𝑥_−𝑄𝑡)𝑔(_−𝑢𝑥_𝑣) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡·𝑣) 𝑑𝑠 𝑑𝑡 𝑑𝑢 𝑑𝑣. (8)

The coproduct Δ, the counit Y and the antipode 𝑆, which are defined on the Hopf algebra of
representative functions of 𝐺 by

Δ 𝑓 (𝑥, 𝑦) := 𝑓 (𝑥𝑦), Y( 𝑓 ) := 𝑓 (1), 𝑆 𝑓 (𝑥) := 𝑓 (𝑥−1), (9)

whereby Δ and Y are algebra homomorphisms and 𝑆 is an antiisomorphism, obviously extend to
algebra maps of𝐶 (𝐺) with the same properties. It is shown in [28,36] that they also satisfy the same
algebraic relations for the twisted product. The formulas (9) make sense for 𝑓 ∈ 𝐶∞(𝐺) or even
𝑓 ∈ 𝐶 (𝐺), although the usual requirementΔ(𝐶∞(𝐺)) ⊆ 𝐶∞(𝐺)⊗𝐶∞(𝐺) holds only if the algebraic
tensor product is replaced by the completed tensor product, which we denote by 𝐶∞(𝐺) ⊗̂ 𝐶∞(𝐺)
and identify with 𝐶∞(𝐺 ×𝐺). We can make a formal check of these homomorphism properties for
smooth functions:

(Δ 𝑓 ×𝐽 Δ𝑔) (𝑥, 𝑦) =
∫
h8
𝑓 (_−𝑄𝑠𝑥_−𝑄𝑡−𝑄𝑠′𝑦_−𝑄𝑡′)𝑔(_−𝑢𝑥_𝑣−𝑢′𝑦_𝑣′) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡·𝑣+𝑠′·𝑢′+𝑡′·𝑣′) 𝑑𝑠 · · · 𝑑𝑣′

=

∫
h8
𝑓 (_−𝑄𝑠𝑥_−𝑄𝑡′′𝑦_−𝑄𝑡′)𝑔(_−𝑢𝑥_−𝑢′′𝑦_𝑣′) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡′′·𝑣+𝑠′·𝑢′′+𝑡′·𝑣′) 𝑑𝑠 · · · 𝑑𝑣′

=

∫
h6
𝑓 (_−𝑄𝑠𝑥_−𝑄𝑡′′𝑦_−𝑄𝑡′)𝑔(_−𝑢𝑥_−𝑢′′𝑦_𝑣′) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡′·𝑣′) 𝛿(𝑡′′) 𝛿(𝑢′′) 𝑑𝑠 · · · 𝑑𝑣′

=

∫
h4
𝑓 (_−𝑄𝑠𝑥𝑦_−𝑄𝑡′)𝑔(_−𝑢𝑥𝑦_𝑣′) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡′·𝑣′) 𝑑𝑠 𝑑𝑡′ 𝑑𝑢 𝑑𝑣′

= ( 𝑓 ×𝐽 𝑔) (𝑥𝑦) = Δ( 𝑓 ×𝐽 𝑔) (𝑥, 𝑦).

6



Similarly,

( 𝑓 ×𝐽 𝑔) (1) =
∫
h4
𝑓 (_−𝑄(𝑠+𝑡))𝑔(_𝑣−𝑢) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡·𝑣) 𝑑𝑠 𝑑𝑡 𝑑𝑢 𝑑𝑣

=

∫
h4
𝑓 (_−𝑄(𝑠′))𝑔(_𝑣′) 𝑒2𝜋𝑖(𝑠′·𝑢+𝑡·𝑣′) 𝑑𝑠′ 𝑑𝑡 𝑑𝑢 𝑑𝑣′

=

∫
h2
𝑓 (_−𝑄(𝑠′))𝑔(_𝑣′) 𝛿(𝑠′) 𝛿(𝑣′) 𝑑𝑠′ 𝑑𝑣′ = 𝑓 (1) 𝑔(1),

so Y( 𝑓 ×𝐽 𝑔) = Y( 𝑓 )Y(𝑔). Next, if 𝑄 is invertible, then

(𝑆 𝑓 ×𝐽 𝑆𝑔) (𝑥) =
∫
h4
𝑓 (_𝑄𝑡𝑥−1_𝑄𝑠)𝑔(_−𝑣𝑥−1_𝑢) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡·𝑣) 𝑑𝑠 𝑑𝑡 𝑑𝑢 𝑑𝑣

