Noncommutative Distributions

This book is concerned with noncommutatiYe distribution theory, more the representation theory of certain infinite-dimensional bit the concepts involYed and motivate the interest in their study.

by a spherical function whose Fourier transform is of the energy Dirichlet form naturally associated with the where E the Maurer-Cartan cocycle, by the the Killing metric on the Lie algebra g of G. The connection with other representations and objects is best understood the case where X is one-dimensional +, or Ssay). In this case, [AHKl], the representation can be identified with the regular representation given by left or right translation by elements

the bibliography in [A, Be]). This theory comprehends structure study of certain types of our presentation we give a unitary representation ob- of loop groups (based on [Te]), starting

GSW, FrGZ, AGO, AHKPS, AJPS]. of such models with braid groups, operator algebras, theta FeFroKI,

of '2ll(X, or

then endow with the Schwartz topology coming that of according to the tensor product by On the other the topology of Frechet space that we have defined on is of @ Moreover, similarly as in the proof of l, it follows that the bracket law in is compatible

+ + A(x{ + on X, (1/t)[A(x + ty) - the partial derivative a(A)(x)y of A in x along the vector y; in the same manner the limit limH(1/t)[L'(x the partial -(au) of derivation, and a recursive argument,

X of :s qi Then :s ;\4(A), max sup faf3U(x)f

allg in G), one has g,gj. REMARK. In the same way, one can prove that the mapping 8 on G). > 0, let us denote by the set

of the The manifolds G) and have been studied quite exten- Chapter 1]). We summarize here the main j;(gthis completes the

Riemannian Manifold 2.1 INTRODUCTION The nuclear group 2ll(X, G) may be viewed as a noncommutative generalization of a

of the continuous irreducible unitary representations of 5!(X, G) of unitary equivalence of continuous irre- of the Lie group G, and let be an clement of the class w. To each clement x of X we can associate a continuous and of X being given, @ @ · · · @ is a continuous irreducible of 5!(X, G), and its class is a G-distribution support {xx 2.2.4 Order of a G-Distribution of the Frechet Chapter I, Section 7) restricted to its dense sub- of 5'l(X, G) because the on 5'l(X, G) induced by

which the above property remains true is called the order of of order k if k is the order of one (and then of all) of the irreducible continuous unitary representa- of G) which are in the class of w. Example 2. In the case of the IR-distributions that Xr is of order k if and only if the distribution Tis of order k in the sense of usual distributions. Example 3. Let k be some positive integer, and let xbe in X; we endow the finite-dimensional Lie group of k-jets Chapter 1, Section with the left-invariant Haar measure dj. Let us consider the left regular = 2, p, let be a continuous and irreducible unitary representation of the k-jets group J!,(X, G); we get a continuous and irreducible unitary representation 1r!, of ®(X, G) such that, for all which is of order the class of unitary equivalence of 1r! @ 1r! ® · · · ® bution on X of order sk and support {xi, x ••• It problem of finding G-distributions of finite order and with finite support

of usual distributions the measures are the distributions of on X, theIR-distribution exp(ij.L(g)) on X and of order zero in the sense of IR- This leads to the follmving generalization. The G-distributions on X of order zero arc called G-

The corresponding nontrivial 1-cocycle g in SO(n; l) by + + I-n but straightforward computation shows that the pair of the corollary of Proposition 10, and then u.r. EXP,;;1T is in the class of a multiplicative The case G SU(n; 1) is quite similar (see [26, Example b]).

E.]. Woods, Pub/. R.Uvf.S., Kyoto Univ., 2:I57 (1966). 2-3 (I972).

