Noncommutative Families of Instantons

Landi, G. et al. (2008) “Noncommutative Families of Instantons,” International Mathematics Research Notices, Vol. 2008, Article ID rnn038, 32 pages. doi:10.1093/imrn/rnn038 Noncommutative Families of Instantons Giovanni Landi1,2 , Chiara Pagani3 , Cesare Reina4 , and Walter D. van Suijlekom5 1 Dipartimento di Matematica e Informatica, Università di Trieste Via A.Valerio 12/1, 34127 Trieste, Italy, 2 INFN, Sezione di Trieste, 34127 Trieste, Italy, 3 Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark, 4 International School for Advanced Studies, Via Beirut 2-4, 34014 Trieste, Italy, and 5 IMAPP, Radboud Universiteit Toernooiveld 1, 6525 ED Nijmegen, the Netherlands Correspondence to be sent to: landi@univ.trieste.it We construct θ -deformations of the classical groups SL(2, H) and Sp(2). Coacting on a basic instanton on a noncommutative four-sphere Sθ4 , we construct a noncommutative family of instantons of charge 1. The family is parameterized by the quantum quotient of SLθ (2, H) by Spθ (2). 1 Introduction Self-dual (and anti-self-dual) solutions of Yang–Mills equations have played an important role both in mathematics and physics. Commonly called (anti-)instantons, they are connections with self-dual curvature on smooth G-bundles over a four-dimensional compact manifold M. In particular, one considers SU(2) instantons on the sphere S4 . General solutions are constructed by the ADHM method of [2] and it is known [3] that the moduli space of SU(2)-instantons, with instanton charge k—the second Chern class of the bundle—is a (8k − 3)-dimensional manifold Mk . For k = 1 the moduli space Received October 30, 2007; Revised October 30, 2007; Accepted March 24, 2008 Communicated by Prof. Yuri Manin
C The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org. 2 G. Landi et al. M1 is isomorphic to an open ball in R5 [11] and, in this construction, generic moduli are obtained from the so-called basic instanton. The latter is though of as a quaternion line bundle over P1 H ≃ S4 with connection induced from H2 by orthogonal projection. All other instantons of charge 1 are obtained from the basic one with the action of the conformal group SL(2, H) modulo the isometry group Sp(2) = Spin(5). The resulting homogeneous space is M1 ; it is also the space of quaternion Hermitean structures in H2 . The attempt to generalize to noncommutative geometry the ADHM construction of SU(2) instantons together with their moduli space is the starting motivation behind papers [13–15] and the present one. A noncommutative principal fibration A(Sθ4 ) ֒→ A(Sθ7 ), which “quantizes” the classical SU(2)-Hopf fibration over S4 , has been constructed in [14] on the toric noncommutative four-sphere Sθ4 . The generators of A(Sθ4 ) are the entries of a projection p, which describes the basic instanton on A(Sθ4 ). That is, p gives a projective module of finite-type p[A(Sθ4 )]4 and a connection ∇ = p ◦ d on it, which has a self-dual curvature and charge 1, in some appropriate sense; this is the basic instanton. In [15] infinitesimal instantons (“the tangent space to the moduli space”) were constructed using infinitesimal conformal transformations, that is elements in a quantized enveloping algebra Uθ (so(5, 1)). In the present paper, we look at a global construction and obtain the other charge 1 instantons by “quantizing” the actions of the Lie groups SL(2, H) and Sp(2) on the basic instanton, which enter the classical construction [1]. The paper is organized as follows. In Section 2, we recall the structure of the SU(2)-principal Hopf fibration Sθ7 → Sθ4 . Section 3 is devoted to the construction of θ -deformations A(SLθ (2, H)) and A(Spθ (2)) of the corresponding classical groups, endowed with Hopf algebra structures. The algebras A(Sθ7 ), A(Sθ4 ) are then described as quantum homogeneous spaces of A(Spθ (2)) as illustrated at the end of the section. In Section 4, we consider the coactions of A(SLθ (2, H)) and A(Spθ (2)) on the Hopf fibration Sθ7 → Sθ4 . We use these coactions in Section 5 to construct a noncommutative family of instantons by means of the algebra given by the quantum quotient of A(SLθ (2, H)) by A(Spθ (2)). This turns out to be a noncommutative algebra generated by six elements subject to a “hyperboloid” relation. We finish by mentioning relations to the notion of quantum families of maps as proposed in [20, 23] and by stressing some open problems. 2 The Principal Fibration The class of deformations that we work with is the one of “toric noncommutative spaces” introduced in [8] and further elaborated in [7]. In [14] a noncommutative principal fibration A(Sθ4 ) ֒→ A(Sθ7 ) was introduced and infinitesimal instantons on it were Noncommutative Families of Instantons 3 constructed in [15] using infinitesimal conformal transformations. We refer to these latter papers for a detailed description of the inclusion A(Sθ4 ) ֒→ A(Sθ7 ) as a noncommutative principal fibration (with classical SU(2) as structure group) and for its use for noncommutative instantons. Here we limit ourself to a brief description. The coordinate algebra A(Sθ7 ) on the sphere Sθ7 is the ∗-algebra generated by elements {z j , z∗j ; j = 1, . . . , 4} with relations z j zk = λ jk zk z j , and spherical relation
z∗j zk∗ = λ jk zk∗ z∗j , z∗j zk = λk j zk z∗j , (2.1) z∗j z j = 1. The deformation matrix (λ jk ) is taken so to allow an action by automorphisms of the undeformed group SU(2) on A(Sθ7 ) and so that the subalgebra of invariants under this action is identified with the coordinate algebra A(Sθ4 ) of a sphere Sθ4 . With deformation parameter λ = e2πiθ , the ∗-algebra A(Sθ4 ) is generated by a central element x and elements α, β, α ∗ , β ∗ with commutation relations αβ = λβα, α ∗ β ∗ = λβ ∗ α ∗ , β ∗ α = λαβ ∗ , βα ∗ = λα ∗ β, (2.2) and spherical relation α ∗ α + β ∗ β + x2 = 1. All this (including the relation between the deformation parameter for Sθ7 and Sθ4 ) is most easily seen by taking the generators of A(Sθ4 ) as the entries of a projection, which yields an “instanton bundle” over Sθ4 . Consider the matrix-valued function on Sθ7 given by u = (|ψ1  , |ψ2 ) = where t  z1 z2 −z2∗ z1∗ z3 z4 −z4∗ z3∗ t , (2.3) denotes matrix transposition, and |ψ1  , |ψ2  are elements in the right A(Sθ7 )- module C4 ⊗ A(Sθ7 ). They are orthonormal with respect to the A(Sθ7 )-valued Hermitean
structure ξ , η = ξ ∗j η j and as a consequence, u∗ u = I2 . Hence the matrix p = uu∗ = |ψ1  ψ1 | + |ψ2  ψ2 | (2.4) 4 G. Landi et al. is a self-adjoint idempotent with entries in A(Sθ4 ); we have explicitly: p= ⎛ 1+x 0 ⎜ 1⎜ ⎜ 0 2⎜ ⎝ α∗ √ β −µβ µα ⎞ ⎟ µ α∗ ⎟ ⎟, ⎟ 0 ⎠ 1−x 1 + x −µβ ∗ β∗ with µ = α 1−x 0 (2.5) λ = eπiθ . The generators of A(Sθ4 ) are bilinear in those of A(Sθ7 ) and given by α = 2 z1 z3∗ + z2 z4∗ , β = 2(−z1 z4 + z2 z3 ), x = z1 z1∗ + z2 z2∗ − z3 z3∗ − z4 z4∗ . (2.6) The defining relation of the algebra A(Sθ7 ) can be given on the entries of the matrix u in (2.3). Writing u = (uia ), with i, j = 1, . . . , 4 and a = 1, 2, one gets uia u jb = η ji u jbuia . (2.7) with η = (ηi j ) the matrix ⎛ 1 ⎜ ⎜1 η=⎜ ⎜ ⎝µ µ 1 µ µ ⎞ ⎟ µ µ⎟ ⎟. ⎟ µ 1 1⎠ µ 1 1 1 (2.8) The deformation matrix (λ jk ) in (2.1) is just the above η with entries rearranged. The finitely generated projective A(Sθ4 )-module E = p[A(Sθ4 )]4 is isomorphic to the module of equivariant maps from A(Sθ7 ) to C2 describing the vector bundle associated via the fundamental representation of SU(2). On E one has the Grassmann connection ∇ := p ◦ d : E → E ⊗A(Sθ4 ) with 1 Sθ4 , (2.9) (Sθ4 ) a natural differential calculus on Sθ4 . There is also a natural Hodge star operator ∗θ (see below). The connection has a self-dual curvature ∇ 2 = p(dp)2 , that is, ∗θ ( p(d p)2 ) = p(d p)2 . Noncommutative Families of Instantons 5 Its “topological charge” is computed to be 1 by a noncommutative index theorem. This “basic” noncommutative instanton has been given a twistor description in [5]. The two algebras A(Sθ7 ) and A(Sθ4 ) can be described in terms of a deformed (a “star”) product on the undeformed algebras A(S7 ) and A(S4 ). Both spheres S7 and S4 carry an action of the torus T2 , which is compatible with the action of