= (det𝑄)−2
∫
h4
𝑓 (_−𝑡′𝑥−1_𝑠

′)𝑔(_−𝑣𝑥−1_−𝑢) 𝑒−2𝜋𝑖(𝑄−1𝑡′·𝑣+𝑄−1𝑠′·𝑢) 𝑑𝑠′ 𝑑𝑡′ 𝑑𝑢 𝑑𝑣

=

∫
h4
𝑓 (_−𝑡′𝑥−1_𝑠

′)𝑔(_−𝑄𝑣′𝑥−1_−𝑄𝑢
′) 𝑒2𝜋𝑖(𝑡′·𝑣′+𝑠′·𝑢′) 𝑑𝑠′ 𝑑𝑡′ 𝑑𝑢′ 𝑑𝑣′

= (𝑔 ×𝐽 𝑓 ) (𝑥−1) = 𝑆(𝑔 ×𝐽 𝑓 ) (𝑥),

using the skewsymmetry of 𝑄 in the third step; on the other hand, if 𝑄 = 0, then 𝑓 ×𝐽 𝑔 = 𝑓 𝑔

and the calculation reduces to (𝑆 𝑓 ×𝐽 𝑆𝑔) (𝑥) = 𝑓 (𝑥−1)𝑔(𝑥−1) = 𝑆(𝑔 ×𝐽 𝑓 ) (𝑥); since we may
integrate separately over the nullspace of 𝑄 and its orthogonal complement, we conclude that
𝑆 𝑓 ×𝐽 𝑆𝑔 = 𝑆(𝑔 ×𝐽 𝑓 ) in all cases.

The defining property of the antipode may also be checked in this manner: indeed, if𝑚( 𝑓 ⊗𝑔) :=
𝑓 ×𝐽 𝑔 for 𝑓 , 𝑔 ∈ 𝐶∞(𝐺), similar formal calculations quickly establish that

𝑚(id ⊗𝑆) (Δ 𝑓 ) = Y( 𝑓 ) 1 = 𝑚(𝑆 ⊗ id) (Δ 𝑓 ) (10)

whenever 𝑓 ∈ 𝐶∞(𝐺). However, it should be pointed out that the previous calculations in fact
involve oscillatory integrals of functions of 𝑠, 𝑡, 𝑢, 𝑣 ∈ ℝ𝑙 which have neither compact support
nor fast decrease; but with some additional careful analysis, it is shown in [27] that they remain
valid for smooth functions which have all derivatives bounded on ℝ𝑙 , as is always the case when
𝑓 , 𝑔 ∈ 𝐶∞(𝐺).

In summary, the product (8) on 𝐶∞(𝐺) is fully compatible with its original coalgebra structure
and antipode. The functoriality of the 𝐴𝐽 construction then lifts Δ and 𝑆 as algebra (anti)homo-
morphisms to the 𝐶∗-level. (Some bookkeeping is necessary because the source and target algebras
carry different actions of h ⊕ h in each case.) With 𝐴 = 𝐶 (𝐺) and 𝐽 = 𝑄 ⊕ (−𝑄) as before, we then
obtain a continuous ∗-homomorphism Δ𝐽 : 𝐴𝐽 → 𝐴𝐽 ⊗ 𝐴𝐽 (with the minimal 𝐶∗-tensor product)
and a continuous ∗-antihomomorphism 𝑆𝐽 : 𝐴𝐽 → 𝐴𝐽 . The counit Y on 𝐶∞(𝐺) also extends to a
character of 𝐴𝐽 .

Note, however, that the twisted product on 𝐶∞(𝐺) generally does not extend to a continuous
linear map from 𝐴𝐽 ⊗ 𝐴𝐽 to 𝐴𝐽 . (For one thing, 𝑚 is not an algebra homomorphism unless 𝐺 is
abelian.) Thus, the relation (10) is not helpful at the 𝐶∗-level. This is an old problem, and for unital
𝐶∗-algebras there is a well-known solution, described in the fundamental paper of Woronowicz [37].

