Representations of four-derivation Lie algebras of Block type

Block introduced certain analogues of the Zassenhaus algebras over a field of characteristic 0. The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras. In this paper, we construct a family of irreducible modules in terms of multiplication and differentiation operators on ＂polynomials＂ for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras. We also find a new series of submodules from which some irreducible quotient modules are obtained.     

On Tractability and Ideal Problem in Non-Associative Operator Algebras

The question of the existence of non-trivial ideals of Lie algebras of compact operators is considered from different points of view. One of the approaches is based on the concept of a tractable Lie algebra, which can be of interest independently of the main theme of the paper. Among other results it is shown that an infinite-dimensional closed Lie or Jordan algebra of compact operators cannot be simple. Several partial answers to Wojtyński’s problem on the topological simplicity of Lie algebras of compact quasinilpotent operators are also given.     

Polynomial Lie Algebras and Growth of Their Finitely Generated Lie Subalgebras

The concept of polynomial Lie algebra of finite rank was introduced by V. M. Buchstaber in his studies of new relationships between hyperelliptic functions and the theory of integrable systems. In this paper we prove the following theorem: the Lie subalgebra generated by the frame of a polynomial Lie algebra of finite rank has at most polynomial growth. In addition, important examples of polynomial Lie algebras of countable rank are considered in the paper. Such Lie algebras arise in the study of certain hyperbolic partial differential equations, as well as in the construction of self-similar infinite-dimensional Lie algebras (such as the Fibonacci algebra).     

Commutative post-Lie algebra structures on Kac–Moody algebras

We determine commutative post-Lie algebra structures on some infinite-dimensional Lie algebras. We show that all commutative post-Lie algebra structures on loop algebras are trivial. This extends the results for finite-dimensional perfect Lie algebras. Furthermore, we show that all commutative post-Lie algebra structures on affine Kac–Moody Lie algebras are “almost trivial”.     

On Ringel–Hall Algebras of Tame Hereditary Algebras

In this paper, the double Ringel–Hall algebras of tame hereditary algebras are decomposed as the quantized enveloping algebras of the infinite-dimensional Lie algebras, which are the central extensions of the affine loop algebras and the infinite-dimensional Heisenberg algebras. The numbers of the generators of the Heisenberg algebras are explicitly given at each dimensional level.     

Graded Lie algebras whose Cartan subalgebra is the algebra of polynomials in one variable

We define a class of infinite-dimensional Lie algebras that generalize the universal enveloping algebra of the algebra sl(2, ℂ) regarded as a Lie algebra. These algebras are a special case of ℤ-graded Lie algebras with a continuous root system, namely, their Cartan subalgebra is the algebra of polynomials in one variable. The continuous limit of these algebras defines new Poisson brackets on algebraic surfaces. In memory of M. V. Saveliev Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 345–352, May, 2000.     

Kostant homology formulas for oscillator modules of Lie superalgebras

We provide a systematic approach to obtain formulas for characters and Kostant u-homology groups of the oscillator modules of the finite-dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite-dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned u-homology computation are worked out.     

The polynomial growth of colength of varieties of Lie algebras

It is proved that the colength of every API-variety of Lie algebras grows polynomially, and we give a number of examples in which the colength grows more rapidly than any polynomial function does. These indicate that for many of the important varieties of Lie algebras, such as varieties of solvable algebras of derived length 3, varieties generated by some infinite-dimensional simple algebras of Cartan type, or by certain Katz-Mudi algebras, the growth of colength will be superpolynomial. Supported by RFFR grants No. 96-01-00146 and No. 96-15-96050. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 161–175, March–April, 1999.     

On complete reducibility for infinite-dimensional Lie algebras

In this paper we study the complete reducibility of representations of infinite-dimensional Lie algebras from the perspective of representation theory of vertex algebras.     

李代数W(2,2)上的Hom-李代数结构

李代数W（2，2）是一类重要的无限维李代数，它是在研究权为2的向量生成的顶点算子代数的过程当中提出来的.Hom-李代数是指同时具备代数结构和李代数结构的一类代数，并且乘法与李代数乘法运算满足Leibniz法则.本文确定了李代数W（2，2）上的Hom-李代数结构.主要结论是李代数W（2，2）上没有非平凡的Hom-李代数结构.本文的研究结果对于W（2，2）代数的进一步研究有一定的帮助作用.     

COMPLETE LIE ALGEBRAS WITH l-STEP NILPOTENT RADICALS

The authors first give a necessary and sufficient condition for some solvable Lie algebras with l-step nilpotent radicals to be complete, and then construct a new class of infinite dimensional complete Lie algebras by using the modules of simple Lie algebras. The quotient algebras of this new constructed Lie algebras are non-solvable complete Lie algebras with l-step nilpotent radicals.     

Witt型与Virasoro型Lie双代数的对偶

本文给出单边的Witt 型、Witt 型和Virasoro 型Lie 双代数的对偶Lie 双代数结构. 由此, 本文得到一系列无限维Lie 代数.     

A Generalization of Extended Affine Lie Algebras

We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and extended affine Lie algebras. Our results generalize well-known properties of these examples.     

Universal Verma modules andW-resolvents over Kač-Moody algebras

We consider new aspects of extremal equations over symmetrizable Kač-Moody algebras. We develop new methods (reproducing classical finite-dimensional results) that can be applied to infinite-dimensional (affine) Lie algebras. We describe special extensions of universal enveloping algebras, investigate the fine structure of W-resolvents, and use these methods to investigate “extremal projectors” over Kač-Moody algebras. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 334–356, March, 2000.     

Nonisothermal filtration and seismic acoustics in porous soil: Thermoviscoelastic equations and Lamé equations

We study applications of a new class of infinite-dimensional Lie algebras, called Lax operator algebras, which goes back to the works by I. Krichever and S. Novikov on finite-zone integration related to holomorphic vector bundles and on Lie algebras on Riemann surfaces. Lax operator algebras are almost graded Lie algebras of current type. They were introduced by I. Krichever and the author as a development of the theory of Lax operators on Riemann surfaces due to I. Krichever, and further investigated in a joint paper by M. Schlichenmaier and the author. In this article we construct integrable hierarchies of Lax equations of that type. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 216–226. Dedicated to S.P. Novikov on the occasion of his 70th birthday     

Banach–Lie groupoids and generalized inversion

We study a few basic properties of Banach–Lie groupoids and algebroids, adapting some classical results on finite dimensional Lie groupoids. As an illustration of the general theory, we show that the notion of locally transitive Banach–Lie groupoid sheds fresh light on earlier research on some infinite-dimensional manifolds associated with Banach algebras.     

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].

     

Categorified central extensions, étale Lie 2-groups and Lie?s Third Theorem for locally exponential Lie algebras

Lie?s Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration procedure for Lie algebra cocycles.This paper remedies the obstructions for integrating cocycles and central extensions from Lie algebras to Lie groups by generalising the integrating objects. Those objects obey the maximal coherence that one can expect. Moreover, we show that they are the universal ones for the integration problem.The main application of this result is that a Mackey-complete locally exponential Lie algebra (e.g., a Banach–Lie algebra) integrates to a Lie 2-group in the sense that there is a natural Lie functor from certain Lie 2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphic Lie algebra.