SU(2) on the total space S7 . In other words, it is an action on the principal SU(2)-bundle S7 → S4 . The action by automorphisms on the algebra A(S4 ) is given simply by σt : (x, α, β) → x, e2πit1 α, e2πit2 β , for t = (t1 , t2 ) ∈ T2 . Now, any polynomial in the algebra A(S4 ) is decomposed into ele- ments, which are homogeneous under this action. An element fr ∈ A(S4 ) is said to be homogeneous of bidegree r = (r1 , r2 ) ∈ Z2 if σt ( fr ) = e2πi(r1 t1 +r2 t2 ) fr , and each f ∈ A(S4 ) is a unique finite sum of homogeneous elements [17]. This decompo- sition corresponds to writing the polynomial f in terms of monomials in the generators. Let now θ = (θ jk = −θk j ) be a real antisymmetric 2 × 2 matrix (thus given by a single real number, θ12 = θ , say). The θ -deformation of A(S4 ) is defined by replacing the ordinary product by a deformed product, given on homogeneous elements by fr ×θ gs := eπiθ(r1 s2 −r2 s1 ) fr gs , (2.10) and extended linearly to all elements in A(S4 ). The vector space A(S4 ) equipped with the product ×θ is denoted by A(Sθ4 ). On the other hand, the algebra A(S7 ) does not carry an action of this T2 but rather a lifted action of a double cover 2-torus [15]. Nonetheless, the lifted action still allows us to define the algebra A(Sθ7 ) by endowing the vector space A(S7 ) with a deformed product similar to the one in (2.10). As the notation suggests, these deformed algebras are shown to be isomorphic to the algebras defined by the relations in Equations (2.1) and (2.2). In fact, the torus action can be extended to forms and one also deforms the exterior algebra of forms via a product like the one in (2.10) on spectral components so producing deformed exterior algebras (Sθ4 ) and (Sθ7 ). As for functions, these are isomorphic 6 G. Landi et al. as vector spaces to their undeformed counterparts but endowed with a deformed product. As mentioned, the spheres Sθ4 and Sθ7 are examples of toric noncommutative manifolds (originally called isospectral deformations [8]). They have noncommutative geometries via spectral triples whose Dirac operator and Hilbert space of spinors are the classical ones: only the algebra of functions and its action on the spinors is changed. In particular, having an undeformed Dirac operator (or, in other words, an undeformed metric structure) one takes as a Hodge operator ∗θ the undeformed operator on each spectral component of the algebra of forms. 3 Deformations of the Groups SL(2, H) and Sp(2) Our interest in deforming the groups SL(2, H) and Sp(2) is motivated by their use for the construction of instantons on S4 . Classically, charge 1 instantons are generated from the basic one by the action of the conformal group SL(2, H) of S4 . Elements of the subgroup Sp(2) ⊂ SL(2, H) leave invariant the basic one, hence to get new instantons one needs to quotient SL(2, H) by the spin group Sp(2) ≃ Spin(5). The resulting moduli space of SU(2) instantons on S4 modulo gauge transformations is identified (cf. [1]) with the five-dimensional quotient manifold SL(2, H)/Sp(2). In a parallel attempt to generate instantons on A(Sθ4 ), we construct a quantum group SLθ (2, H) and its quantum subgroup Spθ (2). An infinitesimal construction was proposed in [15] where a deformed dual enveloping algebra Uθ (so(5, 1)) was used to generate infinitesimal instantons by acting on the basic instanton described above. The construction of Hopf algebras A(SLθ (2, H)) and A(Spθ (2)) is a special case of the quantization of compact Lie groups using Rieffel’s strategy in [18], and studied for the toric noncommutative geometries in [21]. Firstly, a deformed (Moyal-type) product ×θ is constructed on the algebra of (continuous) functions A(G) on a compact Lie group G, starting with an action of a closed connected abelian subgroup of G (usually a torus). The algebra A(G) equipped with the deformed product is denoted by A(G θ ). Keeping the classical expression of the coproduct, counit, and antipode on A(G), but now on the algebra A(G θ ), the latter becomes a Hopf algebra. It is in duality with a deformation of the universal enveloping algebra U(g) of the Lie algebra g of G. The Hopf algebra U(g) is deformed to Uθ (g) by leaving unchanged the algebra structure while twisting the coproduct, counit, and antipode. The deformation from U(g) to Uθ (g) is implemented with a twist of Drinfel’d type [9, 10]—in fact, explicitly constructed in [16] for the cases in hand—as revived in [19]. Noncommutative Families of Instantons 7 The deformed enveloping algebra Uθ (so(5, 1)) was explicitly constructed in [15]. We now briefly discuss the dual construction for the Lie group SL(2, H). The torus T2 is embedded in SL(2, H) diagonally, ρ(t) = diag e2πit1 , e2πit2 , for t = (t1 , t2 ) ∈ T2 , and the group T2 × T2 acts on SL(2, H) by (s, t, g) ∈ T2 × T2 × SL(2, H) → ρ(s) · g · ρ(t)−1 ∈ SL(2, H). (3.1) Similar to the case of the spheres in Section 2, any function f ∈ A(SL(2, H)) expands in a
series f = r fr of homogeneous elements for this action of T4 , but now r = (r1 , r2 , r3 , r4 ) is a multidegree taking values in Z4 . A deformed product ×θ is defined by an analogue of formula (2.10) on homogeneous elements: fr ×θ gs = eπiθ(r1 s2 −r2 s1 +r3 s4 −r4 s3 ) fr gs , and extended by linearity to the whole of A(SL(2, H)). The resulting deformed algebra A(SLθ (2, H)), endowed with the classical (expressions for the) coproduct , counit ǫ, and antipode S becomes a Hopf algebra. In fact, to avoid problems coming from the noncommutativity of quaternions, we shall think of elements in H as 2 by 2 matrices over C via the natural inclusion H ∋ q = c1 + c2 j →  c1 c2 −c2 c1  ∈ Mat2 (C), for c1 , c2 ∈ C. In the present paper, we need not only the Hopf algebra A(SLθ (2, H)) but also its coaction on the principal bundle A(Sθ4 ) ֒→ A(Sθ7 ), and in turn on the basic instanton connection (2.9) on the bundle, in order to generate new instantons. Having this fact in mind we proceed to give an explicit construction of A(SLθ (2, H)) out of its coaction in a way that also shows its quaternionic nature. 3.1 The quantum group SLθ (2, H) For the deformation of the quaternionic special linear group SL(2, H), we start from the algebra of a two-dimensional deformed quaternionic space H2θ . Let A(C4θ ) 8 G. Landi et al. be the ∗-algebra generated by elements {z j , z∗j ; a = 1, . . . , 4} with the relations as in Equation (2.1) (for the specific value of the deformation parameter λ considered in Sec- tion 2 and obtained from (2.8) as mentioned there) but without the spherical relation that defines A(Sθ7 ). We take A(H2θ ) to be the algebra A(C4θ ) equipped with the antilinear ∗-algebra map j : A(C4θ ) → A(C4θ ) defined on generators by j : (z1 , z2 , z3 , z4 ) → (z2 , −z1 , z4 , −z3 ). It is worth stressing that this deformation of the quaternions takes place between the two copies of H while leaving the quaternionic structure within each copy of H undeformed. Since the second column of the matrix u in (2.3) is the image through j of the first one, we may think of u as made of two deformed quaternions. Following a general strategy [22], we now define A(Mθ (2, H)) to be the universal bialgebra for which A(H2θ ) is a comodule ∗-algebra. More precisely, we define a transfor- mation bialgebra of A(H2θ ) to be a bialgebra B such that there is a ∗-algebra map L : A C4θ → B ⊗ A C4θ , which satisfies (id ⊗ j) ◦ L = L ◦ j. (3.2) We then set A(Mθ (2, H)) to be the universal transformation bialgebra in the following sense: for any transformation bialgebra B there exists a morphism of transformation bialgebras (i.e. commuting with the coactions) from A(Mθ (2, H)) onto B. The requirement that A(H2θ ) be a A(Mθ (2, H))-comodule algebra allows us to derive the commutation relations of the latter. A coaction L : z1 , −z2∗ , z3 , −z4∗ t is given by matrix multiplication, L . t → Aθ ⊗ z1 , −z2∗ , z3 , −z4∗ , (3.3) for a generic 4 × 4 matrix Aθ = (Ai j ). Asking for (3.2) we have (Ajk )∗ = (−1) j+k Aj′ k′ , with j ′ = j + (−1) j+1 and the same for k ′ ; this means that Aθ has the form Aθ =  ai j ci j bi j di j  ⎛ a1 ⎜ ∗ ⎜−a2 =⎜ ⎜ ⎝ c1 −c2∗ a2 b1 a1∗ −b2∗ c2 c1∗ d1 −d2∗ b2 ⎞ ⎟ b1∗ ⎟ ⎟. ⎟ d2 ⎠ d1∗ (3.4) Noncommutative Families of Instantons 9 We use “quaternion notations” for the above matrix and write Aθ =  a c b d  with a = (ai j ) = ,  a1 a2 −a2∗ a1∗  , (3.5) and similarly for the remaining parts. The defining matrix in (3.4) has a “classical form.” One readily finds that with respect to the torus action (3.1) its entries Ai j are of multidegree i ⊕ (− j ) in Z4 with  = (i ) = ((1, 0), (−1, 0), (0, 1), (0, −1)). The general strategy exemplified by the deformed product (2.10) would then give the deformed product and in turn, the commutation relations defining the deformed algebra A(Mθ (2, H)). We shall get them directly from the coaction on the algebra A(C4θ ). The transformations induced on the generators of A(C4θ ) read with ∗ L (z j ) =( w1 : = L (z1 ) = a1 ⊗ z1 − a2 ⊗ z2∗ + b1 ⊗ z3 − b2 ⊗ z4∗ w2 : = L (z2 ) = a1 ⊗ z2 + a2 ⊗ z1∗ + b1 ⊗ z4 + b2 ⊗ z3∗ w3 : = L (z3 ) = c1 ⊗ z1 − c2 ⊗ z2∗ + d1 ⊗ z3 − d2 ⊗ z4∗ w4 : = L (z4 ) = c1 ⊗ z2 + c2 ⊗ z1∗ + d1 ⊗ z4 + d2 ⊗ z3∗ L (z j )) ∗ . The condition for L (3.6) to be an algebra map determines the com- mutation relations among the generators of A(Mθ (2, H)): the algebra generated by the ai j is commutative, as well as the algebras generated by the bi j , ci j , and di j . However, the whole algebra is not commutative and there are relations. A straightforward computation allows one to concisely write them as Ai j Akl = ηki η jl Akl Ai j (3.7) with η = (ηki ) the deformation matrix in (2.8). Indeed, imposing that (3.3) defines a ∗-algebra map on the generators of A(C4θ ), and using the relations (2.7), we have
kl (Aik A jl − η ji ηkl A jl Aik ) ⊗ uka ulb = 0. Since for a ≤ b the elements uka ulb could be taken to be all independent, relations Aik Ajl − η ji ηkl Ajl Aik = 0 hold, for all values of a, b. An explicit expression of the above commutation relations is in Appendix A. It is not difficult to see that A(Mθ (2, H)) is indeed the universal transformation bialgebra, 10 G. Landi et al. since the commutation relations (3.7) and the quaternionic structure of Aθ in (3.4) are derived from the minimal requirement of L to be a ∗-algebra map such that (3.2) holds. In order to define the quantum group SLθ (2, H) we need a determinant. This is most naturally introduced via the coaction on forms. There is a natural differential calculus (C4θ ) generated in degree 1 by elements {dz j , a = 1, . . . , 4} and relations z j dzk − λ jk dzk z j = 0, z j dzk∗ − λk j dzk∗ z j = 0, dz j dzk + λ jk dzk dz j = 0, together with their conjugates. The forms z∗j dzk − λk j dzk z∗j = 0, dz j dzk∗ + λk j dzk∗ dz j = 0. (C4θ ) could be obtained from the general procedure mentioned at the end of Section 2. The result is also isomorphic to the one obtained from the general construction [6], which uses the Dirac operator to implement the exterior derivative as a commutator. The coaction L is extended to forms by requiring it to commute with d. Having the action (3.3), it is natural to define a determinant element by setting L dz1 dz2∗ dz3 dz4∗ =: det(Aθ ) ⊗ dz1 dz2∗ dz3 dz4∗ . We find its explicit form by using the relations of (C4θ ): det(Aθ ) = a1 a1∗ d1 d1∗ + d2 d2∗ + b2∗ µc2 d1∗ − d2 c1∗ − b1∗ µc2 d2∗ + d1 c1∗ − a2 − a2∗ d1 d1∗ + d2 d2∗ + b2∗ µc1 d1∗ + d2 c2∗ + b1∗ − µc1 d2∗ + d1 c2∗ + b1 − a2∗ c2 d1∗ − µd2 c1∗ − a1∗ c1 d1∗ + µd2 c2∗ + b1∗ c1 c1∗ + c2 c2∗ − b2 a2∗ c2 d2∗ + µd1 c1∗ + a1∗ c1 d2∗ − µd1 c2∗ − b2∗ c1 c1∗ + c2 c2∗ . (3.8) A more compact form for det( Aθ ) is found to be (see also Appendix B) det(Aθ ) =  (−1)|σ | εσ A1,σ (1) A2,σ (2) A3,σ (3) A4,σ (4) , σ ∈S4 with εσ = εσ (1)σ (2)σ (3)σ (4) . The tensor εi jkl has components ε1324 = εcycl = µ; ε1423 = εcycl = µ, (3.9) Noncommutative Families of Instantons 11 and equal to 1 otherwise. In the limit θ → 0, the element det(Aθ ) reduces to the deter- minant of the matrix Aθ=0 as it should. Additional results on the determinant are in the following lemmata. Lemma 3.1. For each i, l ∈ {1, . . . , 4}, define the corresponding algebraic complement: Âil =  σ ∈S3 (−1)|σ | εσ1 ...σi−1 lσi+1 ...σ4 ησ1 l ησ2 l · · · ησi−1 l A1,σ1 . . . Ai−1,σi−1 Ai+1,σi+1 . . . A4,σ (4) , where σ = (σ1 , . . . σi−1 , σi+1 , . . . σ4 ) = σ (1, . . . l − 1, l + 1, . . . 4) ∈ S3 , the group of permutations of three objects. Then, Ail Âil = Âil Ail for any i and l.
Proof. We use the shorthand clσ = εσ1 ...σi−1 lσi+1 ...σ4 ησ1 l ησ2 l · · · ησi−1 l . Then, the commutation relations (3.7) yield, Ail Âil = =  clσ Ail A1σ1 . . . Ai−1,σi−1 Ai+1,σi+1 . . . A4,σ (4) ˆ σ ∈S3 (l)  ˆ σ ∈S3 (l) clσ η1i · · · ηi−1,i ηi+1,i · · · η4i ηl,σ1 · · · ηlσi−1 ηl,σi+1 · · · ηl4 A1σ1 . . . Ai−1,σi−1 ·Ai+1,σi+1 . . . A4,σ (4) Ail = Âil Ail . Here we used the fact that η1i · · · ηi−1,i ηi+1,i · · · η4i is the product of the elements in the ith column of η excluded the element ηii = 1 and the result is 1 as one deduces from the form of the matrix η in (2.8). Similarly for the other coefficient given by the product of the elements of the lth row. Lemma 3.2. The determinant det(Aθ ) is computed via a Laplace expansion:
1. by rows; for each i ∈ {1, . . . , 4} fixed: det(Aθ ) = l (−1)i+l Ail Âil ;
2. by columns; for each i ∈ {1, . . . , 4} fixed: det(Aθ ) = l (−1)i+l Ali Âli . Proof. These follow from (3.7) after a lengthy but straightforward computation. 
 The particular form of the deformation matrix ηi j defining the relations in A(Sθ7 ) implies that det(Aθ ) is (not surprisingly) a central element in the algebra A(Mθ (2, H)) generated by the entries of Aθ . Hence we can take the quotient of this algebra by the 12 G. Landi et al. two-sided ideal generated by det(Aθ ) − 1; we will denote this quotient by A(SLθ (2, H)). The image of the elements Ai j in the quotient algebra will again be denoted by Ai j . In order to have a quantum group we need more structure. On the algebra A(SLθ (2, H)) we define a coproduct by (Ai j ) := a counit by  k Aik ⊗ Ak j , ǫ(Ai j ) := δi j , whereas the antipode S is defined by S(Ai j ) := (−1)i+ j Âji . Here Âli are the algebraic complements introduced in Lemma 3.1. Indeed, from

Lemma 3.2, l Ail S(Ali ) = l (−1)i+l Ail Âil = det(Aθ ) = 1, and similarly, using also Lemma


3.1, l S(Ail )Ali = l (−1)i+l Âli Ali = det(Aθ ) = 1. Moreover, if i = j, l Ail S(Al j ) = 0 as one shows by explicit computation. The definitions above are collected in the following proposition. Proposition 3.3. The datum (A(SLθ (2, H)), , ǫ, S) constitutes a Hopf algebra.
in (3.3) passes to a coaction of A(SLθ (2, H)) on A(H2θ ) and it is
still a ∗-algebra map. However, the spherical relation j z∗j z j = 1 is no longer invariant The coaction under L. L Thus, the algebra A(Sθ7 ) is not an A(SLθ (2, H))-comodule algebra but only a A(SLθ (2, H))-comodule. We shall elaborate more on this in Section 4 below. 3.2 The quantum group Spθ (2) Motivated by the classical picture, we next introduce the symplectic group A(Spθ (2)). Recall that a two-sided ∗-ideal I in a Hopf algebra (A, , ǫ, S) is a Hopf ideal if (I ) ⊆ I ⊗ A + A ⊗ I , ǫ(I ) = 0, S(I ) ⊆ I. Then the quotient A/I is a Hopf algebra with induced structures ((π ⊗ π ) ◦ where π : A → A/I is the natural projection. (3.10) , ǫ, π ◦ S), Proposition 3.4. Let I denote the two-sided ∗-ideal in A(SLθ (2, H)) generated by the

elements k (Aki )∗ Ak j − δi j for i, j = 1, . . . , 4. Then I ⊂ A(SLθ (2, H)) is a Hopf ideal. Noncommutative Families of Instantons 13 Proof. The first two conditions in (3.10) follow easily from the definition of and ǫ for A(SLθ (2, H)) in Proposition 3.3. For the third, we observe that if J is an ideal in A(SLθ (2, H)) such that the classical counterpart J (0) is an ideal in A(SL(2, H)), which is generated by homogeneous elements, then J = J (0) as vector spaces. Indeed, the deformed product of a generator with any homogeneous function merely results in multiplication by a complex
phase. In our case, the classical counterparts for the generators k (Aki )∗ Ak j − δi j are indeed homogeneous (if i = j, they are of bidegree (0, 0), otherwise of bidegree i −  j ) and the above applies. In particular, S(I ) = S(I (0) ) ⊆ I (0) = I .  Corollary 3.5. The quotient A(Spθ (2)) := A(SLθ (2, H))/I is a Hopf algebra with the induced Hopf algebra structure.