7



Given a unital 𝐶∗-algebra 𝐴 and a unital ∗-homomorphism Δ : 𝐴→ 𝐴 ⊗ 𝐴 which is coassociative,
define linear maps𝑊 ,𝑊′ on the algebraic tensor product of 𝐴 with itself by

𝑊 (𝑎 ⊗ 𝑏) := (Δ𝑎) (1 ⊗ 𝑏) and 𝑊′(𝑎 ⊗ 𝑏) := (𝑎 ⊗ 1) (Δ𝑏).

(These are the Kac–Takesaki or “fundamental unitary” operators.) Woronowicz’ postulate is that
the maps𝑊 ,𝑊′ have dense range. Then (𝐴,Δ) is called a compact quantum group. The counit and
antipode are automatically defined on a dense ∗-subalgebra, and 𝐴 has a unique state (the “Haar
state”) which is both left and right invariant [37]. For 𝐴 = 𝐶 (𝐺), these maps are

𝑊 ( 𝑓 ⊗ 𝑔) (𝑥, 𝑦) := 𝑓 (𝑥𝑦)𝑔(𝑦), 𝑊′( 𝑓 ⊗ 𝑔) := 𝑓 (𝑥)𝑔(𝑥𝑦),

which have dense range in 𝐶 (𝐺 × 𝐺). After deformation, these become

𝑊 ( 𝑓 ⊗ 𝑔) := (Δ 𝑓 ) ×𝐽 (1 ⊗ 𝑔), 𝑊′( 𝑓 ⊗ 𝑔) := ( 𝑓 ⊗ 1) ×𝐽 (Δ𝑔),

for 𝑓 , 𝑔 ∈ 𝐶∞(𝐺), and these extend to invertible maps on 𝐶∞(𝐺 × 𝐺). Concretely, for ℎ ∈
𝐶∞(𝐺 × 𝐺),

𝑊ℎ(𝑥, 𝑦) =
∫
h4
ℎ(𝑥_−𝑄𝑠𝑦_−𝑄𝑡 , _−𝑢𝑦_𝑣) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡·𝑣) 𝑑𝑠 𝑑𝑡 𝑑𝑢 𝑑𝑣,

𝑊−1ℎ(𝑥, 𝑦) =
∫
h4
ℎ(𝑥_𝑄𝑡𝑦−1_−𝑄𝑠, _−𝑢𝑦_𝑣) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡·𝑣) 𝑑𝑠 𝑑𝑡 𝑑𝑢 𝑑𝑣,

as may be verified directly. It follows that𝑊 and likewise𝑊′ have dense range in 𝐶 (𝐺 × 𝐺).

5 Noncommutative spheres as homogeneous spaces
The standard 4-sphere is a homogeneous space of the 5-dimensional rotation group, namely:
𝕊4 ≈ SO(5)/SO(4). Note that SO(5) is a compact simple Lie group of rank two. More generally,
we may consider homogeneous spaces of the form 𝑀 = 𝐺/𝐾 , where 𝐺 is a compact Lie group
(which need not be semisimple) and 𝐾 is a closed subgroup. Let 𝐻 be a closed abelian subgroup
of 𝐾; then we can deform both𝐶 (𝐺) and𝐶 (𝐾) by the same action (7) of 𝐻×𝐻. Note in passing that
a maximal torus in SO(2𝑙) is carried onto a maximal torus of SO(2𝑙 + 1) by the standard inclusion
SO(2𝑙) ⊂ SO(2𝑙 + 1), so that even-dimensional spheres 𝕊2𝑙 = SO(2𝑙 + 1)/SO(2𝑙) fall under this
heading.

