Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of ‘transverse symmetries,’ by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a ‘quantum group’ counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the étale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.     

Morphisms between complete Riemannian pseudogroups

We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a closure and its maps are C^∞ along the orbit closures. Here, completeness and closure are versions for morphisms of concepts introduced by Haefliger for pseudogroups. This result is applied to approximate foliated maps by smooth ones in the case of transversely complete Riemannian foliations, yielding the foliated homotopy invariance of their spectral sequence. This generalizes the topological invariance of their basic cohomology, shown by El Kacimi-Alaoui-Nicolau. A different proof of the spectral sequence invariance was also given by the second author.     

On fixed points of conformal pseudogroups

We show in this paper Theorem 2 that if (H, H ₁) is a pseudogroup generated by a finite numberH ₁ of germs of conformal diffeomorphisms of defined on a sufficiently small discD, which is not linearizable and such that the linear group (L,H ₁)={g(0)/g(H,H ₁)}* is dense in *, then the set of fixed points of the pseudogroup (H, H ₁) is dense inD. This implies the abundance of distinct homotopy classes of loops in leaves of foliations defined in ² by generic polynomial vector fields as well as for germs of holomorphic vector fields in ² beginning with generic jets, both of degree at least 2. These homotopy classes may be realized arbitrarily close to the line at infinity or to 0, respectively. This shows the genericity of polynomial vector fields with infinite Petrovsky-Landis genus ([5]).The idea of the proof is very simple. Ifg is a non-linear conformal diffeomorphism with multiplier =g'(0), then the map obtained by the composition ofg and the linear map with multiplier ^–1 will have at 0 a fixed point of multiplicity at least 2. Since we may approximate ^–1 by elementsh in the pseudogroup and the multiplicity of fixed points satisfy a law of conservation of number, we obtain thath o g has fixed points close to 0. These fixed points appear as a by product of the relative nonlinearity of the generators of the pseudogroup, since linearizable pseudogroups have 0 as an isolated fixed point. The fixed points obtained are not conjugate since they have distinct multipliers.The main technical tool is the angular derivative introduced in [8]. It allows one to split the search for fixed points into two parts: One is to obtain a contraction and the other is to return arbitrarily close to the starting point without modifying the property of contraction. This is carried out since the angular derivative is multiplicative for compositions and is identically 1 for linear maps.Supported by CONACYT-CNRS and CONACYT 3398-E9307.     

Plane poisson manifolds and finite-dimensional pseudogroups

Translated from Matematicheskie Zametki, Vol. 45, No. 3, pp. 53–65, March, 1989.     

Lie structure of smash products

We investigate the conditions under which the smash product of an (ordinary or restricted) enveloping algebra and a group algebra is Lie solvable or Lie nilpotent.     

Nilpotent pseudogroups of functions on an interval

A near-identity nilpotent pseudogroup of order m 1 is a family f ₁, . . . , f _n : (-1, 1) of C ² functions for which: for some small positive real number < 1/10 ^m+1 and commutators of the functions f _i of order at least m equal the identity. We present a classification of near-identity nilpotent pseudogroups: our results are similar to those of Plante, Thurston, Farb and Franks. As an application, we classify certain foliations of nilpotent manifolds.     

Tannaka-Krein duality for compact matrix pseudogroups. TwistedSU(N) groups

Summary The notion of concrete monoidalW ^*-category is introduced and investigated. A generalization of the Tannaka-Krein duality theorem is proved. It leads to new examples of compact matrix pseudogroups. Among them we have twistedSU(N) groups denoted byS U(N). It is shown that the representation theory forS U(N) is similar to that ofSU(N): irreducible representations are labeled by Young diagrams and formulae for dimensions and multiplicity are the same as in the classical case.On leave from Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoza 7400-682, Warsaw, Poland.     

Lie algebras admitting a unique quadratic structure

We prove that any Lie algebra g over a field K of characteristic zero admitting a unique up to a constant quadratic structure is necessarily a simple Lie algebra. If the field K is algebraically closed, such condition is also sufficient.

Further, a real Lie algebra g admits a unique quadratic structure if and only if its complexification g^C is a simple Lie algebra over C     

Lie algebraic structure of akns matrix system

For an AKNS matrix system, Lie algebraic structure and its mastersymmetry are obtained by a purely algebraic approach; and by using the reduced technique, two similar algebraic structures for MKdV and KdV matrix systems are given.This project is supported by the National Education Foundation of China.     

Local structure of Koszul-Vinberg and of Lie algebroids

In this short note we continue our study of Koszul-Vinberg algebroids which form a subcategory of the category of Lie algebroids, and which appear naturally in the study of affine structures, affine and transversally affine foliations [N. Nguiffo Boyom, R. Wolak, J. Geom. Phys. 42 (2002) 307-317]. We prove a local decomposition theorem for KV-algebroids. Using the notion of KV-algebroids we introduce a new class of singular foliations: affine singular foliations. In the last section we study the holonomy of these foliations and prove a stability theorem.     

Quantum group structure of the q-deformed Lie superalgebra

In this paper a general procedure is given to get the quantum derformation of the Lie superalgebra spl(2,1) and its corresponding algebraic structure of a quantum group i.e. non-commutative, non-co-commutative Hopf superalgebra. This procedure would be suitable for another Lie superalgebra.     

Algebre de Lie attachee a la structure presque-tangente

Résumé On se propose d'étudier l'algèbre de Lie des champs de vecteurs qui laissent invariante la structure presque tangente sur le fibré tangent. On montre que cette algèbre de Lie est égale à son algèbre dérivée et on détermine toutes ses dérivations. On en déduit que cette algèbre de Lie caractérise la structure différentiable de la variété de base du fibré tangent.     

On the generalized Lie structure of associative algebras

We study the structure of Lie algebras in the category ^H MA ofH-comodules for a cotriangular bialgebra (H, 〈|〉) and in particular theH-Lie structure of an algebraA in ^H MA. We show that ifA is a sum of twoH-commutative subrings, then theH-commutator ideal ofA is nilpotent; thus ifA is also semiprime,A isH-commutative. We show an analogous result for arbitraryH-Lie algebras whenH is cocommutative. We next discuss theH-Lie ideal structure ofA. We show that ifA isH-simple andH is cocommutative, then any non-commutativeH-Lie idealU ofA must contain [A, A]. IfU is commutative andH is a group algebra, we show thatU is in the graded center ifA is a graded domain. Dedicated to the memory of S. A. Amitsur Supported by a Fulbright grant. Supported by NSF grant DMS-9203375.