Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras

We consider a wide class of unital involutive topological algebras provided with aC ^*-norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebra sare taking a prominent position in noncommutative differential geometry, where they are often called unital smooth algebras. In this paper we prove that the group of invertible elements of such a unital solution smooth algebra and the subgroup of its unitary elements are regular analytic Fréchet-Lie groups of Campbell-Baker-Hausdorff type and fulfill a nice infinite-dimensional version of Lie's second fundamental theorem.     

Projective limits of Banach-Lie groups

In this paper we characterize commutative Fréchet-Lie groups using the exponential map. In particular we prove that if a commutative Fréchet-Lie groupG has an exponential map, which is a local diffeomorphism, thenG is the limit of a projective system of Banach-Lie groups.     

Universal Central Extensions of Lie Groups

We call a central Z-extension of a group G weakly universal for an Abelian group A if the correspondence assigning to a homomorphism ZA the corresponding A-extension yields a bijection of extension classes. The main problem discussed in this paper is the existence of central Lie group extensions of a connected Lie group G which is weakly universal for all Abelian Lie groups whose identity components are quotients of vector spaces by discrete subgroups. We call these Abelian groups regular. In the first part of the paper we deal with the corresponding question in the context of topological, Fréchet, and Banach–Lie algebras, and in the second part we turn to the groups. Here we start with a discussion of the weak universality for discrete Abelian groups and then turn to regular Lie groups A. The main results are a Recognition and a Characterization Theorem for weakly universal central extensions.     

Derivations of differential forms on jet bundles

Summary The theory of derivations of differential forms, originally formulated by Frölicher and Nijenhuis [2], is generalized to include derivations which connect differential forms on two different manifolds. This generalized theory is then used to analyze derivations of direct limits of exterior algebras of forms on jet bundles. Such derivations have found applications in the analysis of Lie equations [3], [4] and in the inverse problem of the calculus of variations [6], [7], [8], [9].

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Résumé On présente une généralization de la théorie des dérivations des formes extérieures, formulée par Frölicher et Nijenhuis [2], au cas des dérivations qui relient les formes extérieurs sur deux variétés différentes. On utilize ensuite cette théorie généralizée pour étudier les dérivations des limites directes des algèbres des formes extérieures sur les fibrés des jets. Ces dérivations ont été appliquées dans l'analyse des équations de Lie [3], [4] et dans le problème inverse du calcul des variations [6], [7], [8], [9].

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This work is a part of a program conducted jointly with ProfessorBenenti at Istituto di Fisica Matematica «J. L. Lagrange» of Torino. The authors are greatly indebted to the Director of the Institute Professor D.Galletto for his interest and encouragement.This work has been supported by Gruppo Nazionale per la Fisica Matematica del Consiglio Nazionale delle Ricerche.     

Necklace rings and logarithmic functions

In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor Nr from the category of special λ-rings into the category of special λ-rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant's paper [Free Lie algebras and formal power series, J. Algebra 253(1) (2002) 167-188] to the case of graded Lie (super)algebras with a group action by applying the Euler-Poincaré principle.     

Narrow Positively Graded Lie Algebras

The present paper is devoted to the classification of infinite-dimensional naturally graded Lie algebras that are narrow in the sense of Zelmanov and Shalev [9]. Such Lie algebras are Lie algebras of slow linear growth. In the theory of nonlinear hyperbolic partial differential equations the notion of the characteristic Lie algebra of equation is introduced [3]. Two graded Lie algebras n₁ and n₂ from our list, that are positive parts of the affine Kac–Moody algebras A₁⁽¹⁾ and A₂⁽²⁾, respectively, are isomophic to the characteristic Lie algebras of the sinh-Gordon and Tzitzeika equations [6]. We also note that questions relating to narrow and slowly growing Lie algebras have been extensively studied in the case of a field of positive characteristic [2].

     

Drinfeld–Sokolov reduction for quantum groups and deformations of W-algebras

We define deformations of W-algebras associated to comple semisimple Lie algebras by means of quantum Drinfeld-Sokolov reduction procedure for affine quantum groups. We also introduce Wakimoto modules for arbitrary affine quantum groups and construct free field resolutions and screening operators for the deformed W-algebras. We compare our results with earlier definitions of q-W-algebras and of the deformed screening operators due to Awata, Kubo, Odake, Shiraishi [60],[6], [7], Feigin, E. Frenkel [22] and E. Frenkel, Reshetikhin [34]. The screening operator and the free field resolution for the deformed W-algebra associated to the simple Lie algebra sl₂ coincide with those for the deformed Virasoro algebra introduced in [60]. The author is supported by the Swiss National Science Foundation.     

Spectra for solvable Lie algebras of bundle endomorphisms

The aim of the paper is to investigate spectral properties of the Lie algebras corresponding to the symmetry groups of certain flags of vector bundles over a compact space. Under natural hypotheses, such Lie algebras are solvable, being in general infinite dimensional. The spectral theory of finite-dimensional solvable Lie algebras of operators is extended to this natural class of infinite-dimensional solvable Lie algebras. The discussion uses the language of continuous fields of -algebras. The flag manifolds in -algebraic framework are naturally involved here, they providing the basic method for obtaining flags of vector bundles. Received: 8 October 2001 / Revised version: 4 February 2002 / Published online: 6 August 2002 Research supported from the contract ICA1–CT–2000–70022 with the European Commission.     