The left action of 𝐺 on 𝐺/𝐾 yields a ∗-homomorphism 𝜌 : 𝐶 (𝐺/𝐾) → 𝐶 (𝐺) ⊗ 𝐶 (𝐺/𝐾) by
𝜌 𝑓 (𝑥, 𝑦𝐾) := 𝑓 (𝑥𝑦𝐾). Restricted to smooth functions, this can be viewed as a left coaction of
𝐶∞(𝐺) on 𝐶∞(𝐺/𝐾). Let 𝐶 (𝐺)𝐾 denote the subalgebra of 𝐶 (𝐺) consisting of right-invariant
functions under the action of 𝐾 , so 𝑓 ∈ 𝐶 (𝐺)𝐾 if 𝑓 (𝑥𝑤) = 𝑓 (𝑥) whenever 𝑤 ∈ 𝐾 , 𝑥 ∈ 𝐺; and let
𝐶∞(𝐺)𝐾 := 𝐶 (𝐺)𝐾 ∩ 𝐶∞(𝐺). There is an obvious ∗-isomorphism Z : 𝐶 (𝐺)𝐾 → 𝐶 (𝐺/𝐾) given
by Z 𝑓 (𝑥𝐾) := 𝑓 (𝑥), and Z (𝐶∞(𝐺)𝐾) = 𝐶∞(𝐺/𝐾). The coproduct Δ of 𝐶∞(𝐺) maps 𝐶∞(𝐺)𝐾
into 𝐶∞(𝐺) ⊗̂ 𝐶∞(𝐺)𝐾 , the space of smooth functions ℎ on 𝐺 × 𝐺 for which ℎ(𝑥, 𝑦𝑤) ≡ ℎ(𝑥, 𝑦)
when 𝑤 ∈ 𝐾 . Moreover, if 𝑓 ∈ 𝐶∞(𝐺)𝐾 , then

[𝜌Z 𝑓 ] (𝑥, 𝑦𝐾) = Z 𝑓 (𝑥𝑦𝐾) = 𝑓 (𝑥𝑦) = Δ 𝑓 (𝑥, 𝑦) = [(id ⊗Z)Δ 𝑓 ] (𝑥, 𝑦𝐾),

8



so Z intertwines the coactions 𝜌 and Δ. In short, the algebra 𝐶∞(𝐺/𝐾), together with its isomor-
phism onto 𝐶∞(𝐺)𝐾 , is an embedded homogeneous space in the Hopf algebra 𝐶∞(𝐺).

Now we come to the main point. Since 𝐻 ⊆ 𝐾 , the left-right action (6) of 𝐻 × 𝐻 on both
𝐺 and 𝐾 induces a left action of 𝐻 on 𝐺/𝐾 , since the right action of 𝐻 is absorbed in the right
𝐾-cosets. If we deform𝐶 (𝐺) and𝐶 (𝐾) via the 𝐻 ×𝐻 action along the direction 𝐽 = 𝑄 ⊕ (−𝑄), the
corresponding effect on 𝐶 (𝐺/𝐾) should be a deformation under an 𝐻-action along the direction𝑄.
And so it proves.

To see that, we first notice that for 𝑓 , 𝑔 ∈ 𝐶∞(𝐺)𝐾 , (5) yields

( 𝑓 ×𝐽 𝑔) (𝑥) =
∫
h4
𝑓 (_−𝑄𝑠𝑥_−𝑄𝑡)𝑔(_−𝑢𝑥_𝑣) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡·𝑣) 𝑑𝑠 𝑑𝑡 𝑑𝑢 𝑑𝑣

=

∫
h4
𝑓 (_−𝑄𝑠𝑥)𝑔(_−𝑢𝑥) 𝑒2𝜋𝑖(𝑠·𝑢+𝑡·𝑣) 𝑑𝑠 𝑑𝑡 𝑑𝑢 𝑑𝑣

=

∫
h2
𝑓 (_−𝑄𝑠𝑥)𝑔(_−𝑢𝑥) 𝑒2𝜋𝑖𝑠·𝑢 𝑑𝑠 𝑑𝑢,

or

𝑓 ×𝐽 𝑔 =

∫
h2
𝛾𝑄𝑠 ( 𝑓 )𝛾𝑢 (𝑔) 𝑒2𝜋𝑖𝑠·𝑢 𝑑𝑠 𝑑𝑢, (11)

where (𝛾𝑡 𝑓 ) (𝑥) := 𝑓 (_𝑡𝑥) for 𝑓 ∈ 𝐶 (𝐺)𝐾 . The action of h on 𝐶 (𝐺/𝐾) may be defined as
(𝛽𝑡ℎ) (𝑥𝐾) := ℎ(_𝑡𝑥𝐾), so that Z intertwines the actions 𝛽 and 𝛾 of 𝐻. Then (11) becomes simply

Z 𝑓 ×𝑄 Z𝑔 = Z ( 𝑓 ×𝐽 𝑔) for all 𝑓 , 𝑔 ∈ 𝐶∞(𝐺/𝐾).