We still use the symbols ( , ǫ, S) for the induced structures. The “defining matrix” Aθ of A(Spθ (2)) has the form (3.4) (or (3.5)) with the additional condition that A∗θ Aθ = 1, coming from the very definition of A(Spθ (2)). A little algebra shows also that Aθ A∗θ = 1. These conditions are equivalent to the statement that S(Aθ ) = A∗θ . In the quaternionic form, the conditions A∗θ Aθ = Aθ A∗θ = 1 become  a ∗ a + c∗ c a ∗ b + c∗ d b∗ a + d ∗ c b∗ b + d ∗ d  =  aa ∗ + bb∗ ac∗ + bd ∗ ca ∗ + db∗ cc∗ + dd ∗    1 0 = 0 1 . 3.3 The quantum homogeneous spaces A(Sθ7) and A(Sθ4) Using the same notations as in (3.4), let us consider the two-sided ideal in A(Spθ (2)) given by Iθ := bi j , ci j , a2 , a2∗ , a1 − 1, a1∗ − 1. This is a ∗-Hopf ideal, that is, (Iθ ) ⊆ A(Spθ (2)) ⊗ Iθ + Iθ ⊗ A(Spθ (2)), S(Iθ ) ⊆ Iθ , ε(Iθ ) = 0, and we can take the quotient Hopf algebra A(Spθ (1)) := A(Spθ (2))/Iθ with corresponding projection map π Iθ . By projecting with π Iθ , the algebra reduces to the commutative one generated by the entries of the diagonal matrix π Iθ (Aθ ) = diag(I2 , di j ) = diag(I2 , d), with d ∗ d = dd ∗ = I2 or d1 d1∗ + d2 d2∗ = 1; hence, A(Spθ (1)) = A(Sp(1)). There is a coaction A(Spθ (2)) → A(Spθ (2)) ⊗ A(Sp(1)),  a c b d  →  a c b d  . ⊗  I 0 0 d  . The corresponding algebra of coinvariants A(Spθ (2))co(A(Sp(1)) is clearly generated by the first two columns (a, c) = {ai j , ci j } of Aθ . An algebra isomorphism between the algebra of 14 G. Landi et al. coinvariants and A(Sθ7 ) is provided by the ∗-map sending these columns to the matrix u in (2.3). On the generators, this is given by a1 → z1 , a2 → z2 , c1 → z3 , c2 → z4 (3.11) and the spherical relation corresponds to the condition (A∗θ Aθ )11 =
(ai∗ ai + ci∗ ci ) = 1. Summarizing, we have that A(Spθ (2))co(A(Sp(1)) ≃ A(Sθ7 ). It follows from the general theory of noncommutative principal bundles over quantum homogeneous spaces [4] that the inclusion A(Sθ7 ) ֒→ A(Spθ (2)) is a noncommutative principal bundle with the classical group Sp(1) as structure group. Next, we consider the ideal in A(Spθ (2)) given by Jθ := bi j , ci j —again easily shown to be a Hopf ideal. Denote by π Jθ the projection map onto the quotient Hopf algebra A(Spθ (2))/Jθ generated, as an algebra, by the entries of π Jθ (Aθ ) = diag(ai j , di j ) = diag(a, d). The conditions A∗θ Aθ = Aθ A∗θ = 1 give that both {ai j } and {di j } generate a copy of the algebra A(Sp(1)). However, from the explicit relations in Appendix A, we see that in general ai j dmn = dmn ai j and the quotient algebra is not commutative. The algebra of coinvariants for the right coaction (id ⊗ π Jθ ) ◦ A(Spθ (2)), is A(Sθ4 ). . : Aθ → Aθ ⊗ π Jθ (Aθ ) given by Indeed, with the map (id ⊗ π Jθ ) ◦  a c b d  →  . a⊗a . c⊗a of π Jθ (A(Spθ (2))) on . b⊗d . d ⊗d  , one finds that the ∗-algebra of coinvariants is generated by the elements (aa ∗ )11 = a1 a1∗ + a2 a2∗ , (ca ∗ )11 = c1 a1∗ + c2 a2∗ , (ca ∗ )12 = −c1 a2 + c2 a1 . The ∗-map (3.11) combined with (2.6) then provides the identification with the generators of A(Sθ4 ). Again, the general theory of noncommutative principal bundles over quantum homogeneous spaces of [4] gives that the inclusion A(Sθ4 ) ֒→ A(Spθ (2)) is a noncommutative principal bundle with π Jθ (A(Spθ (2)) as structure group. It is a deformation of the principal bundle over S4 with total space Sp(2) and structure group Sp(1) × Sp(1). 4 Noncommutative Conformal Transformations There is a natural coaction fibration A(Sθ4 ) ֒→ A(Sθ7 ) L of A(SLθ (2, H)) on the SU(2) noncommutative principal of Section 2. Since the matrix u in (2.3) consists of two deformed Noncommutative Families of Instantons 15 quaternions, the left coaction L L of A(SLθ (2, H)) in (3.6) can be written on A(Sθ7 ) as : A Sθ7 → A(SLθ (2, H)) ⊗ A Sθ7 , u→ u := or, in components, uia := uia →  L (uia ) =  j L (u) . = Aθ ⊗ u, Ai j ⊗ u ja . (4.1) (4.2) We have already mentioned that the left coaction L of A(SLθ (2, H)) as in (3.6) does not
leave invariant the spherical relation: L ( j z∗j z j ) = 1 ⊗ 1. We will denote by A( Sθ7 ) the image of A(Sθ7 ) under the left coaction of A(SLθ (2, H)): it is a subalgebra of A(SLθ (2, H)) ⊗
Sθ7 ) as a θ -deformation of a family of “inflated” spheres. Since j z∗j z j A(Sθ7 ). We think of A( is central in A(Sθ7 ) its image ρ 2 := L  j  z∗j z j , (4.3) is a central element in A( Sθ7 ) that parameterizes a family of noncommutative 7-spheres 7  Sθ7 . By evaluating ρ 2 as any real number r 2 ∈ R, we obtain an algebra A(Sθ,r ) which is a deformation of the algebra of polynomials on a sphere of radius r. Remark 4.1. As expected, the coaction of the quantum subgroup A(Spθ (2)) does not “inflate the spheres,” i.e. ρ 2 = 1 ⊗ 1 in this case. Indeed, if Aθ = (Ai j ) is the defining matrix of A(Spθ (2)), one gets (u∗ u)ab → which gives
j  i jl (A∗ )li Ai j ⊗ (u∗ )al u jb = z∗j z j → 1 ⊗
j  jl δl j ⊗ (u∗ )al u jb = 1 ⊗ (u∗ u)ab, z∗j z j . Hence, both A(Sθ7 ) and A(Sθ4 ) are A(Spθ (2))-comodule ∗-algebras. Using the identification (3.11) one sees that the coaction of A(Spθ (2)) on A(Sθ7 ) is the restriction of the coproduct of A(Spθ (2)) to the first column of Aθ , i.e. to the algebra of coinvariants A(Spθ (2))co(A(Sp(1)) .
Next, we define a right action of SU(2) on A( Sθ7 ) in such a way that the correspond- ing algebra of invariants describes a family of noncommutative 4-spheres. It is natural to require that the above left coaction of A(SLθ (2, H)) on A(Sθ7 ) intertwines the right action of SU(2) on A( Sθ7 ) with the action of SU(2) on A(Sθ7 ). The algebra A( Sθ7 ) is generated by elements {w j , w ∗j , j = 1, . . . , 4}, the w j s being as
in (3.6) but with “coefficients” in A(SLθ (2, H)). Clearly, j w ∗j w j = ρ 2 . Then, the algebra of 16 G. Landi et al. invariants of the action of SU(2) on A( Sθ7 ) is generated by  x = w1 w1∗ + w2 w2∗ − w3 w3∗ − w4 w4∗ ,  = 2(−w1 w4 + w2 w3 ), β  α = 2(w1 w3∗ + w2 w4∗ ), (4.4) together with ρ 2 . This is so because the elements (4.4) are the images under the map L of the elements (2.6) that generate the algebra of invariants under the action of SU(2) on A(Sθ7 ). This correspondence also gives for their commutation relations the same expressions as the ones in (2.2) for the generators of A(Sθ4 ). The difference is that we do not have the spherical relation of A(Sθ4 ) any longer but rather we find that +  ∗ β x2 =  α ∗ α+β  j w ∗j w j 2 = ρ4. (4.5) We denote by A( Sθ4 ) ⊂ A(SLθ (2, H)) ⊗ A(Sθ4 ) the algebra of invariants and conclude that of A(SLθ (2, H)) on the SU(2) principal fibration A(Sθ4 ) ֒→ A(Sθ7 ) generates a family of SU(2) principal fibrations A( Sθ4 ) ֒→ A( Sθ7 ). Evaluating the central element ρ 2 , for the coaction L 4 7 ) ֒→ A(Sθ,r ) of spheres of radius r 2 and any r ∈ R we get an SU(2) principal fibration A(Sθ,r r, respectively. Motivated by the interpretation of the Hopf algebra SLθ (2, H) as a parameter space (see Section 5), the coaction of A(SLθ (2, H)) is extended to the forms (Sθ4 ) by requiring that it commutes with d, i.e. L (dω) = (id ⊗ d) L (ω), thus extending the differential of A(Sθ4 ) to A(SLθ (2, H)) ⊗ A(Sθ4 ) as (id ⊗ d). Having these, we have the following characterization of A(SLθ (2, H)) as conformal transformations. Proposition 4.2. With ∗θ the natural Hodge operator on Sθ4 , the algebra A(SLθ (2, H)) coacts by conformal transformations on L (∗θ ω) Proof. The map L = (id ⊗ ∗θ ) (Sθ4 ), that is L (ω) , ∀ω∈ Sθ4 . is given by the classical coaction of A(SL(2, H)) on