The Jordan socle and finitary Lie algebras

In this paper we introduce the notion of Jordan socle for nondegenerate Lie algebras, which extends the definition of socle given in [A. Fernández López et al., 3-Graded Lie algebras with Jordan finiteness conditions, Comm. Algebra, in press] for 3-graded Lie algebras. Any nondegenerate Lie algebra with essential Jordan socle is an essential subdirect product of strongly prime ones having nonzero Jordan socle. These last algebras are described, up to exceptional cases, in terms of simple Lie algebras of finite rank operators and their algebras of derivations. When working with Lie algebras which are infinite dimensional over an algebraically closed field of characteristic 0, the exceptions disappear and the algebras of derivations are computed.     

Pseudo-differential operators and Markov semigroups on compact Lie groups

We extend the Ruzhansky-Turunen theory of pseudo-differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N. Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by Lévy processes. We study a family of Lévy-type linear operators on general Lie groups that are pseudo-differential operators when the group is compact and find conditions for them to give rise to symmetric Dirichlet forms.     

Norm closure and extension of the symbolic calculus for the cone algebra

We investigate the symbolic structure of an algebra of pseudodifferential operators on manifolds with conical singularities which has been introduced by B.-W. Schulze. Our main objective is the extension of the symbolic calculus of this algebra to its norm closure in an adapted scale of Sobolev spaces. This procedure yields Banach algebras and Fréchet algebras of singular integral operators with continuous principal symbols.     

A quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups

Summary In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of instruments on groups and the associated semigroups of probability operators. In this paper the case is considered of a finite-dimensional Hilbert space (n-level quantum system) and of instruments defined on a finite-dimensional Lie group. Then, the generator of a continuous semigroup of (quantum) probability operators is characterized. In this way a quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained.     

Representations of a Class of Associative Algebras Related to the Quantum Torus

1 Introduction The mitivation of this paper comes from the work of[5].We kmow that vertex algebratheory is one of the importans parts in the study of infinte dimensional Lie theory,while thelattice vertex algebras form one of the most important and fundamental classes of vertexalgebras.In they studied the representation theory for certain” half lattice”vertex algebra     

算子李代数的一些性质

算子群作为群的推广,算子群在群论里有许多应用.类似地,作为算子群和李代数的推广,算子李代数将会有许多应用.给出了算子李代数的一些性质,得到了算子李代数半单性的充分必要条件.同时得到算子李代数半单性与非退化killing型的关系.     

着色李超代数与左着色对称结构

本文研究了着色李超代数上的左着色对称结构问题.利用着色李超代数的两种仿射表示和1-上同调群,得出左着色对称结构存在的几个充分或必要条件,推广了文[2]的结论.     

无扭算子李代数g(G,M)的子代数

由算子构成的李代数在李代数理论中具有重要的应用,因而研究算子李代数及其子代数的代数结构就显得尤为重要.首先构造了无扭算子李代数g(G,M)的子代数L_1,L_2,g1,g2,然后给出了这些子代数的代数结构及一些重要应用.     

Lax operator algebras

In this paper, we develop the general approach, introduced in [l], to Lax operators on algebraic curves. We observe that the space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct orthogonal and symplectic analogs of Lax operators, prove that they form almost graded Lie algebras, and construct local central extensions of these Lie algebras.     

Multiplicative Lie algebras and Schur multiplier

In this paper, we attempt to study the structure of multiplicative Lie algebras, the theory of extensions, the second cohomology groups of multiplicative Lie algebras, and in turn the Schur multipliers. The Schur–Hopf formula is established for multiplicative Lie algebras. We also introduce the group of nontrivial relations satisfied by the Lie product in a multiplicative Lie algebra, and study it as a functor arising from the presentations of multiplicative Lie algebras. Some applications in K-theory are also discussed.     

A class of angelic sequential non-Fréchet-Urysohn topological groups

Fréchet-Urysohn (briefly F-U) property for topological spaces is known to be highly non-multiplicative; for instance, the square of a compact F-U space is not in general Fréchet-Urysohn [P. Simon, A compact Fréchet space whose square is not Fréchet, Comment. Math. Univ. Carolin. 21 (1980) 749-753. [27]]. Van Douwen proved that the product of a metrizable space by a Fréchet-Urysohn space may not be (even) sequential. If the second factor is a topological group this behaviour improves significantly: we have obtained (Theorem 1.6(c)) that the product of a first countable space by a F-U topological group is a F-U space. We draw some important consequences by interacting this fact with Pontryagin duality theory. The main results are the following:

(1)

If the dual group of a metrizable Abelian group is F-U, then it must be metrizable and locally compact.

(2)

Leaning on (1) we point out a big class of hemicompact sequential non-Fréchet-Urysohn groups, namely: the dual groups of metrizable separable locally quasi-convex non-locally precompact groups. The members of this class are furthermore complete, strictly angelic and locally quasi-convex.

(3)

Similar results are also obtained in the framework of locally convex spaces.

Another class of sequential non-Fréchet-Urysohn complete topological Abelian groups very different from ours is given in [E.G. Zelenyuk, I.V. Protasov, Topologies of Abelian groups, Math. USSR Izv. 37 (2) (1991) 445-460. [32]].     

On a Type of Fréchet Principal Bundles over Banach Bases

In this paper we study a certain class of Fréchet principal bundles. Those which have structural groups obtained as projective limits of Banach Lie groups. In particular, we prove that each bundle of the previous type can be thought of as a projective limit of Banach principal bundles and any connection of them is a generalized limit of Banach connections. Using the previous, we achieve to translate in the Fréchet case basic geometric properties known so far only for Banach bundles.