Finite vertex algebras and nilpotence

I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (= without nilpotent elements) finite vertex algebra is nilpotent.     

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.     

Construction process for simple Lie algebras

We give here a construction process for the complex simple Lie algebras and the non-Hermitian type real forms which intersect the minimal nilpotent complex adjoint orbit, using a finite dimensional irreducible representation of the conformal group, or of some two-fold covering of it, with highest weight vector a semi-invariant of degree four. This process leads to a five-graded simple complex Lie algebra and the underlying semi-invariant is intimately related to the structure of the minimal nilpotent orbit. We also describe a similar construction process for the simple real Lie algebras of Hermitian type.     

Modules for map Virasoro conformal algebras

Archiv der Mathematik - Any nontrivial finite irreducible module for the map Lie conformal algebra $$\hbox {Vir}\bigotimes {\mathcal {A}}$$ , where $$\hbox {Vir}$$ is the Virasoro conformal algebra...     

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

Biderivations of finite-dimensional complex simple Lie algebras

In this paper, we prove that a biderivation of a finite-dimensional complex simple Lie algebra without the restriction of being skewsymmetric is an inner biderivation. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also obtain the forms of the linear commuting maps on the finite-dimensional complex simple Lie algebra or general linear Lie algebra.     

Locally finite dimensional lie algebras and their derivation algebras

The derivation algebras of all locally finite dimensional locally simple Lie algebras over a field of characteristic 0 are determined. Every locally finite dimensional Lie algebra of countable dimension is a subalgebra of the outer derivation algebra outder (ℒ) for every Lie algebra ℒ, which is the direct limit of diagonally embedded classical Lie algebras. These outer derivation algebras have dimension ℒ and are never locally finite dimensional. Dedicated to Prof. H. Petersson on the occasion of his 60th birthday     

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     

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.     

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.     

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.     

Simple left-symmetric algebras with solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie groupG correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we studysimple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied.     

Hom-Lie algebra structures on semi-simple Lie algebras

Hom-Lie algebras can be considered as a deformation of Lie algebras. In this note, we prove that the hom-Lie algebra structures on finite-dimensional simple Lie algebras are trivial. We find when a finite-dimensional semi-simple Lie algebra admits non-trivial hom-Lie algebra structures and the isomorphic classes of non-trivial hom-Lie algebras are determined.     

Derivation algebras of JB-algebras

Let X be a JB-algebra without spin factor representation of dimension 3. It is shown that the Banach Lie algebra aut(X) of all derivations of X is semi-simple, and aut(X) is (topologically) simple if X is simple. This extends results of Jacobson and Meyberg to the infinite dimensional case. Applications to the Lie algebra of all infinitesimal automorphisms of a symmetric Banach tube domain are given.Research supported by the Deutsche Forschungsgemeinschaft     

Cartan matrices for restricted Lie algebras

Consider a finite dimensional restricted Lie algebra over a field of prime characteristic. Each linear form on this Lie algebra defines a finite dimensional quotient of its universal enveloping algebra, called a reduced enveloping algebra. This leads to a Cartan matrix recording the multiplicities as composition factors of the simple modules in the projective indecomposable modules for such a reduced enveloping algebra. In this paper we show how to compare such Cartan matrices belonging to distinct linear forms. As an application we rederive and generalise the reciprocity formula first discovered by Humphreys for Lie algebras of reductive groups. For simple Lie algebras of Cartan type we see, for example, that the Cartan matrices for linear forms of non-positive height are submatrices of the Cartan matrix for the zero linear form.     

Cohomology of Lie Conformal Algebra Vir × Cur g

In this article,we compute cohomology groups of the semisimple Lie conformal algebra S =Vir × Cur g with coefficients in its irreducible modules for a finite-dimensional simple Lie algebra g.     

Tubular algebras and affine Kac-Moody algebras

The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.     

An Artinian theory for Lie algebras

Complemented Lie algebras are introduced in this paper (a notion similar to that studied by O. Loos and E. Neher in Jordan pairs). We prove that a Lie algebra is complemented if and only if it is a direct sum of simple nondegenerate Artinian Lie algebras. Moreover, we classify simple nondegenerate Artinian Lie algebras over a field of characteristic 0 or greater than 7, and describe the Lie inner ideal structure of simple Lie algebras arising from simple associative algebras with nonzero socle.     

Symmetry algebras of linear differential equations

The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.Tomsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 3–12, July, 1992.     

Simple left-symmetric algebras with¶solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie group {G} correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we study simple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied. Received: 10 June 1997 / Revised version: 29 September 1997