3.1 INTRODUCTION This chapter is devoted to the construction and the study of a class of of ®(X, G) when G is a

that the "vacuum vector" 1, X ~ 1(x) Neumann algebra of Ur contains the bicommutant of From the assumption made in Theorem 4, it follows from Theorem 2 that flr has the f-property; con- the corollary of Lemma 5, L(F) contains the operators of of As is the union of all its Cartan subalgebras, it follows that of [7] one knO\vs that the set { V*(g) dui(g, It follows that the functions

The value \I81T for corresponds to the value a of the disjointness of Gaussian measures \vith Laplacian- of Theorems 6 and 7 was The recent result [II, 20] on tri,·iality of the = \,/8TI indicates that of the energy representation could also be expected for To handle irreducibility or reducibility for ilall of a quantized (Euclidean) nonlinear Theorem 4, one easily prmTs F' be two different Riemannian tlags of X. Under of Theorem 3, the basic energy representation L'r and Ur Ur defines a one- of Riemannian tlags of X into the unitary of D(X; G), and, more precisely, into the set of nonlocated and order on X. 3.6 RINGS OF GENERALIZED ENERGY REPRESENTATIONS

4.1 INTRODUCTION Chapter 3 we showed how to obtain in certain cases irreducibility the basic energy representations (Theorem 4). Such results the problem of exhibiting candidates for a noncommutativc (mul- of the introduction to Chapter 1 ..\ s

PROPERTIES OF THE RIGHT AND LEFT VERSIONS UR, UL OF THE ENERGY REPRESENTATION of the energy representation discussed in the pre- with a connection with :Jx + x:J as \'ertical subspace and Then: path in C(l, G), there is a decomposition (the

of Borel sets in There is a conull set .N'' in C(/, a, are functions with T, T, and G respectively) gen- the a-algebra of Borel subsets of .N''. Chapter 3. We that this result is stronger than classical results (e.g., [5],[8]) on of Gaussian measures. The first lemma was proved originally our object, and for the sake of

Current Algebras 5.1 INTRODUCTION The previous chapters were concerned with unitary representations of some infinite-dimensional groups, i.e., the gauge groups. The techniques used were basically functional analytic and probabilistic. Actually, it is

Lemma 1. The Lie algebra generated by e;J;, i 0, jj] r,; ;jj, t,j = h,h' E = jj =

The classification of the finite indecomposable G.C.M., and of the as- the simple Lie algebras, is a classical work going Theorem 2. Let A be any indecomposable G.C.M. of affine type. that the ith and jth vertices are connected by an ayfold arrow pointing toward the jth (resp. the ith) vertex if The labels a; appearing in Table l arc the coordinates of the an_,) such that A · 8 Table 1

One easily checks that: Lemma 4. dim H = and 'I' In other words, 'I'

of belongs to ;£. 0 Any affine Kac-Moody algebra of type (see Table I) The multiplicity of any isotropic root of is equal to N. of the type A,, Dx, or E(see Table 2). The Dynkin diagrams of these Lie algebras possess remarkable symmetry under the following transformations: I-2); the action of p, order of p (k 2 or 3), and exp( -2i1r1)x} One can then form the Lie algebra = @ Cd of C[t, byE;, F;, i N, the Chevalley generators of and let H; [E;, F;]. Introduce

= 3), by the corresponding Cartan subalgebra. The rest of the untwisted case: choose EfE _ ., that the Set:£ where the complex C§-subalgebra generated by f, hl=o. :£ has (B Cc (B Cd as a Cartan subalgebra, and if 8 E f)' is defined by Then (8, 8) = 0. The root basis {a , aJ

This gradation is called the gradation of type s. of Section 5. 2. 2. The Lie algebra constructed there has a natural gradation given by the of the d = t(dldt) element. Clearly one has

