An explicit *-product on the cotangent bundle of a Lie group

We give explicit formulas for a ^*-product on the cotangent bundle T ^* G of a Lie group G; these formulas involve on the one hand the multiplicative structure of the universal enveloping algebra U(G) of the Lie algebra G of G and on the other hand bidifferential operators analogous to the ones used by Moyal to define a ^*-product on IR²ⁿ.Chargé de recherches au FNRS, on leave of absence from Université libre de Bruxelles.     

Some Graded Bialgebras Related to Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras. We denote this algebra by wU_q^t(G)wU_{q}^{\tau}(\mathcal{G}). It is a noncommutative and noncocommutative weak graded Hopf algebra under some additional condition. It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra U_q(G)U_{q}(\mathcal{G}) of G\mathcal{G}.     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

Modular Categories and Orbifold Models

In this paper, we try to answer the following question: given a modular tensor category ? with an action of a compact group G, is it possible to describe in a suitable sense the “quotient” category ?/G? We give a full answer in the case when ?=?ℯ? is the category of vector spaces; in this case, ?ℯ?/G turns out to be the category of representation of Drinfeld's double D(G). This should be considered as the category theory analog of the topological identity {pt}/G=BG. This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if ? is a vertex operator algebra which has a unique irreducible module, ? itself, and G is a compact group of automorphisms of ?, and some not too restrictive technical conditions are satisfied, then G is finite, and the category of representations of the algebra of invariants, ? ^G , is equivalent as a tensor category to the category of representations of Drinfeld's double D(G). We also get some partial results in the non-holomorphic case, i.e. when ? has more than one simple module. Received: 27 August 2001 / Accepted: 1 March 2002     

The number of independent traces and supertraces on symplectic reflection algebras

It is shown that A:= H₁, _η (G), the sympectic reflection algebra over ?, has T_G independent traces, where T_G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ? Sp(2N) ? End(?^2N) generated by the system of symplectic reflections.

Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S_G independent supertraces, where S_G is the number of conjugacy classes of elements without eigenvalue -1 belonging to G.

We consider also A as a Lie algebra A^L and as a Lie superalgebra A^S.

It is shown that if A is a simple associative algebra, then the supercommutant [A^S, A^S] is a simple Lie superalgebra having at least S_G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A^L, A^L]/([A^L, A^L] ∩ ?) is a simple Lie algebra having at least T_G independent symmetric invariant non-degenerate bilinear forms.     

On Lie groups with left-invariant symplectic or Kählerian structures

Left-invariant symplectic structure on a group G; properties of the corresponding Lie algebra g. A unimodular symplectic Lie algebra has to be solvable (see [1]). Symplectic subgroups and left-invariant Poisson structures on a group. Affine Poisson structures: an affine Poisson structure associated to g and admitting g ^* as a unique leaf corresponds to a unimodular symplectic Lie algebra and the associate group is right-affine. If G is unimodular and endowed with a left-invariant metric g, harmonic theory for the left-invariant forms. Kählerian group is metabelian and Riemannianly flat. Decomposition of a simply connected Kählerian group. A symplectic group admitting a left-invariant metric with a nonnegative Ricci curvature is unimodular and admits a left-invariant flat Kählerian structure.     

Regular * representations of lie algebras

We prove the existence of a * product on the cotangent bundle of a parallelizable manifold M. When M is a Lie group the properties of this * product allow us to define a linear representation of the Lie algebra of this group on L ²(G), which is, in fact, the one corresponding to the usual regular representation of G.Chargé de recherches au FNRS.     

Topological gauge theories and group cohomology

We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH ⁴(BG,Z). In a similar way, possible Wess-Zumino interactions of such a groupG are classified byH ³(G,Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models involves a certain natural map fromH ⁴(BG,Z) toH ³(G,Z). We generalize this correspondence to topological spin theories, which are defined on three manifolds with spin structure, and are related to what might be calledZ ₂ graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.     

Extension of K.M.S. states with respect to some locally compact groups

Given a state ω on the C^*-system { , R×G, α} which is K.M.S. with respect to time on the sub-algebra of of fixed points under the action of some locally compact group G, we study its K.M.S.-extension to the whole algebra. In this work, we study more precisely the cases where G is a class of locally compact abelian groups (such that { , G, α|G} is almost periodic) or a product of these groups with a compact group.     

On the nilpotent * - Fourier transform

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transform was introduced by D. Arnal and J. C. Cortet,:L(G) C (V,L(^2d )), whereL(G) is the Schwartz space ofG andV × ^2k is aG-invariant Zariski open set ing ^* the dual of the Lie algebra ofG. We prove the surjectivity of this transformation, which allows us to extend it to distribution spaces.     

A duality theorem for the automorphism group of a covariant system

In this paper we examine the covariant representation theory of a covariant system (A, G) introduced by Doplicher, Kastler and Robinson. (A is aC*-algebra andG is a locally compact group of automorphisms ofA.) We define the concept of left tensor product of two covariant representations. Loosely stated, our duality theorem says thatG is canonically isomorphic to the set of bounded operator valued maps on the set of covariant representations of the covariant system (A, G) which preserve direct sums, unitary equivalence and left tensor products. We further show that the enveloping von Neumann algebraA(A, G) of the covariant system (A, G) admits a (not necessarily injective) comultiplicationd such that (A(A, G),d) is a Hopf von Neumann algebra. The intrinsic group of this Hopf von Neumann algebra is canonically isomorphic and (strong operator topology) homeomorphic toG.     

On G-Trace and G-Index in Deformation Quantization

For a compact group G of symplectomorphisms we define a G-trace on the algebra of quantum observables by postulating its properties. We give an explicit construction of such a trace and prove a G-index theorem similar to the Atiyah–Segal–Singer equivariant index theorem for elliptic operators.     

Operator Algebras and Poisson Manifolds¶Associated to Groupoids

It is well known that a measured groupoid G defines a von Neumann algebra W ^*(G), and that a Lie groupoid G canonically defines both a C ^*-algebra C ^*(G) and a Poisson manifold A ^*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C ^*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W ^*(G), G↦C ^*(G), and G↦A ^*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence. Received: 6 December 2000 / Accepted: 19 April 2001     

Fractional Loop Group and Twisted K-Theory

We study the structure of abelian extensions of the group L _q G of q-differentiable loops (in the Sobolev sense), generalizing from the case of the central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of the supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on G is discussed.     

Unitary positive energy representations of the gauge group

We investigate the positive energy representations (also called highest weight representations) of the gauge groupC (T ^v,G ₀),G ₀ being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions in it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations.Allocataire du MRT.     

Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Given a finite dimensional C ^*-Hopf algebra H and its dual Ĥ we construct the infinite crossed product and study its superselection sectors in the framework of algebraic quantum field theory. is the observable algebra of a generalized quantum spin chain with H-order and Ĥ-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If is a group algebra then becomes an ordinary G-spin model. We classify all DHR-sectors of – relative to some Haag dual vacuum representation – and prove that their symmetry is described by the Drinfeld double . To achieve this we construct localized coactions and use a certain compressibility property to prove that they are universal amplimorphisms on . In this way the double can be recovered from the observable algebra as a universal cosymmetry. Received: 4 September 1995\,/\,Accepted: 3 December 1996     

Non-compact quantum groups associated with Abelian subgroups

LetG be a Lie group. For any Abelian subalgebra of the Lie algebra g ofG, and any , the difference of the left and right translates ofr gives a compatible Poisson bracket onG. We show how to construct the corresponding quantum group, in theC ^*-algebra setting. The main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onC ^*-algebras.The research reported on here was supported in part by National Science Foundation grant DMS-9303386.