(S4 ) as vector spaces and only the two products on A(SL(2, H)) and (S4 ) are deformed. Since ∗θ coincides with the undeformed Hodge operator ∗ on (Sθ4 ) ≃ (S4 ) as vector spaces, the result follows from the fact that SL(2, H) acts by conformal transformations on S4 .  Noncommutative Families of Instantons 17 4.1 The quantum group SOθ (5, 1) ,  By construction, the generators  α, β x of A( Sθ4 ) are the images under sponding α, β, x of A(Sθ4 ). L of the corre- Some algebra yields 2 x = a1 a1∗ + a2 a2∗ + b1 b1∗ + b2 b2∗ − c1 c1∗ − c2 c2∗ − d1 d1∗ − d2 d2∗ ⊗ 1 + a1 a1∗ + a2 a2∗ − b1 b1∗ − b2 b2∗ − c1 c1∗ − c2 c2∗ + d1 d1∗ + d2 d2∗ ⊗ x + a1 b1∗ + µb2 a2∗ − c1 d1∗ − µd2 c2∗ ⊗ α + b1 a1∗ + µa2 b2∗ − d1 c1∗ − µc2 d2∗ ⊗ α ∗ + a1 b2∗ − µb1 a2∗ − c1 d2∗ + µd1 c2∗ ⊗ β + b2 a1∗ − µa2 b1∗ − d2 c1∗ + µc2 d1∗ ⊗ β ∗ ,  α = a1 c1∗ + a2 c2∗ + b1 d1∗ + b2 d2∗ ⊗ 1 + a1 c1∗ + a2 c2∗ − b1 d1∗ − b2 d2∗ ⊗ x + a1 d1∗ + µb2 c2∗ ⊗ α + b1 c1∗ + µa2 d2∗ ⊗ α ∗ + a1 d2∗ − µb1 c2∗ ⊗ β + b2 c1∗ − µa2 d1∗ ⊗ β ∗ ,  = (a2 c1 − a1 c2 + b2 d1 − b1 d2 ) ⊗ 1 + (a2 c1 − a1 c2 − b2 d1 + b1 d2 ) ⊗ x β + (−a1 d2 + µb2 c1 ) ⊗ α + (−b1 c2 + µa2 d1 ) ⊗ α ∗ + (a1 d1 − µb1 c1 ) ⊗ β + (−b2 c2 + µa2 d2 ) ⊗ β ∗ . (4.6) From the definition of ρ in (4.3), using the commutation relations (2.7), it follows that 2ρ 2 =  (Ail )∗ Aik ⊗ ((ul1 )∗ uk1 + (ul2 )∗ uk2 )   = (Ail )∗ Aik ηlk ⊗ (uu∗ )kl , (Ail )∗ Aik ⊗ ηlk (uk1 (ul1 ) + uk2 (ul2 )∗ ) = ∗ L ((u u)11 ilk + (u∗ u)22 ) = ilk ∗ ilk and, being (uu∗ )kl the component pkl of the defining projector p in (2.4), an explicit computation yields 2ρ 2 = a1 a1∗ + a2 a2∗ + c1 c1∗ + c2 c2∗ + b1 b1∗ + b2 b2∗ + d1 d1∗ + d2 d2∗ ⊗ 1 + a1 a1∗ + a2 a2∗ + c1 c1∗ + c2 c2∗ − b1 b1∗ − b2 b2∗ − d1 d1∗ − d2 d2∗ ⊗ x + a1 b1∗ + µb2 a2∗ + c1 d1∗ + µd2 c2∗ ⊗ α + b1 a1∗ + µa2 b2∗ + d1 c1∗ + µc2 d2∗ ⊗ α ∗ + a1 b2∗ − µb1 a2∗ + c1 d2∗ − µd1 c2∗ ⊗ β + b2 a1∗ − µa2 b1∗ + d2 c1∗ − µc2 d1∗ ⊗ β ∗ . (4.7) In the expressions (4.6) and (4.7), the elements of A(SLθ (2, H)) appear only quadratically. Rather than a coaction of A(SLθ (2, H)), on A(Sθ4 ) there is a coaction of the Z2 -invariant subalgebra. We denote this by A(SOθ (5, 1)), a notation that will become clear presently. In A(C4θ ), let us consider the vector-valued function X := (r, x, α, α ∗ , β, β ∗ )t , with r := z1 z1∗ + z2 z2∗ + z3 z3∗ + z4 z4∗ and x, α, β are the quadratic elements, with the same formal 18 G. Landi et al. expression as in (2.6), but with the zµ ’s in A(C4θ ) (that is we do not impose any spherical relations). A little algebra shows that they satisfy the condition  ij gi j Xi X j = 0, or X t gX = 0, where g is the metric on R6 with signature (5, 1), i.e. on R5,1 . In terms of the basis {Xi }i=1,...,6 and with the natural identification of R5,1 with R1,1 ⊕ C2 , this metric becomes g= 1 ij g 2 = 1 2 diag       0 1 0 1 . , , 1 0 1 0 2 −2 0 0 (4.8) The coaction in (3.3) can be given on these quadratic elements and summarized by
2 L (Xi ) = j C i j ⊗ X j . Here the C i j ’s—assembled in a matrix C θ —are (Z -invariant) el- ements in A(SLθ (2, H)) whose expression can be read off from Equations (4.6) and (4.7) simply reading r instead of 1. Their commutation relations are obtained from the (3.7): C il C jm = νi j νml C jm C il , where the matrix ν = (νi j ) has entries all equal to 1 except for ν35 = ν46 = ν54 = ν63 = λ, ν36 = ν45 = ν53 = ν64 = λ. There are two additional properties of the matrix C θ . The first one is that C θ t g C θ = g, (4.9) as we shall now prove. In order to simplify computations for this, we shall rearrange the generators and use, instead of X, the vector Y = (π12 , π34 , π14 , π23 , π13 , π24 ), where the πi j ’s are the 2-minors of the matrix u in (2.3): πi j := ui1 u j2 − ui2 u j1 , i < j, i, j = 1, . . . 4. The relations with the Xi ’s are X1 = Y1 + Y2 , X2 = Y1 − Y2 , X4 = −2µ Y4 , X5 = −2Y5 , X3 = 2Y3 , X6 = −2µ Y6 . Noncommutative Families of Instantons 19 On the generators πi j ’s, the coaction L in (4.2) simply reads L (πi j ) πi j → =  l,s=1...4 mi j ls ⊗ πls , l<s with the m’s given by the 2-minors of the matrix Aθ : mi j ls = Ail Ajs − ηls Ais Ajl , l < s, i < j. (4.10) With the generators πi j s the condition X t gX = 0 translates to π12 π34 + µ π14 π23 − µ π13 π24 = 0, (4.11) which, at the classical value of the deformation parameter, µ = λ = 1, is the Plücker quadric [1]. In turn, by rewriting the metric, this condition can be written as t Y hY = 0, with h= 1 IJ h 2 = 1 2       0 −µ 0 µ 0 1 , , diag . µ 0 1 0 −µ 0 (4.12) Here and in the following, we use capital letters to denote indices I ∈ {1 = (12), 2 = (34), 3 = (14), 4 = (23), 5 = (13), 6 = (24)}. The statement in (4.9) that C θ t g C θ = g is equivalent to the following proposition whose proof is given in Appendix B. Proposition 4.3. The minors in (4.10) and the metric in (4.12) satisfy the condition,  IJ hI J m I K m J L = hK L .
The second relevant property of the matrix C θ concerns its determinant. An element det(C θ ) can be defined using a differential calculus, now on A(R5,1 θ ) (with relations dictated by those in A(Sθ4 ) except for the spherical relation) as L (dX1 · · · dX6 ) = det(C θ ) ⊗ dX1 · · · dX6 . 20 G. Landi et al. One expects that det(C θ ) can be expressed in terms of det(Aθ ) as defined in (3.8) and that it should indeed be equal to 1. Instead of checking this via a direct computation, we observe that dX1 · · · dX6 is a central element in the differential calculus on A(R5,1 θ ) and has a classical limit which is invariant under the torus action. In the framework of the deformation described at the end of Section 2, the result of L on it remains undeformed and coincides with the classical coaction of A(SO(5, 1)) giving indeed det(C θ ) = 1. Remark 4.4. With the two properties above, we could have defined the algebra A(SOθ (5, 1)) without reference to A(SLθ (2, H)). The entries of the matrix C θ are its genera
tors with relations derived as in Section 3.1 by imposing that Xi → j C i j ⊗ X j respects the commutation relations of A(Sθ4 ), except the spherical relation. In addition, one im- poses the conditions C θt gC θ = g and det(C θ ) = 1. Of course, this algebra is isomorphic to the Hopf subalgebra of Z2 -invariants in A(SLθ (2, H)) discussed above. It could also be obtained along the lines of [18, 21] (see also the beginning of Section 3) by deforming the product on A(SO(5, 1)) with respect to the adjoint action of the torus T2 ⊂ SO(5, 1).