Finally, we can lift this isomorphism to the𝐶∗-level, using the functoriality of𝐶∗-deformations.
First, since Z : 𝐶 (𝐺/𝐾) → 𝐶 (𝐺)𝐾 is a ∗-isomorphism intertwining 𝛽 and 𝛾, its restriction to
𝐶∞(𝐺/𝐾) extends to a ∗-isomorphism of 𝐶 (𝐺/𝐾)𝑄 to 𝐶 (𝐺)𝐾

𝑄
, where the latter comes from the

action of 𝛾 on𝐶∞(𝐺)𝐾 . Of course, 𝛾 can be regarded as an action of 𝐻×𝐻 where the second factor
acts trivially; since elements of 𝐶 (𝐺)𝐾 are right-invariant under 𝐻, 𝛾 is just the restriction of the
action 𝛼 to 𝐶 (𝐺)𝐾 . This means that the inclusion 𝐶 (𝐺)𝐾 ↩→ 𝐶 (𝐺) is equivariant for the actions 𝛾
and 𝛼, and so its restriction to 𝐶∞(𝐺)𝐾 extends to a ∗-homomorphism from 𝐶 (𝐺)𝐾

𝑄
to 𝐶 (𝐺)𝐽 ; by

Proposition 5.8 of [27], this is still injective. In summary,

𝐶 (𝐺/𝐾) ≃ 𝐶 (𝐺)𝐾 ↩→ 𝐶 (𝐺) leads to 𝐶 (𝐺/𝐾)𝑄 ≃ 𝐶 (𝐺)𝐾𝑄 ↩→ 𝐶 (𝐺)𝐽 .

If the subgroup 𝐻 is not a maximal torus in either 𝐾 or 𝐺, the space of smooth elements for the
action of 𝐻×𝐻 will be strictly larger than𝐶∞(𝐺) (for instance, if the action is trivial, all continuous
functions are smooth in this sense); however, as clarified in Sect. 1 of [28], we may continue to
use 𝐶∞(𝐺) instead, because it will be dense in the Fréchet topology of the space of all smooth
elements, and therefore will remain dense in the deformed 𝐶∗-algebra 𝐶 (𝐺)𝐽 . The same applies,
mutatis mutandis, to 𝐶∞(𝐺/𝐾) and 𝐶 (𝐺/𝐾)𝑄 .

We have thus proved the following result.

Theorem 1. The deformed 𝐶∗-algebra 𝐶 (𝐺/𝐾)𝑄 is an embedded homogeneous space for the
compact quantum group 𝐶 (𝐺)𝐽 . □

9



Example 1. The even-dimensional noncommutative spheres 𝕊2𝑙
\

of Connes and Landi come directly
from this framework, for 𝑙 ⩾ 2. Just take𝐺 = SO(2𝑙 +1), 𝐾 = SO(2𝑙) and let 𝐻 ≃ 𝕋 𝑙 be a maximal
torus for 𝐾; then let 𝑄 = 1

2\, where \ is a skewsymmetric 𝑙 × 𝑙 matrix.
The odd-dimensional spheres 𝕊2𝑙+1 = SO(2𝑙 + 2)/SO(2𝑙 + 1) have somewhat different defor-

mations, since the 𝑙-dimensional maximal torus of SO(2𝑙 + 1) is not maximal in SO(2𝑙 + 2), so the
twisted product reduces to the ordinary commutative product along some directions.
Example 2. Our construction yields several new examples of homogeneous spaces. For instance,
if 𝑇 is a maximal torus of 𝐺, the flag manifold 𝐺/𝑇 may be deformed in any direction 𝑄 = −𝑄𝑡
in 𝑀𝑙 (ℝ) provided 𝑙 = dim𝑇 ⩾ 2. In particular, it yields a family of 6-dimensional quantized
manifolds 𝐶 (SU(3)/𝕋 2)𝑄 . It would be of interest to classify these up to isomorphism or Morita
equivalence.

At the algebraic level, there are other deformations of flag manifolds [21] which go beyond those
considered here, in that more general solutions of the classical Yang–Baxter equation are used for
the deformation directions. These could yield further examples of quantum homogeneous spaces.

