Geometric Models for Noncommutative Algebras Ana Cannas da Silva1 Alan Weinstein2 University of California at Berkeley December 1, 1998 [email protected], [email protected] [email protected] Contents Preface xi Introduction xiii I Universal Enveloping Algebras 1 1 Algebraic Constructions 1 1.1 Universal Enveloping Algebras . . . . . . . . . . . . . . . . . . . . . 1 1.2 Lie Algebra Deformations . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 The Graded Algebra of U(g). . . . . . . . . . . . . . . . . . . . . . . 3 2 The Poincar´e-Birkhoff-Witt Theorem 5 2.1 Almost Commutativity of U(g) . . . . . . . . . . . . . . . . . . . . . 5 2.2 Poisson Bracket on Gr U(g) . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 The Role of the Jacobi Identity . . . . . . . . . . . . . . . . . . . . . 7 2.4 Actions of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Proof of the Poincar´e-Birkhoff-Witt Theorem . . . . . . . . . . . . . 9 II Poisson Geometry 11 3 Poisson Structures 11 3.1 Lie-Poisson Bracket. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Almost Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Structure Functions and Canonical Coordinates . . . . . . . . . . . . 13 3.5 Hamiltonian Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . 14 3.6 Poisson Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Normal Forms 17 4.1 Lie’s Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 A Faithful Representation of g . . . . . . . . . . . . . . . . . . . . . 17 4.3 The Splitting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 Special Cases of the Splitting Theorem . . . . . . . . . . . . . . . . . 20 4.5 Almost Symplectic Structures . . . . . . . . . . . . . . . . . . . . . . 20 4.6 Incarnations of the Jacobi Identity . . . . . . . . . . . . . . . . . . . 21 5 Local Poisson Geometry 23 5.1 Symplectic Foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Transverse Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.3 The Linearization Problem . . . . . . . . . . . . . . . . . . . . . . . 25 5.4 The Cases of su(2) and sl(2;R) . . . . . . . . . . . . . . . . . . . . . 27 III Poisson Category 29 v vi CONTENTS 6 Poisson Maps 29 6.1 Characterization of Poisson Maps . . . . . . . . . . . . . . . . . . . . 29 6.2 Complete Poisson Maps . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.3 Symplectic Realizations . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.4 Coisotropic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.5 Poisson Quotients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.6 Poisson Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7 Hamiltonian Actions 39 7.1 Momentum Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.2 First Obstruction for Momentum Maps . . . . . . . . . . . . . . . . 40 7.3 Second Obstruction for Momentum Maps . . . . . . . . . . . . . . . 41 7.4 Killing the Second Obstruction . . . . . . . . . . . . . . . . . . . . . 42 7.5 Obstructions Summarized . . . . . . . . . . . . . . . . . . . . . . . . 43 7.6 Flat Connections for Poisson Maps with Symplectic Target . . . . . 44 IV Dual Pairs 47 8 Operator Algebras 47 8.1 Norm Topology and C∗-Algebras . . . . . . . . . . . . . . . . . . . . 47 8.2 Strong and Weak Topologies . . . . . . . . . . . . . . . . . . . . . . 48 8.3 Commutants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8.4 Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 9 Dual Pairs in Poisson Geometry 51 9.1 Commutants in Poisson Geometry . . . . . . . . . . . . . . . . . . . 51 9.2 Pairs of Symplectically Complete Foliations . . . . . . . . . . . . . . 52 9.3 Symplectic Dual Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 53 9.4 Morita Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 9.5 Representation Equivalence . . . . . . . . . . . . . . . . . . . . . . . 