5 A Noncommutative Family of Instantons on Sθ4 We mentioned in Section 2 that out of the matrix-valued function u in (2.3) one gets a projection p = u∗ u, given explicitly in (2.5), whose Grassmannian connection ∇ = p ◦ d has self-dual curvature: ∗θ ∇ 2 = ∇ 2 . The corresponding instanton connection 1-form— acting on equivariant maps—is expressed in terms of u as well and it is an su(2)valued 1-form on Sθ7 . Indeed, the A(Sθ4 )-module E determined by p is isomorphic to the A(Sθ4 )-module of equivariant maps for the defining representation π of SU(2) on C2 :   E ≃ A(Sθ7 ) ⊠π C2 := f ∈ A(Sθ7 ) ⊗ C2 : (id ⊗ π (g)−1 )( f) = (αg ⊗ id)( f) , whose elements we write as f =
a fa ⊗ ea by means of the standard basis {e1 , e2 } of C2 . The connection ∇ = p ◦ d : E → E ⊗A(S4 ) ∇( fa ) = d fa + (Sθ4 ) becomes on the equivariant maps:  b ωab fb, a, b = 1, 2, where the connection 1-form ω = (ωab) is found to be given by ωab = 1 2  k (u∗ )ak dukb − d(u∗ )ak ukb . (5.1) Noncommutative Families of Instantons 21 One has ωab = −(ω∗ )ba and
a 1 ωaa = 0 so that ω is in (Sθ7 ) ⊗ su(2). Out of the coaction of the quantum group SLθ (2, H) on the Hopf fibration on Sθ4 , we shall get a family of such connections in the sense that we explain in the next sections. 5.1 A family of projections We shall first describe a family of vector bundles over Sθ4 . This is done by giving a family u of suitable projections. We know from (4.1) or (4.2) the transformation of the matrix u to 
for the coaction of A(SLθ (2, H)):  uia = L (uia ) = j Ai j ⊗ u ja , with Aθ = (Ai j ) the defining matrix of A(SLθ (2, H)). The fact that the latter does not preserve the spherical relations is also the statement that  k ( u∗ )ak  ukb = L  k  (u∗ )ak ukb = L  k or ( u)∗  u = ρ 2 I2 . Then, we define P = (Pi j ) ∈ Mat4 (A( Sθ4 )) by P :=  u ρ −2 ( u)∗ , or Pi j = ρ −2  a  zk∗ zk δab = ρ 2 δab,  uia ( u∗ )a j . (5.2) (5.3) The condition ( u)∗  u = ρ 2 I2 gives that P is an idempotent; being ∗-self-adjoint it is a projection. Remark 5.1. For the above definition, we need to enlarge the algebra A( Sθ4 ) by adding the extra element ρ −2 , the inverse of the positive self-adjoint central element ρ 2 . In fact,  we shall also presently need the element ρ −1 = ρ −2 . At the smooth level this is not problematic. The algebra C ∞ ( Sθ4 ) can be defined as a fixed point algebra as in [7] and one finds that the spectrum of ρ 2 is positive and does not contain the point 0.
Explicitly, one finds for the projection P the expression ⎛ ⎞  ρ2 +  x 0  α β ⎜ ⎟ ∗ µ α∗ ⎟ x −µ β 0 ρ2 +  1 −2 ⎜ ⎟, ⎜ P = ρ ⎜ ⎟  ρ2 −  2 x 0 ⎠ α∗ −µ β ⎝  ∗ β µ α 0 ρ2 −  x a matrix strikingly similar to the matrix (2.5) for the basic projection. The entries of the projection P are in A( Sθ4 ), that is A(SLθ (2, H)) ⊗ A(Sθ4 ): we interpret P as a noncommutative 22 G. Landi et al. family of projections parameterized by the noncommutative space SLθ (2, H). This is the analogue for projections of the noncommutative families of maps that were introduced and studied in [20, 23]. The interpretation as a noncommutative family is justified by the classical case: at θ = 0, there are evaluation maps evx : A(SL(2, H)) → C and for each point x in SL(2, H), (evx ⊗ id)P is a projection in Mat4 (A(S4 )), that is a bundle over S4 . Although there need not be enough evaluation maps available in the noncommutative case, we can still work with the whole family at once. As mentioned, we think of the Hopf algebra SLθ (2, H) as a parameter space and we extend to A(SLθ (2, H)) ⊗ A(Sθ4 ) the differential of A(Sθ4 ) as (id ⊗ d) (and similarly, the Hodge star operator of A(Sθ4 ) as (id ⊗ ∗θ )). Having these, out of the projection P one gets a noncommutative family of instantons.  := P ◦ (id ⊗ d) has self-dual curvature Proposition 5.2. The family of connections ∇ 2 = P ((id ⊗ d)P )2 , that is, ∇ (id ⊗ ∗θ )P ((id ⊗ d)P )2 = P ((id ⊗ d)P )2 .
Proof. From Proposition 4.2 we know that A(SLθ (2, H)) coacts by conformal transforma2 = P ((id ⊗ d)P )2 is the image of the curvature p(d p)2 under the tions and the curvature ∇ coaction of A(SLθ (2, H)).  It was shown in [8] that the charge of the basic instanton p is 1. This charge was given as a pairing between the second component of the Chern character of p—an element in the cyclic homology group HC4 (A(Sθ4 ))—with the fundamental class of Sθ4 in the cyclic cohomology HC4 (A(Sθ4 )). The zeroth and first components of the Chern character were shown to vanish identically in HC0 (A(Sθ4 )) and HC2 (A(Sθ4 )), respectively. We will reduce the computation of the Chern character for the family of projections P to this case by proving that P is equivalent to the projection 1 ⊗ p. Hence, we conclude that P represents the same class as 1 ⊗ p in the K-theory of the algebra A(SLθ (2, H)) ⊗ A(Sθ4 ). Recall that two projections p, q are Murray–von Neumann equivalent if there exists a partial isometry V such that p = V V ∗ and q = V ∗ V. Lemma 5.3. The projection P is Murray–von Neumann equivalent to the projection 1 ⊗ p in the algebra M4 A(SLθ (2, H)) ⊗ A(Sθ4 ) . Proof. Define the matrix V = (Vik ) ∈ M4 (A(SLθ (2, H)) ⊗ A(Sθ4 )) by Vik = ρ −1 Ai j ⊗ pjk = ρ −1 Ai j ⊗ u ja (u∗ )ak = ρ −1  uia (1 ⊗ (u∗ )ak ),
Noncommutative Families of Instantons 23 with ũ = (ũia ) as in (4.1). For its adjoint, we have u∗ )ak . (V ∗ )ik = ρ −1 (1 ⊗ uia )( Then, using (5.2), one obtains   (1 ⊗ uia )( u∗ )ak  ukb(1 ⊗ (u∗ )bl ) (V ∗ )ik Vkl = ρ −2 kab k   = ρ −2 uia (u∗ )al = 1 ⊗ pil (1 ⊗ uia )(ρ 2 δab)(1 ⊗ (u∗ )bl ) = 1 ⊗ (V ∗ V)il = a ab and   uia (1 ⊗ (u∗ )ak )(1 ⊗ ukb)( u∗ )bl Vik (V ∗ )kl = ρ −2 kab k    uia (1 ⊗ δab)( u∗ )bl  uia (1 ⊗ (u∗ )ak ukb)( u∗ )bl = ρ −2 = ρ −2 ab kab   uia ( u∗ )al = Pil , = ρ −2 (V V ∗ )il =  a which finishes the proof.  It follows from this lemma that the components chn (P ) ∈ HC2n (A(SLθ (2, H)) ⊗ A(Sθ4 )), with n = 0, 1, 2, of the Chern character of P coincide with the pushforwards φ∗ chn ( p) of chn ( p) ∈ HC2n (A(Sθ4 )) under the algebra map φ : A(Sθ4 ) → A(SLθ (2, H)) ⊗ A(Sθ4 ), a → 1 ⊗ a. As a consequence, both ch0 (P ) and ch1 (P ) are zero since ch0 ( p) and ch1 ( p) vanish [8]. Next, we would like to compute the charge of the family of instantons by pairing ch2 (P ) with the fundamental class [Sθ4 ] ∈ HC4 (A(Sθ4 )); classically, this corresponds to an integration over S4 giving a value 1 of the charge which is constant over SL(2, H). As said, the Chern character ch2 (P ) is an element in HC4 (A(SLθ (2, H)) ⊗ A(Sθ4 )), which at first sight seems unsuitable to pair with an element in HC4 (A(Sθ4 )). However, there is a pairing between the K-theory group K0 (A(SLθ (2, H)) ⊗ A(Sθ4 )) and the K-homology group K0 (A(Sθ4 )). Recall that for any algebra A from Kasparov’s KK-theory, one has that K0 (A) = KK(A, C) and K0 (A) = KK(C, A). As described in [6, Appendix IV.A], for algebras A, B, and C, there is a map τC : KK(A, B) → KK(C ⊗ A, C ⊗ B) which simply tensors a Kasparov A − B module by C on the left. In our case we get an element τA(SLθ (2,H)) [Sθ4 ] ∈ KK(A(SLθ (2, H)) ⊗ A(Sθ4 ), A(SLθ (2, H))), which can be paired with [P ] via the cup product [12]. Thus we obtain the desired pairing: KK C, A(SLθ (2, H)) ⊗ A Sθ4 [P ], Sθ4 × KK A Sθ4 , C → KK(C, A(SLθ (2, H))),   → [P ], τA(SLθ (2,H)) Sθ4 . 24 G. Landi et al. Having [P ] = [1 ⊗ p], we obtain   [P ], τA(SLθ (2,H)) Sθ4 = [1] ⊗ [ p], [Sθ4 ] = [1] ∈ K0 (A(SLθ (2, H))), where in the last line we used the equality [ p], [Sθ4 ] = 1 proved in [8] too. The above is the statement that the value 1 of the topological charge is constant over the family. 5.2 A family of connections When transforming u by the coaction of SLθ (2, H) in (4.1), one transforms the connection ω = ( ωab) with, 1-form ω in (5.1) as well to  Since L  ωab := L (ωab ) = 1 (A∗ )ik Ak j ⊗ (u∗ )ai du jb − d(u∗ )ai u jb . ki j 2 (5.4)
is linear,  ω is still traceless ( a  ωaa = 0) and skew-Hermitean ( ωab = −( ω∗ )ba ). Proposition 5.4. The instanton connection 1-form ω is invariant under the coaction of the quantum group Spθ (2), that is for this quantum group one has L (ωab ) = 1 ⊗ ωab.
Proof. This is a simple consequence of the fact that for Spθ (2) one has from which (5.4) reduces to  ωab = 1 ⊗ ωab.