6 Homogeneous noncommutative spin geometries
These new homogeneous spaces give rise to spectral triples, by the isospectral deformation procedure
of [10]. We may start from the manifold𝐺/𝐾 with, say, the normalized𝐺-invariant metric. Suppose
that 𝐺/𝐾 also has a homogeneous spin structure (if not, a homogeneous spinc structure will do).
Let 𝐷 be the corresponding Dirac operator, let 𝑋1, . . . , 𝑋𝑙 be the chosen basis of h, and let 𝑝 𝑗 be
the selfadjoint operator representing 𝑋 𝑗 on the spinor space H, for 𝑗 = 1, . . . , 𝑙. Since the action
of h integrates to a representation of 𝐻 on spinors, the operators 𝑝 𝑗 have integer or half-odd-integer
spectra, and for each 𝑟 ∈ ℤ𝑙 , there is a unitary operator 𝜎(𝑝, 𝑟) := exp

{
−2𝜋𝑖

∑
𝑗 ,𝑘 𝑝 𝑗𝑄 𝑗 𝑘𝑟𝑘

}
, using

the notation of (4); its inverse is 𝜎(𝑟, 𝑝). These operators commute with each other and also with 𝐷,
although not with the representation of 𝐶∞(𝐺/𝐾) on H. Any bounded operator 𝑇 in the common
smooth domain of the transformations 𝑇 ↦→ 𝜎(𝑝, 𝑟)𝑇𝜎(𝑟, 𝑝) has a decomposition 𝑇 =

∑
𝑟∈ℤ𝑙 𝑇𝑟 ,

where 𝜎(𝑝, 𝑟) 𝑇𝑠 = 𝑇𝑠 𝜎(𝑝 + 𝑠, 𝑟) for 𝑟, 𝑠 ∈ ℤ𝑙 ; define

𝐿 (𝑇) :=
∑︁
𝑟∈ℤ𝑙

𝑇𝑟 𝜎(𝑝, 𝑟).

The cocycle property of 𝜎 immediately gives 𝐿 ( 𝑓 )𝐿 (𝑔) = 𝐿 ( 𝑓 ×𝑄 𝑔), so that 𝐿 yields a represen-
tation of (𝐶∞(𝐺/𝐾),×𝑄) on H, while [𝐷, 𝐿( 𝑓 )] = ∑

𝑟 [𝐷, 𝑓𝑟] 𝜎(𝑝, 𝑟) = 𝐿 ( [𝐷, 𝑓 ]) is a bounded
operator for all 𝑓 ∈ 𝐶∞(𝐺/𝐾). The charge conjugation operator 𝐶 on spinors [17, Chap. 9] com-
mutes with all 𝜎(𝑝, 𝑟) and therefore 𝐶𝑝 𝑗𝐶−1 = −𝑝 𝑗 for each 𝑗 . It follows that 𝑅(𝑇) := 𝐶𝐿 (𝑇)∗𝐶−1

is given by
𝑅(𝑇) =

∑︁
𝑟∈ℤ𝑙

𝜎(𝑟, 𝑝) 𝐶𝑇∗
𝑟 𝐶

−1 =
∑︁
𝑟∈ℤ𝑙

𝐶𝑇∗
𝑟 𝐶

−1 𝜎(𝑟, 𝑝).

Since 𝐶 𝑓 ∗𝐶−1 = 𝑓 for 𝑓 in the commutative algebra 𝐶∞(𝐺/𝐾), this reduces to the relation
𝑅( 𝑓 ) = ∑

𝑟∈ℤ𝑙 𝑓𝑟 𝜎(𝑟, 𝑝), and therefore 𝑅( 𝑓 )𝑅(𝑔) = 𝑅( 𝑓 ×−𝑄 𝑔). (Our use of the skewsymmetrized
cocycle 𝜎 obviates the need to twist the conjugation as in [10].) It is easy to see – compare [16] –

10



that 𝑅 gives an antirepresentation of (𝐶∞(𝐺/𝐾),×𝑄) on H, which commutes with 𝐿 because