55 9.6 Topological Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . 56 10 Examples of Symplectic Realizations 59 10.1 Injective Realizations of T3 . . . . . . . . . . . . . . . . . . . . . . . 59 10.2 Submersive Realizations of T3 . . . . . . . . . . . . . . . . . . . . . . 60 10.3 Complex Coordinates in Symplectic Geometry . . . . . . . . . . . . 62 10.4 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 63 10.5 A Dual Pair from Complex Geometry . . . . . . . . . . . . . . . . . 65 V Generalized Functions 69 11 Group Algebras 69 11.1 Hopf Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 11.2 Commutative and Noncommutative Hopf Algebras . . . . . . . . . . 72 11.3 Algebras of Measures on Groups . . . . . . . . . . . . . . . . . . . . 73 11.4 Convolution of Functions . . . . . . . . . . . . . . . . . . . . . . . . 74 11.5 Distribution Group Algebras . . . . . . . . . . . . . . . . . . . . . . 76 CONTENTS vii 12 Densities 77 12.1 Densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 12.2 Intrinsic Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 12.3 Generalized Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 12.4 Poincar´e-Birkhoff-Witt Revisited . . . . . . . . . . . . . . . . . . . . 81 VI Groupoids 85 13 Groupoids 85 13.1 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . 85 13.2 Subgroupoids and Orbits . . . . . . . . . . . . . . . . . . . . . . . . 88 13.3 Examples of Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . 89 13.4 Groupoids with Structure . . . . . . . . . . . . . . . . . . . . . . . . 92 13.5 The Holonomy Groupoid of a Foliation . . . . . . . . . . . . . . . . . 93 14 Groupoid Algebras 97 14.1 First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 14.2 Groupoid Algebras via Haar Systems . . . . . . . . . . . . . . . . . . 98 14.3 Intrinsic Groupoid Algebras . . . . . . . . . . . . . . . . . . . . . . . 99 14.4 Groupoid Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 14.5 Groupoid Algebra Actions . . . . . . . . . . . . . . . . . . . . . . . . 103 15 Extended Groupoid Algebras 105 15.1 Generalized Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 15.2 Bisections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 15.3 Actions of Bisections on Groupoids . . . . . . . . . . . . . . . . . . . 107 15.4 Sections of the Normal Bundle . . . . . . . . . . . . . . . . . . . . . 109 15.5 Left Invariant Vector Fields . . . . . . . . . . . . . . . . . . . . . . . 110 VII Algebroids 113 16 Lie Algebroids 113 16.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 16.2 First Examples of Lie Algebroids . . . . . . . . . . . . . . . . . . . . 114 16.3 Bundles of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 116 16.4 Integrability and Non-Integrability . . . . . . . . . . . . . . . . . . . 117 16.5 The Dual of a Lie Algebroid . . . . . . . . . . . . . . . . . . . . . . . 119 16.6 Complex Lie Algebroids . . . . . . . . . . . . . . . . . . . . . . . . . 120 17 Examples of Lie Algebroids 123 17.1 Atiyah Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 17.2 Connections on Transitive Lie Algebroids . . . . . . . . . . . . . . . 124 17.3 The Lie Algebroid of a Poisson Manifold . . . . . . . . . . . . . . . . 125 17.4 Vector Fields Tangent to a Hypersurface . . . . . . . . . . . . . . . . 127 17.5 Vector Fields Tangent to the Boundary . . . . . . . . . . . . . . . . 128 viii CONTENTS 18 Differential Geometry for Lie Algebroids 131 18.1 The Exterior Differential Algebra of a Lie Algebroid . . . . . . . . . 131 18.2 The Gerstenhaber Algebra of a Lie Algebroid . . . . . . . . . . . . . 