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Noncommutative Geometry Andrew Lesniewski lesniewski.qxp 3/25/98 9:11 AM Page 800 Noncommutative Geometry Andrew Lesniewski Noncommutative Spaces is a locally compact space, and A C0(M), where ' 0(M) denotes the C∗ -algebra of continuous It was noticed a long time ago that various prop- C erties of sets of points can be restated in terms functions on M vanishing at infinity. of properties of certain commutative rings of Recall that a C∗-algebra is an algebra over C, functions over those sets. In particular, this ob- equipped with an involutive operation and a servation proved to be extremely fruitful in al- ∗ norm , which satisfies the condition gebraic geometry and has led to tremendous k·k SS = S 2. In other words, the category of progress in this subject over the past few k ∗k k k decades. In these developments the concept of locally compact spaces is equivalent to the cat- a point in a space is secondary and overshadowed egory of abelian C∗ -algebras. The points of a by the algebraic properties of the (sheaves of) topological space can be characterized in purely rings of functions on those spaces. algebraic terms as the maximal ideals of an al- This idea also underlies noncommutative gebra of functions on the space. geometry, a new direction in mathematics initi- Another important source of inspiration for ated by the French mathematician Alain Connes noncommutative geometry is quantum physics. and outlined in his recent book [3]. In noncom- It has been known since the heroic days of quan- mutative geometry one goes one step further: it tum mechanics (Heisenberg, Born, Jordan, is no longer required that the algebra of func- Schrödinger, Dirac, von Neumann, … ) that or- tions be commutative! Furthermore, while alge- dinary concepts of classical mechanics and sym- braic geometry did not entirely rid itself of the plectic geometry do not apply to the subatomic concept of a point, noncommutative geometry world. In order to understand the physical phe- does not use this concept at all. In fact, a point nomena taking place at the atomic scale, one in a noncommutative space is often a contra- needs to replace the concepts of classical geom- diction in terms. etry by other, noncommutative structures. The One of the sources of noncommutative geom- notion of a function on phase space needs to be etry is the following classic theorem due to replaced by an operator acting on a Hilbert space Gelfand and Naimark. of states or a quantum observable . In Dirac’s H Theorem 1. Let A be a commutative C∗-alge- parlance, c-numbers get replaced by q-numbers. bra, and let M denote the set of maximal ideals This procedure is called quantization. of A. Then, equipped with a natural topology, M The simplest example is that of a flat space R2 which is the phase space of a particle mov- ing in one dimension. After quantization, the co- Andrew Lesniewski was associate professor of physics ordinates q and p of a point in R2 are replaced at Harvard University. Currently he is vice president of by operators q and p which obey the Heisenberg- Acadian Asset Management in Boston, MA. His e-mail Born-Jordan commutation relation address is [email protected]. Research supported by NSF Grant DMS-94-24344. (1) [q,p]=i~I, 800 NOTICES OF THE AMS VOLUME 44, NUMBER 7 lesniewski.qxp 3/25/98 9:11 AM Page 801 where ~ is a fundamental constant of nature, an antihomomorphism S, which satisfies the Planck’s constant. Explicitly, one takes the Hilbert three conditions stated above. For the case of space of states to be = L2(R) , and SU(2), the corresponding quantum group (or H q(x)=x(x), p(x)= i~0(x). This quanti- q-deformation) is constructed as follows. We − zation procedure results in a structure which can let a, b denote the complex valued functions on be thought of as a noncommutative deformation SU(2) assigning to a group element g the cor- of a classical phase space. Heisenberg’s uncer- responding matrix entry in tainty principle implies that there is no natural concept of a point on this quantum deformed ab SU(2) g = . phase space: all we have is a nonabelian algebra 3 b a − of “functions on the noncommutative plane”. One can also quantize classical phase spaces The algebra generated by these functions is with more complicated geometry; the simplest abelian. The (C∗ -algebra of functions on the) quantum group SU (2) is defined as a defor- of them is the torus T2. Here the algebra of ob- q servables is generated by two unitary operators mation of the algebra of functions on SU(2). u and v, such that We will denote the generators of the deformed algebra by the same symbols, a, b, a, b, and im- (2) uv = e2πiλvu, pose the relations where λ =2π . One can think of