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.     

Universal central extensions of Lie conformal algebras,Part 2: Supercase

In the previous part of this study we considered universal central extensions of Lie conformal algebras and conditions for existence of such extensions. The aim of this paper is some refinement of previous results and extension of these results to Lie conformal superalgebras.     

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.     

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.     

Deformations and generalized derivations of Hom-Lie conformal algebras

The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.     

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.     

Twisted Hamiltonian Lie algebras and their multiplicity-free representations

We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated with Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu’s generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.     

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.     

Some General Results on Conformal Embeddings of Affine Vertex Operator Algebras

We give a general criterion for conformal embeddings of vertex operator algebras associated to affine Lie algebras at arbitrary levels. Using that criterion, we construct new conformal embeddings at admissible rational and negative integer levels. In particular, we construct all remaining conformal embeddings associated to automorphisms of Dynkin diagrams of simple Lie algebras. The semisimplicity of the corresponding decompositions is obtained by using the concept of fusion rules for vertex operator algebras.     

Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic

We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a 1–1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted Lie algebra extensions.     

Two Kinds of Solvable Complete Lie Algebras

In this paper,we will give the definition of completable nilpotent Lie algebras,discuss its decomposition and prove that the heisenberg algebras and extensions of abelian quadratic Lie algebras are all completable nilpotent Lie algebras.     

Locally Vacant Double Lie Groupoids and the Integration of Matched Pairs of Lie Algebroids

We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method used is an extension of a method introduced by Lu and Weinstein in the case of Poisson Lie groups, and yields double groupoids which satisfy an étale form of the vacancy condition.     

Varieties of dialgebras and conformal algebras

We introduce and study the concept of a variety of dialgebras which is closely related to the concept of a variety of conformal algebras: Each dialgebra of a given variety embeds into an appropriate conformal algebra of the same variety. In particular, the Leibniz algebras are exactly Lie dialgebras, and each Leibniz algebra embeds into a conformal Lie algebra.     

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.     

Gröbner–Shirshov Bases for Universal Enveloping Conformal Algebras of Simple Conformal Lie Superalgebras of Type WN

For simple conformal Lie superalgebras of type W_N, Gröbner–Shirshov bases of their universal enveloping associative conformal algebras are found. The universal enveloping algebras considered correspond to a minimal locality function for which there is an injective embedding.     

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H²_Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.     

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.     

Conformal representations of Leibniz algebras

We study the embedding construction of Lie dialgebras (Leibniz algebras) into conformal algebras. This construction leads to the concept of a conformal representation of Leibniz algebras. We prove that each (finite-dimensional) Leibniz algebra possesses a faithful linear representation (of finite type). As a corollary we give a new proof of the Poincaré-Birkhoff-Witt theorem for Leibniz algebras.