𝐿 ( 𝑓 )𝑅(𝑔) =
∑︁
𝑟,𝑠

𝑓𝑟 𝜎(𝑝, 𝑟) 𝑔𝑠 𝜎(𝑠, 𝑝) =
∑︁
𝑟,𝑠

𝑓𝑟𝑔𝑠 𝜎(𝑝 + 𝑠, 𝑟) 𝜎(𝑠, 𝑝)

=
∑︁
𝑟,𝑠

𝑔𝑠 𝑓𝑟 𝜎(𝑠, 𝑝 + 𝑟) 𝜎(𝑝, 𝑟) =
∑︁
𝑟,𝑠

𝑔𝑠 𝜎(𝑠, 𝑝) 𝑓𝑟 𝜎(𝑝, 𝑟) = 𝑅(𝑔)𝐿 ( 𝑓 ).

This verifies the reality property of the spin geometry. It is readily checked that

[[𝐷, 𝐿( 𝑓 )], 𝑅(𝑔)] =
∑︁
𝑟,𝑠∈ℤ𝑙

𝜎(𝑝, 𝑟) [[𝐷, 𝑓𝑟], 𝑔𝑠] 𝜎(𝑠, 𝑝) = 0,

so the first-order property of the spin geometry holds, too.
Such a spin geometry (𝐿 (𝐶∞(𝐺/𝐾)),H, 𝐷, 𝐶, 𝜒) has maximal symmetry; we may indeed refer

to the quantum group 𝐶 (𝐺)𝐽 as its “noncommutative symmetry group”. They provide examples of
spectral triples with noncommutative symmetries as discussed, for instance, in [23]. However, only
the invariance of 𝐷 under the abelian subgroup 𝐻 is actually used, so we are free to build other spin
geometries by deforming the commutative ones obtained from any 𝐻-invariant metric on 𝐺/𝐾 .

More elaborate examples of deformed geometries can also be built, starting from commutative
spin geometries wherein the spin connection is replaced by a Clifford superconnection (as in [15],
for instance), provided the latter is also 𝐻-invariant.

Finally, we consider whether the noncommutative homogeneous spaces constructed here may
play the same role as noncommutative tori in quantum field theory. Recall that Seiberg and
Witten [30] and Konechny and Schwarz [19] have extensively explored noncommutative gauge
theories based on tori. In general, the divergent ultraviolet behaviour for field theories based on
noncommutative tori [34] is no better than in the commutative case. This divergence holds also for
field theories obtained by second-quantizing the spin geometries constructed here.

Without going into the detailed analysis, the matter may be summed up as follows. The action
of a 𝐺-invariant Dirac operator over 𝐺/𝐾 decomposes into matrix actions on finite-dimensional
subspaces of smooth spinors, for which explicit formulas are available [1, 33]. The sign operator
𝐹 := 𝐷 |𝐷 |−1 preserves these subspaces, which are permuted by the representation of the algebra
(𝐶∞(𝐺/𝐾),×𝑄). For any unitary 𝑢 in this algebra, we can decompose the operator [𝐹, 𝑢] as
in [34] or [17, Sect. 13.A] and estimate its Schatten class, which measures the degree of ultraviolet
divergence of the theory. The norms ∥ [𝐹, 𝑢] ∥𝑝 turn out to be independent of the cocycle 𝜎
defining the product, provided 𝜎(𝑟, 𝑟 + 𝑠) = 𝜎(𝑟, 𝑠); in view of (4), this is immediate from the
skewsymmetry of the parameter matrix 𝑄. Therefore, the overall UV behaviour remains the same
as in the commutative case when 𝑄 = 0: our deformations never soften the ultraviolet divergence.

The ubiquity of the Moyal product in noncommutative field theory is already familiar. While
the present work cannot pretend to explain its pervasiveness, we have at any rate shown that
noncommutative geometries with a high degree of symmetry are easy to deform along (at least two)
commuting directions, leading always to Moyal products with a few parameters; thus the emphasis
on noncommutative tori is by no means misplaced. Whether this is in the nature of things remains
to be seen.

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Acknowledgments

We thank Alain Connes, Ludwik Da̧browski, Héctor Figueroa, José M. Gracia-Bondı́a, Piotr M.
Hajac and Mario Paschke for helpful discussions on several matters. Support from the Vicerrectorı́a
de Investigación of the University of Costa Rica and the Abdus Salam ICTP, Trieste, is gratefully
acknowledged.

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