∗ k (A )ik Ak j = δi j  Hence, the relevant space that parameterizes the connection one-forms is not SLθ (2, H) but rather the quotient of SLθ (2, H) by Spθ (2). Denoting by π the natural quotient map from A(SLθ (2, H)) to A(Spθ (2)), the algebra of the quotient is the algebra of coinvariants of the natural left coaction L = (π ⊗ id) ◦ A(Mθ ) := {a ∈ A(SLθ (2, H)) | L (a) of Spθ (2) on SLθ (2, H): = 1 ⊗ a}. Since Spθ (2) is a quantum subgroup of SLθ (2, H) the quotient is well defined: the algebra A(Mθ ) is a quantum homogeneous space and the inclusion A(Mθ ) ֒→ A(SLθ (2, H)) is a noncommutative principal bundle with A(Spθ (2)) as structure group. Lemma 5.5. The quantum quotient space A(Mθ ) is generated as an algebra by the


elements mi j := k (A∗ )ik Ak j = k (Aki )∗ Ak j . Noncommutative Families of Instantons 25 Proof. Since the relations in the quotient A(Spθ (2)) are quadratic in the matrix elements Ai j and (Ai j )∗ , the generators of A(Mθ ) have to be at least quadratic in them. For the first leg of the tensor product
a = i (Aik )∗ Ail , so that (π ⊗ id) (a) = = (a) to involve these relations in A(Spθ (2)), we need to take  imn π ((Aim )∗ Ain ) ⊗ (Amk )∗ Anl imn π ((Aim )∗ )π (Ain ) ⊗ (Amk )∗ Anl =   mn δmn ⊗ (Amk )∗ Anl , giving the desired result.  We will think of the transformed  ω in (5.4) as a family of connection one-forms parameterized by the noncommutative space Mθ . At the classical value θ = 0, we get the moduli space Mθ=0 = SL(2, H)/Sp(2) of instantons of charge 1. For each point x in Mθ=0 , the evaluation map evx : A(Mθ=0 ) → C gives an instanton connection (i.e. one with self-dual curvature) (evx ⊗ id) ω on the bundle over S4 described by (evx ⊗ id)P . 5.3 The space Mθ of connections and its geometry The structure of the algebra A(Mθ ) is deduced from that of A(SLθ (2, H)). We collect the
generators mi j = k (Aki )∗ Ak j into a matrix M := (mi j ). Explicitly, one finds ⎛ m ⎜ ⎜0 M=⎜ ⎜ ∗ ⎝g1 g2 0 g1 m −µ g2 −µ g2∗ µ g1 n 0 g2∗ ⎞ ⎟ µ g1∗ ⎟ ⎟ ⎟ 0 ⎠ n (5.5) with its entries related to those of the defining matrix Aθ in (3.4) of A(SLθ (2, H)) by m = m∗ = a1∗ a1 + a2∗ a2 + c1∗ c1 + c2∗ c2 , n = n∗ = b1∗ b1 + b2∗ b2 + d1∗ d1 + d2∗ d2 , g1 = a1∗ b1 + µ b2∗ a2 + c1∗ d1 + µ d2∗ c2 , (5.6) g2 = b2∗ a1 − µ a2∗ b1 + d2∗ c1 − µ c2∗ d1 . As for the commutation relations, one finds that both m and n are central: m x = x m, n x = x n ∀x ∈ Mθ ; (5.7a) 26 G. Landi et al. that g1 and g2 are normal: g1 g1∗ = g1∗ g1 , g2 g2∗ = g2∗ g2 ; (5.7b) g1 g2∗ = µ2 g2∗ g1 . (5.7c) and that g1 g2 = µ2 g2 g1 , together with their conjugates. There is also a quadratic relation, mn − (g1∗ g1 + g2∗ g2 ) = 1, (5.8) coming from the condition det(Aθ ) = 1. Indeed, one first establishes that besides the product mn, also g1∗ g1 + g2∗ g2 is a central element in A(Mθ ), and then computes mn − g1∗ g1 + g2∗ g2 = a1∗ a1 d1∗ d1 + a1∗ a1 d2∗ d2 + a2∗ a2 d1∗ d1 + a2∗ a2 d2∗ d2 + b1∗ b1 c1∗ c1 + b1∗ b1 c2∗ c2 + b2∗ b2 c1∗ c1 + b2∗ b2 c2∗ c2 − a1∗ c1 d1∗ b1 − a1∗ c1 d2∗ b2 − a2∗ c2 d1∗ b1 − a2∗ c2 d2∗ b2 − b1∗ d1 c1∗ a1 − b1∗ d1 c2∗ a2 − b2∗ d2 c1∗ a1 − b2∗ d2 c2∗ a2 − a1∗ c2∗ d2 b1 + a1∗ c2∗ d1 b2 + a2∗ c1∗ d2 b1 − a2∗ c1∗ d1 b2 − b1∗ d2∗ c2 a1 + b1∗ d2∗ c1 a2 + b2∗ d1∗ c2 a1 − b2∗ d1∗ c1 a2 = det(Aθ ) by a direct comparison with the expression (3.8). Elements (mi j ) of the matrix M enter the expression for ρ 2 . With pkl the components of the defining projector p in (2.4) and having formula (4.7), one finds that 1 ηi j mi j ⊗ pji ij 2 1 (m + n) ⊗ 1 + (m − n) ⊗ x + µ g1∗ ⊗ α + µ g2 ⊗ β + µ g1 ⊗ α ∗ + µ g2∗ ⊗ β ∗ . = 2 ρ2 = In particular, for Ai j ∈ A(Spθ (2)) one gets ρ 2 = 12 (1 ⊗ tr( p)) = 1 ⊗ 1, as already observed in Remark 4.1. 5.4 The boundary of Mθ The defining matrix M of Mθ in (5.5), with the commutation relations among its entries, is strikingly similar to the defining projection p of A(Sθ4 ) in (2.5) with the corresponding commutation relations. Clearly, the crucial difference is that while for A(Sθ4 ) we have a spherical relation, for Mθ we have the relation (5.8), which makes Mθ a θ -deformation Noncommutative Families of Instantons 27 of a hyperboloid in six dimensions. This becomes more clear if we introduce two central elements w and y, given in terms of m, n by w := 12 (m + n); y := 21 (m − n). Relation (5.8) then reads w 2 − (y2 + g1∗ g1 + g2∗ g2 ) = 1, (5.9) making evident the hyperboloid structure. Let us examine its structure at “infinity.” We first adjoin the inverse of w to A(Mθ ), and stereographically project onto the coordinates, Y := w −1 y, G 1 := w −1 g1 , G 2 := w −1 g2 . The relation (5.9) becomes, Y2 + G ∗1 G 1 + G ∗2 G 2 = 1 − w −2 . Evaluating w as a real number, and taking its “limit to infinity” we get a spherical relation, Y2 + G ∗1 G 1 + G ∗2 G 2 = 1. By combining this with relations (5.7), we can conclude that at the “boundary” of Mθ , we reencounter the noncommutative 4-sphere A(Sθ4 ) via the identification Y ↔ x, G1 ↔ α G 2 ↔ β. The above construction is the analogue of the classical structure, in which 4-spheres are found at the boundary of the moduli space. 6 Outlook We have constructed a noncommutative family of instantons of charge 1 on the noncommutative 4-sphere Sθ4 . The family is parameterized by a noncommutative space Mθ , which reduces to the moduli space of charge 1 instantons on S4 in the limit when θ → 0. Although this means that Mθ is a quantization of the moduli space Mθ=0 , it does not imply that it is itself a space of moduli. In order to call this the moduli space of charge 1 instantons on Sθ4 a few things must be clarified. We mention in particular two important points that for the moment lack a proper understanding. 28 G. Landi et al. First of all, we are confronted with the difficulty of finding a proper notion of gauge group and gauge transformations. A naive dualization of the undeformed construction would lead one to consider the group of A(SU(2))-coequivariant algebra maps from the algebra A(SU(2)) to A(Sθ7 ), equipped with the convolution product. However, since the algebra A(SU(2)) is commutative as opposed to A(Sθ7 ), one quickly realizes that there are not so many elements in this group (an interesting open problem is to find in general a correct noncommutative analogue of the group of maps from a space X to a group G). The second open problem is related to the fact that one would need some sort of universality for the noncommutative family of instantons to call it a moduli space. A possible notion of universality could be defined as follows. A family of instantons parameterized by A(M) is said to be universal if for any other noncommutative family of instantons parameterized by, say, an algebra B, there exists an algebra map φ : M → B such that this family can be obtained from the universal family via the map φ. Again, this is the analogue of the notion of universality for noncommutative families of maps as in [20, 23]. But it appears that, in order to prove universality for the actual family that we have constructed in the present paper, an argument along the classical lines—involving a local construction of the moduli space from its tangent bundle [3]—fails here, due to the fact that there is no natural notion of a tangent space to a noncommutative space. Progress on both of these problems must await another time. Appendix A Explicit Commutation Relations For convenience, we list the explicit commutation relations (3.7) of the elements of the matrix (3.4). The not trivial ones are the following: a1 b1 = µ b1 a1 a2 b1 = µ b1 a2 a1 c1 = µ c1 a1 a2 c1 = µ c1 a2 a1 b2 = µ b2 a1 a2 b2 = µ b2 a2 a1 c2 = µ c2 a1 a2 c2 = µ c2 a2 a1 b1∗ = µ b1∗ a1 a2 b1∗ = µ b1∗ a2 a1 c1∗ = µ c1∗ a1 a2 c1∗ = µ c1∗ a2 a1 b2∗ = µ b2∗ a1 a2 b2∗ = µ b2∗ a2 a1 c2∗ = µ c2∗ a1 a2 c2∗ = µ c2∗ a2 a2 d1 = µ2 d1 a2 b1 c1 = µ2 c1 b1 b2 c1 = c1 b2 a1 d1 = d1 a1 a1 d2 = µ2 d2 a1 a1 d1∗ = d1∗ a1 a1 d2∗ = µ2 d2∗ a1 a2 d2 = d2 a2 a2 d1∗ = µ2 d1∗ a2 a2 d2∗ = d2∗ a2 b1 c2 = c2 b1 b1 c1∗ = µ2 c1∗ b1 b1 c2∗ = c2∗ b1 b2 c2 = µ2 c2 b2 b2 c1∗ = c1∗ b2 b2 c2∗ = µ2 c2∗ b2 Noncommutative Families of Instantons 29 b1 d1 = µ d1 b1 b2 d1 = µ d1 b2 c1 d1 = µ d1 c1 c2 d1 = µ d1 c2 b1 d2 = µ d2 b1 b2 d2 = µ d2 b2 c1 d2 = µ d2 c1 c2 d2 = µ d2 c2 b1 d1∗ = µ d1∗ b1 b2 d1∗ = µ d1∗ b2 c1 d1∗ = µ d1∗ c1 c2 d1∗ = µ d1∗ c2 b1 d2∗ = µ d2∗ b1 b2 d2∗ = µ d2∗ b2 c1 d2∗ = µ d2∗ c1 c2 d2∗ = µ d2∗ c2 together with their conjugates. B Explicit Proof of Proposition 4.3 We prove here that the minors in (4.10) and the metric in (4.12) satisfy the condition  IJ hI J m I K m J L = hK L (B.1) of Proposition 4.3. As said in the main text, this is equivalent to the fact that the defining matrix C θ of A(SOθ (5, 1)) satisfy C θ t g C = g with g the metric in (4.8). We first prove (B.1) when in the right-hand side h K L = 0, namely for the cases (K, L) = (1, 2), (3, 4), (5, 6). For this we need the formula (3.9) for the determinant of Aθ . A little algebra show that the determinant can also be written as det(Aθ ) =  (−1)|σ | ε σ Aσ (1),1 Aσ (2),2 Aσ (3),3 Aσ (4),4 (B.2) σ ∈S4 where ε σ = εσ . Also, for the tensor ε we find relations: εi jkl = η ji ε jikl ; εi jkl = ηlk εi jlk ; εi jkl = ηk j εik jl , (B.3) εi jkl = ηi j ε jikl ; εi jkl = ηkl εi jlk , εi jkl = η jk εik jl . (B.4) and analogues for ε, Given σ ∈ S4 , we let σ ′ = (12)σ and σ ′′ = (34)σ , and compute det(Aθ ) = = =  (−1)|σ | εσ Aσ (1),1 Aσ (2),2 Aσ (3),3 Aσ (4),4 σ ∈S4  σ ∈S4 \σ ′  σ ∈S4 \σ ′ ′ (−1)|σ | εσ Aσ (1),1 Aσ (2),2 − ε σ Aσ (2),1 Aσ (1),2 Aσ (3),3 Aσ (4),4 ′ (−1)|σ | εσ Aσ (1),1 Aσ (2),2 − ε σ ησ (1)σ (2) η12 Aσ (1),2 Aσ (2),1 Aσ (3),3 Aσ (4),4 30 G. Landi et al. = = =  (−1)|σ | εσ mσ (1)σ (2) 12 Aσ (3),3 Aσ (4),4 σ ∈S4 \σ ′  σ ∈S4 \{σ ′ ,σ ′′ } σ ∈S4 \{σ ′ σ ′′ }  (−1)|σ | ε σ mσ (1)σ (2) 12 (Aσ (3),3 Aσ (4),4 − η34 Aσ (3),4 Aσ (4),3 ) (−1)|σ | εσ mσ (1)σ (2) 12 mσ (3)σ (4) 34 . (B.5) Since the mi j kl where defined for i < j, k < l, we choose σ ∈ S4 \ {σ ′ σ ′′ } such that σ1 < σ2 and σ3 < σ4 . Hence the sum above runs over σ = (σ1 , σ2 ), (σ3 , σ4 ) ∈ I, where I :=   (1, 2), (3, 4) ; ((1, 3), (2, 4)); ((1, 4), (2, 3)); ((2, 3), (1, 4)); ((2, 4), (1, 3)); ((3, 4), (1, 2)) . Finally, using the explicit form of the εs, the above formula (B.5) reads det(Aθ ) =  IJ hI J mI 1m J 2, and the condition det(Aθ ) = 1 proves (B.1) for K = 1, L = 2. The relation above coincides with the “hyperboloid” relation (5.8) for the generators of the matrix Mθ . For the other two cases we use different orders for the Aσ (i)i in (B.2). Similar procedures to the one in (B.5)—and using the properties (B.3) and (B.4)—lead to det(Aθ ) = η24  (−1)|σ | εσ mσ (1)σ (4) 14 mσ (2)σ (3) 23 , σ ∈S4 which gives µdet(Aθ ) =  IJ hI J mI 3m J 4, hence proving (B.1) for K = 3, L = 4; and to det(Aθ ) = −η23  (−1)|σ | ε σ mσ (1)σ (3) 13 mσ (2)σ (4) 24 , σ ∈S4 which gives µdet(Aθ ) = −  IJ hI J mI 5m J 6, hence proving (B.1) for K = 5, L = 6. Finally, we have (B.1) when h K L = 0 in the right-hand side; for these cases (B.1) is m12 i j m34 kl + m34 i j m12 kl + µm23 i j m14 kl + µm14 i j m23 kl − µm24 i j m13 kl − µm13 i j m24 kl = 0. Noncommutative Families of Instantons 31 These can be proved with the explicit expressions of the mi j kl in (4.10) and observing that the hypothesis I = (i j), K = (kl) such that h I K = 0 implies four possibilities: i = k, i = l, j = k or j = l. In addition, the relation mi j kl = −ηkl mi j lk reduces the computations to just one case, say i = l. Acknowledgments We thank Simon Brain, Eli Hawkins, Mark Rieffel, Lech Woronowicz, and Makoto Yamashita for useful discussions and remarks. G.L. and C.R. were partially supported by the “Italian project PRIN06 - Noncommutative geometry, quantum groups and applications.” C.P. gratefully acknowledges support from MRTN-CT 2003-505078, INDAM, MKTD-CT 2004-509794, and SNF. References [1] Atiyah, M. The Geometry of Yang–Mills Fields. Lezioni Fermiane. Pisa, Italy: Scuola Normale Superiore, 1979. [2] Atiyah, M., V. G. Drinfeld, N. J. Hitchin, and Yu. I. Manin. “Construction of instantons.” Physical Letters 65A (1978): 185–7. [3] Atiyah, M., N. J. Hitchin, and I. M. Singer. “Deformations of instantons.” Proceedings of the [4] Brzeziński, T., and S. Majid. “Quantum group gauge theory on quantum spaces.” Communi- [5] Brain, S., and S. Majid. “Quantisation of twistor theory by cocycle twist.” (2007): preprint National Academy of Sciences, USA 74A (1977): 2662–3. cations in Mathematical Physics 157 (1993): 591–638. Erratum 167 (1995): 235. arXiv:math/0701893. [6] [7] Connes, A. Noncommutative Geometry. San Diego, CA: Academic Press, 1994. Connes, A., and M. Dubois-Violette. “Noncommutative finite-dimensional manifolds 1: Spherical manifolds and related examples.” Communications in Mathematical Physics 230 (2002): 539–79. [8] Connes, A., and G. Landi. “Noncommutative manifolds: the instanton algebra and isospectral deformations.” Communications in Mathematical Physics 221 (2001): 141–59. [9] Drinfel’d, V. G. “Constant quasiclassical solutions of the Yang–Baxter quantum equation.” Soviet Mathematics Doklady 28 (1983): 667–71. [10] [11] Drinfel’d, V. G. “Quasi-Hopf algebras.” Leningrad Mathematical Journal 1 (1990): 1419–57. Hartshorne, R. “Stable vector bundles and instantons.” Communications in Mathematical Physics 59 (1978): 1–15. [12] Kasparov, G. “The operator K-functor and extensions of C ∗ -algebras.” Izvestiya Rossiiskoi Akademii Nauk 44 (1980): 571–636. [13] Landi, G., C. Pagani, and C. Reina. “A Hopf bundle over a quantum four-sphere from the symplectic group.” Communications in Mathematical Physics 263 (2006): 65–88. 32 G. Landi et al. [14] Landi, G., and W. van Suijlekom. “Principal fibrations from noncommutative spheres.” Communications in Mathematical Physics 260 (2005): 203–25. [15] Landi, G., and W. van Suijlekom. “Noncommutative instantons from twisted conformal symmetries.” Communications in Mathematical Physics 271 (2007): 591–639. [16] Reshetikhin, N. “Multiparameter quantum groups and twisted quasitriangular Hopf algebras.” Letters in Mathematical Physics 20 (1990): 331–5. [17] Rieffel, M. A. Deformation Quantization for Actions of Rd . Memoirs of the American Mathematical Society 506. Providence, RI: American Mathematical Society, 1993. [18] Rieffel, M. A. “Compact quantum groups associated with toral subgroups.” Contemporary Mathematics 145 (1993): 465–91. [19] Sitarz, A. “Twists and spectral triples for isospectral deformations.” Letters in Mathematical Physics 58 (2001): 69–79. [20] Soltan, P. M. “Quantum families of maps and quantum semigroups on finite quantum spaces.” (2006): preprint arXiv:math/0610922. [21] Várilly, J. C. “Quantum symmetry groups of noncommutative spheres.” Communications in Mathematical Physics 221 (2001): 511–23. [22] Wang, S. “Quantum symmetry groups of finite spaces.” Communications in Mathematical Physics 195 (1998): 195–211. [23] Woronowicz, S. L. “Pseudospaces, Pseudogroups and Pontryagin Duality.” In Proceedings of the International Conference on Mathematical Physics, edited by K. Osterwalder, 407–12. Lecture Notes in Physics 116. Berlin: Springer, 1980.