The conjugation w being assumed consistent (i.e., w 'Vex E the first step in this analysis is to find all the possible sets of positive {a,, any basis of simple roots define the standard set of It then that of Then taking for any Borel subalgebra of the form:

that any set of positive roots is Weyl-conjugate to the corresponding Borel subalgebra: ® f, ® h If =

of the compact case, it follows that OU(C[t, r of the contravariant Hermitian form Moreover, < 0; one then usc the following Lemma 6 (Vergne, Rossi) In the case where the real rank is greater than number .\ ••• of of the m measure

of the definition and of some important properties of M): G.C.M. of is symmetric or k > l, then M Q, the orthogonal of the root lattice Q on the complex-linear span of the M is the canonical image (under the pairing ~ f)') of the fjv of the coroot lattice Qv on its "gradient The character formula then takes the form pis the Weyl element, h number, 5.3 SOME CONNECTED PROBLEMS AND APPLICATIONS 5.3.1 The Virasoro Algebra

+ + The of that is actually an irreducible highest weight of the characters).

Q-sublattices: R M be the radical of Sin L: R {a E L, S(a, Denote he associated The next lemma then follows Lemma 10. There exists a unique homomorphism x: the character x that @ 'f,

the special property of the X character of then led to + + j)]

the rootspace attached to a. (;'\Jote that = the end of Definition 5, for which we need the REMARKS. The following properties are easily proven. The root system is symmetric: - a, E {0} if a by h) V h E The Killing form can thus be carried onto as follows: for any E on and will REMARK. Assumption (iii) is not fulfilled by indefinite Kac-Moody

_, 1 - the a-string of roots through j3. PROOF. The proposition basically follows from the observation that if is nonisotropic, then C§_" and [C§"' C§_"] span a complex Lie thus a consequence of

of Lemma 15; fix any 8 E f and let a be any short root in '2lt; the smallest element in R 8 such that 2a One then has Lemma 19. (i) 2a number p/4 < m < p/2. Then + + (4 m - p 1, which proves the lemma. o To completely characterize it is now necessary to have more about Pick E such that 2a, + E and with respect to basis

Lemma 21. The bilinear form ( ·, ·) is symmetric, invariant and non- It follows that ( ·, ·) is a relevant Killing form for Note that I, . that the root system ffi of of the form {a +

/o = Fo Hj, e, = E, = F, = E2, = F2 (G2). = Ejll: Then = 0, I, 2 Proposition 15. is an elliptic quasisimplc Lie algebra, whose root

rent algebras, starting from automorphisms of twist-two algebras, ho\\'- of such automorphisms. that for v > 2, a given elliptic quasisimple root system is not the root of a unique quasisimple Lie algebra, but of many of them. This = I case, inherits a canonical projccti\T action of the > I cases, the Lie the studv of 5.4.3 Real Quasisimple Lie Algebras of any quasisimple Lie algebra are defined by tv)@ bar denoting complex

Abelian nuclear group, 74 Borel Spaces, 7 5 Adjoint representation, 6 Bose-Fermi correspondence, Affine (Kac-,\1oody) Lie 3, 9 algebras, 4, 108, 116 Bosonic string, 82 Araki-Woods theorem, 61 Braid groups, 9

Equivariant loop, 89, 98 Gauge groups, 1, 4, 38 Equivariant loops representation, Gaussian measure (associated Equivariant paths, 88 with a symmetric Equivariant representation, 9

representation, 10, Integral weight lattice, 126 107, 108, 125, 145, 174 Invariant, 80 Hilbertian triads, 21 Irreducibility, 79, 80, 88 Hilbert-Lie algebra, 26 Irreducible, 81, 157 Hilbert manifold, 3 3 Irreducibility Hilbert-Sobolev Lie algebras Isotropic root, 114, 116 G), 26, 36 Hilbert-Sobolev-Lie groups, Jacobi triple product identity, 38 135 H0egh-Krohn model, 81 Jacobson-Kac, 131

Lie algebras, infinite dimensional, Nilpotent, 54, 64 Lie peralgebras, 145 Locally lear pace, Noncommutative convex nuc s distribution, 1

on C(l, Killing form symmetric Fock space, Symmetrizable, 109 a-property, 78, 80, 82 U}, 73