132 18.3 Poisson Structures on Lie Algebroids . . . . . . . . . . . . . . . . . . 134 18.4 Poisson Cohomology on Lie Algebroids . . . . . . . . . . . . . . . . . 136 18.5 Infinitesimal Deformations of Poisson Structures . . . . . . . . . . . 137 18.6 Obstructions to Formal Deformations . . . . . . . . . . . . . . . . . 138 VIII Deformations of Algebras of Functions 141 19 Algebraic Deformation Theory 141 19.1 The Gerstenhaber Bracket . . . . . . . . . . . . . . . . . . . . . . . . 141 19.2 Hochschild Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 142 19.3 Case of Functions on a Manifold . . . . . . . . . . . . . . . . . . . . 144 19.4 Deformations of Associative Products . . . . . . . . . . . . . . . . . 144 19.5 Deformations of the Product of Functions . . . . . . . . . . . . . . . 146 20 Weyl Algebras 149 20.1 The Moyal-Weyl Product . . . . . . . . . . . . . . . . . . . . . . . . 149 20.2 The Moyal-Weyl Product as an Operator Product . . . . . . . . . . 151 20.3 Affine Invariance of the Weyl Product . . . . . . . . . . . . . . . . . 152 20.4 Derivations of Formal Weyl Algebras . . . . . . . . . . . . . . . . . . 152 20.5 Weyl Algebra Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 153 21 Deformation Quantization 155 21.1 Fedosov’s Connection. . . . . . . . . . . . . . . . . . . . . . . . . . . 155 21.2 Preparing the Connection . . . . . . . . . . . . . . . . . . . . . . . . 156 21.3 A Derivation and Filtration of the Weyl Algebra . . . . . . . . . . . 158 21.4 Flattening the Connection . . . . . . . . . . . . . . . . . . . . . . . . 160 21.5 Classification of Deformation Quantizations . . . . . . . . . . . . . . 161 References 163 Index 175 Preface Noncommutativegeometryisthestudyofnoncommutativealgebrasasiftheywere algebras of functions on spaces, like the commutative algebras associated to affine algebraicvarieties,differentiablemanifolds,topologicalspaces,andmeasurespaces. In this book, we discuss several types of geometric objects (in the usual sense of sets with structure) which are closely related to noncommutative algebras. Central to the discussion are symplectic and Poisson manifolds, which arise when noncommutative algebras are obtained by deforming commutative algebras. We also make a detailed study of groupoids, whose role in noncommutative geom- etry has been stressed by Connes, as well as of Lie algebroids, the infinitesimal approximations to differentiable groupoids. Thesenotesarebasedonatopicscourse,“GeometricModelsforNoncommuta- tive Algebras,” which one of us (A.W.) taught at Berkeley in the Spring of 1997. We would like to express our appreciation to Kevin Hartshorn for his partic- ipation in the early stages of the project – producing typed notes for many of the lectures. Henrique Bursztyn, who read preliminary versions of the notes, has provided us with innumerable suggestions of great value. We are also indebted to Johannes Huebschmann, Kirill Mackenzie, Daniel Markiewicz, Elisa Prato and Olga Radko for several useful commentaries or references. Finally,wewouldliketodedicatethesenotestothememoryoffourfriendsand colleagues who, sadly, passed away in 1998: Mosh´e Flato, K. Guruprasad, Andr´e Lichnerowicz, and Stanisl(cid:32)aw Zakrzewski. Ana Cannas da Silva Alan Weinstein xi Introduction We will emphasize an approach to algebra and geometry based on a metaphor (see Lakoff and Nun˜ez [100]): An algebra (over R or C) is the set of (R- or C-valued) functions on a space. Strictly speaking, this statement only holds for commutative algebras. We would like to pretend that this statement still describes noncommutative algebras. Furthermore, different restrictions on the functions reveal different structures on the space. Examples of distinct algebras of functions which can be associated to a space are: • polynomial functions, • real analytic functions, • smooth functions, • Ck, or just continuous (C0) functions, • L∞, or the set of bounded, measurable functions modulo the set of functions vanishing outside a set of measure 0. So we can actually say, An algebra (over R or C) is the set of good (R- or C-valued) functions on a space with structure. Reciprocally, we would like to be able to recover the space with structure from thegivenalgebra. In algebraicgeometrythatisachievedbyconsideringhomomor- phisms from the algebra to a field or integral domain. Examples. 1. Take the algebra C[x] of complex polynomials in one complex variable. All homomorphismsfromC[x]toCaregivenbyevaluationatacomplexnumber. We recover C as the space of homomorphisms. 2. Take the quotient algebra of C[x] by the ideal generated by xk+1 C[x] (cid:104)xk+1(cid:105) = {a +a x+...+a xk |a ∈C} . (cid:46) 0 1 k i The coefficients a ,...,a may be thought of as values of a complex-valued 0 k function plus its first, second, ..., kth derivatives at the origin. The corre- sponding “space” is the so-called kth infinitesimal neighborhood of the point0. Eachofthese“spaces”hasjustonepoint: evaluationat0. Thelimit as k gets large is the space of power series in x. 3. The algebra C[x ,...,x ] of polynomials in n variables can be interpreted as 1 n thealgebraPol(V)of“good”(i.e.polynomial)functionsonann-dimensional complex vector space V for which (x ,...,x ) is a dual basis. If we denote 1 n the tensor algebra of the dual vector space V∗ by T(V∗)=C⊕V∗⊕(V∗⊗V∗)⊕...⊕(V∗)⊗k⊕... , xiii xiv INTRODUCTION where (V∗)⊗k is spanned by {x ⊗...⊗x | 1 ≤ i ,...,i ≤ n}, then we realize the symmetric algebra S(iV1 ∗)=Pol(iVk) as 1 k S(V∗)=T(V∗)/C , where C is the ideal generated by {α⊗β−β⊗α|α,β ∈V∗}. ThereareseveralwaystorecoverV anditsstructurefromthealgebraPol(V): • LinearhomomorphismsfromPol(V)toCcorrespondtopointsofV. We thus recover the set V. • Algebra endomorphisms of Pol(V) correspond to polynomial endomor- phisms of V: An algebra endomorphism f :Pol(V)−→Pol(V) is determined by the f(x ),...,f(x )). Since Pol(V) is freely generated 1 n by the x ’s, we can choose any f(x )∈Pol(V). For example, if n=2, f i i could be defined by: x (cid:55)−→ x 1 1 x (cid:55)−→ x +x2 2 2 1 which would even be invertible. We are thus recovering a polynomial structure in V. • Graded algebra automorphisms of Pol(V) correspond to linear isomor- phisms of V: As a graded algebra ∞ Pol(V)= Polk(V) , (cid:77) k=0 where Polk(V) is the set of homogeneous polynomials of degree k, i.e. symmetric tensors in (V∗)⊗k. A graded automorphism takes each x to i anelementofdegreeone,thatis,alinearhomogeneousexpressioninthe x ’s. Hence, by using the graded algebra structure of Pol(V), we obtain i a linear structure in V. 4. For a noncommutative structure, let V be a vector space (over R or C) and define Λ•(V∗)=T(V∗)/A , where A is the ideal generated by {α⊗β+β⊗α|α,β ∈V∗}. We can view this as a graded algebra, ∞ Λ•(V∗)= Λk(V∗) , (cid:77) k=0 whose automorphisms give us the linear structure on V. Therefore, as a graded algebra, Λ•(V∗) still “represents” the vector space structure in V. ThealgebraΛ•(V∗)isnotcommutative,butisinsteadsuper-commutative, i.e. for elements a∈Λk(V∗),b∈Λ(cid:96)(V∗), we have ab=(−1)k(cid:96)ba . INTRODUCTION xv Super-commutativity is associated to a Z -grading:1 2 Λ•(V∗) = Λ[0](V∗)⊕Λ[1](V∗) , where Λ[0](V∗) =Λeven(V∗) := Λk(V∗) , and (cid:76) k even Λ[1](V∗) =Λodd(V∗) := Λk(V∗) . (cid:76) k odd Therefore, V is not just a vector space, but is called an odd superspace; “odd”becauseallnonzerovectorsinV haveodd(=1)degree. TheZ -grading 2 allows for more automorphisms, as opposed to the