u and v as ~ ab = qba, ab = qab, quantizations of the classical functions e2πip and e2πiq, respectively. The resulting noncom- mutative algebra is called the (algebra of func- bb = bb, ba = qab, tions on the) quantum torus or the irrational ro- 1 tation algebra. This algebra also appears ba = qab, aa aa = q− q bb, − − naturally in other contexts (periodic structures in magnetic fields, matrix models of string the- where q is a real parameter such that q < 1. ory). Quantization of more complicated geome- | | The abelian case, that of the algebra of con- tries leads to noncommutative structures which tinuous functions on SU(2), corresponds to cannot be described as easily. q =1. For the construction of the relevant Hopf Related noncommutative structures arise as algebra structure we refer the reader to the lit- q-deformations of Lie groups (see, e.g., [5, 8]). erature. These structures are often called quantum Voiculescu’s free probability theory [7] is an- groups.1 Here the noncommutative algebras carry other example of a noncommutative structure the additional structure of a Hopf algebra. This motivated by physics applications. Here the con- extra structure on the algebra of functions en- cept of probability space is replaced by a non- codes the fact that the underlying noncommu- commutative structure leading to noncommut- tative space is a group-like object. To illustrate ing random variables. One of the main results this, let us consider the abelian case first. Let F of this theory, Voiculescu’s central limit theorem, be an algebra (whose topological structure we ig- yields the Wigner semicircle law, which arises in nore) of complex functions on a group G with the theory of random matrices. Related fields of the identity element e. For f F we set ∈ quantum ergodic theory and quantum informa- f (g ,g )=f (g g ) , εf(g)=f (e) , and Sf(g)= 1 2 1 2 tion theory have recently been the focus of a great f (g 1). This defines algebra homomorphisms − deal of attention. They play a pivotal role in the ∆ : F F F (“coproduct”) and ε : F C → ⊗ → emerging field of quantum computation. (“counit”) and an algebra antihomomorphism Interesting examples of noncommutative ∆S : F F (“antipode”). The usual properties defin- → spaces abound, and they are thoroughly dis- ing a group can equivalently be formulated in cussed in Connes’ book. In fact, it turns out that the language of these homomorphisms without noncommutative geometry also provides a con- ever referring to the notion of an element of G. venient framework for studying “commutative” Explicitly, we verify easily that ( Id) = ⊗ ◦ but highly singular structures. These include (Id ) , (ε Id) = (Id ε) =Id, and ⊗ ◦ ⊗ ◦ ⊗ ◦ fractal sets and products of smooth manifolds m (S Id) = Iε. In the last of these∆ identi-∆ ◦ ⊗ ◦ by finite sets. ties, m∆ denotes∆ the multiplication∆ on∆ F, and I is the identity∆ element of F. A Hopf algebra is K-Theory and Fredholm Modules an algebra (not necessarily commutative) which An important part of Connes’ program is the no- is equipped with homomorphisms and ε and tion of a vector bundle over a noncommutative 1 space. The well-known Swan’s theorem states The term “quantum groups” is somewhat∆ misleading, as these structures are not quantizations of classical that the algebraic K0 group of the algebra C(M) groups in the physical sense described above. (defined in terms of stable isomorphism classes AUGUST 1997 NOTICES OF THE AMS 801 lesniewski.qxp 3/25/98 9:11 AM Page 802 of projective C(M)-modules), where M is com- graded concept, that of a Lie supergroup. The lat- pact, coincides with the Grothendieck group ter is a noncommutative space whose coordinate K(M) (defined in terms of locally trivial, finite- algebra contains anticommuting elements. A dimensional complex vector bundles over M) particular combination of the generators of su- of the underlying manifold. In the framework of persymmetry acting on the Hilbert space of noncommutative geometry one can thus regard states of the system is an infinite-dimensional, the elements of the group K0(A), where A is Dirac-type operator. now a nonabelian algebra of functions on a non- Other examples of Fredholm modules include commutative space, as (equivalence classes of) models associated with physical phenomena of vector bundles over the noncommutative space. quantum Hall effect and quantum chaos. The K-theory of operator algebras was orig- Indeed, much of classical differential geom- inated by Brown, Douglas, and Fillmore [1] in the etry can be encoded into the concept of a Fred- mid-seventies—and, incidentally, they gave the holm module. The length of a curve on a mani- first look into the Connes program.
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