秩为 2 的 Virasoro-型李共形代数

作者对秩为2的无挠的李共形代数进行了刻画.在这些代数中,作者主要关注Virasoro-型李共形代数.并且,作者描述了一种特殊Virasoro-型李共形代数的共形导子、秩为1的自由共形模和中心扩张.     

Steinberg unitary Lie conformal algebras

In this paper, we study Steinberg unitary Lie conformal algebras, which are universal central extensions of unitary Lie conformal algebras. We describe the kernels of these extensions by means of skew-dihedral homology. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 135–155, 2005.     

Structure of a Class of Rank Two Lie Conformal Algebras

Let β be a complex number,and a class of non-semisimple and non-solvable rank two Lie conformal algebras-(L)(β) are introduced.In this paper,conformal derivations,conformal quasiderivations,generalized conformal derivations and conformal biderivations of (L)(β) are studied.Besides,central extensions and conformal modules of rank one of (L)(β) are determined.     

Simplicity of quadratic Lie conformal algebras

In this paper, simplicity of quadratic Lie conformal algebras is investigated. From the view point of the corresponding Gel’fand–Dorfman bialgebras, some su?cient conditions and necessary conditions to ensure simplicity of quadratic Lie conformal algebras are presented. By these observations, we present several new classes of infinite simple Lie conformal algebras. These results will be useful for classification purposes.     

On Schrödinger-Virasoro type Lie conformal algebras

In this paper, two classes of Schrödinger-Virasoro type Lie conformal algebras TSV(a,b) and TSV(c) which are non-simple are introduced for some a, b, c∈?. Moreover, central extensions, conformal derivations and free conformal modules of rank 1 of TSV(a,b) and TSV(c) are determined.     

Universal enveloping conformal algebras

The main objective of this paper is to study embeddings of Lie conformal algebras into associative conformal algebras. We prove that not all Lie conformal algebras admit such embeddings. However, in many important cases, including semisimple Lie conformal algebras of finite type, embeddings of this form exist and sometimes we can even describe universal enveloping associative conformal algebras of Lie conformal algebras and prove an analogue of the classical Poincaré-Birkhoff-Witt theorem.     

一类Block型李代数的分类和中心扩张

本文我们决定了一类Block型李代数$\BB$在同构意义下的分类和中心扩张, 其中$q$是一个非零复数. 我们的结果推广了以前的一些结果.     

Conformal biderivations of loop W (a,b) Lie conformal algebra

We study conformal biderivations of a Lie conformal algebra. First, we give the definition of a conformal biderivation. Next, we determine the conformal biderivations of loop W(a, b) Lie conformal algebra, loop Virasoro Lie conformal algebra, and Virasoro Lie conformal algebra. Especially, all conformal biderivations on Virasoro Lie conformal algebra are inner conformal biderivations.     

Towards a Lie theory of locally convex groups

In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.     

On stem covers and the universal central extensions of Lie crossed modules

In this article, we investigate stem covers and the universal central extensions of Lie crossed modules. In particular, a cover for the direct sum of two Lie crossed modules in term of given covers of them will be constructed. Using the results obtained, we give some properties of covers and the universal relative central extensions of pairs of Lie algebras.     

Lie symmetries and invariants for a 2D nonlinear heat equation

The paper generalizes previous results on the 2D Ricci flow equation in a conformal gauge. It investigates a general form of the 2D nonlinear heat equation and it points out all possible cases where Lie type symmetries, associated invariant quantities and similarity solutions can appear. The connection with the already known results for the 1D problem is obtained.     

Gröbner–Shirshov Bases: From their Incipiency to the Present

In this paper, the history and the main results of the theory of Gröbner–Shirshov bases are given for commutative, noncommutative, Lie, and conformal algebras from the beginning (1962) to the present time. The problem of constructing a base of a free Lie algebra is considered, as well as the problem of studying the structure of free products of Lie algebras, the word problem for Lie algebras, and the problem of embedding an arbitrary Lie algebra into an algebraically closed one. The modern form of the composition-diamond lemma (the CD lemma) is presented. The rewriting systems for groups are considered from the point of view of Gröbner–Shirshov bases. The important role of conformal algebras is treated, the statement of the CD lemma for associative conformal algebras is given, and some examples are considered. An analog of the Hilbert basis theorem for commutative conformalalgebras is stated. Bibliography: 173 titles.     

经典N=2李共形超代数的导子和第二上同调群

研究了经典N=2李共形超代数的导子和第二上同调群的结构,并应用第二上同调群的结果确定了该李共形超代数的泛中心扩张.     

Extensions of Schrödinger–Virasoro conformal modules

In this paper, we study extensions between two finite irreducible conformal modules over the Schrödinger–Virasoro conformal algebra and the extended Schrödinger–Virasoro conformal algebra. Also, we classify all finite nontrivial irreducible conformal modules over the extended Schrödinger–Virasoro conformal algebra. As a byproduct, we obtain a classification of extensions of Heisenberg–Virasoro conformal modules.     

A remark on simplicity of vertex algebras and Lie conformal algebras

I give a short proof of the following algebraic statement: in a simple vertex algebra, the underlying Lie conformal algebra is either abelian, or it is an irreducible central extension of a simple Lie conformal algebra. This provides many examples of non-finite simple Lie conformal algebras, and should prove useful for classification purposes.     

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting

We extend classical results of Kostant et al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.     

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.     

Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on M are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.