Z-grading. For instance, x (cid:55)−→ x 1 1 x (cid:55)−→ x +x x x 2 2 1 2 3 x (cid:55)−→ x 3 3 is legal; this preserves the relations since both objects and images anti- commute. Although there is more flexibility, we are still not completely free to map generators, since we need to preserve the Z -grading. Homomor- 2 phisms of the Z -graded algebra Λ•(V∗) correspond to “functions” on the 2 (odd) superspace V. We may view the construction above as a definition: a superspace is an object on which the functions form a supercommutative Z -gradedalgebra. Repeateduseshouldconvinceoneofthevalueofthistype 2 of terminology! 5. The algebra Ω•(M) of differential forms on a manifold M can be regarded as a Z -graded algebra by 2 Ω•(M)=Ωeven(M)⊕Ωodd(M) . We may thus think of forms on M as functions on a superspace. Locally, the tangentbundleTM hascoordinates{x }and{dx },whereeachx commutes i i i with everything and the dx anticommute with each other. (The coordinates i {dx }measurethecomponentsoftangentvectors.) Inthisway,Ω•(M)isthe i ◦ ◦ algebra of functions on the odd tangent bundle TM; the indicates that here we regard the fibers of TM as odd superspaces. The exterior derivative d:Ω•(M)−→Ω•(M) has the property that for f,g ∈Ω•(M), d(fg)=(df)g+(−1)degff(dg) . Hence,disaderivationofasuperalgebra. Itexchangesthesubspacesofeven ◦ and odd degree. We call d an odd vector field on TM. 6. Consider the algebra of complex valued functions on a “phase space” R2, with coordinates (q,p) interpreted as position and momentum for a one- dimensional physical system. We wish to impose the standard equation from quantum mechanics qp−pq =i(cid:126) , 1Theterm“super”isgenerallyusedinconnectionwithZ2-gradings. xvi INTRODUCTION which encodes the uncertainty principle. In order to formalize this condition, wetakethealgebrafreelygeneratedbyq andpmodulotheidealgeneratedby qp−pq−i(cid:126). As(cid:126)approaches0,werecoverthecommutativealgebraPol(R2). Studyingexampleslikethisnaturallyleadsustowardtheuniversalenvelop- ing algebra of a Lie algebra (here the Lie algebra is the Heisenberg algebra, where (cid:126) is considered as a variable like q and p), and towards symplectic geometry (here we concentrate on the phase space with coordinates q and p). ♦ Each of these latter aspects will lead us into the study of Poisson algebras, andtheinterplaybetweenPoissongeometryandnoncommutativealgebras, inpar- ticular, connections with representation theory and operator algebras. Inthesenoteswewillbealsolookingatgroupoids,Liegroupoidsandgroupoid algebras. Briefly,agroupoidissimilartoagroup,butwecanonlymultiplycertain pairs of elements. One can think of a groupoid as a category (possibly with more than one object) where all morphisms are invertible, whereas a group is a category with only one object such that all morphisms have inverses. Lie algebroids are the infinitesimal counterparts of Lie groupoids, and are very close to Poisson and symplectic geometry. Finally, wewilldiscussFedosov’sworkindeformationquantizationofarbitrary symplectic manifolds. All of these topics give nice geometric models for noncommutative algebras! Of course, we could go on, but we had to stop somewhere. In particular, these notes contain almost no discussion of Poisson Lie groups or symplectic groupoids, both of which are special cases of Poisson groupoids. Ample material on Poisson groups can be found in [25], while symplectic groupoids are discussed in [162] as well as the original sources [34, 89, 181]. The theory of Poisson groupoids [168] is evolvingrapidlythankstonewexamplesfoundinconjunctionwithsolutionsofthe classical dynamical Yang-Baxter equation [136]. The time should not be long before a sequel to these notes is due.