Lie Conformal Algebras Research Articles

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 $\alpha^k$-derivations of multiplicative Hom-Lie conformal algebras and study their properties.

Deformations and generalized derivations of Lie conformal superalgebras

In this paper, we study deformations and generalized derivations of Lie conformal superalgebras. First, we study conformal derivations of the semidirect product of a Lie conformal superalgebra and its conformal module. Second, we extend the cohomology theory of Lie conformal algebras to the super case and discuss some applications to deformations of Lie conformal superalgebras. Finally, we introduce generalized derivations of Lie conformal superalgebras and study their properties.

A new Composition-Diamond lemma for associative conformal algebras

Let [Formula: see text] be the free associative conformal algebra generated by a set [Formula: see text] with a bounded locality [Formula: see text]. Let [Formula: see text] be a subset of [Formula: see text]. A Composition-Diamond lemma for associative conformal algebras is first established by Bokut, Fong and Ke in 2004 [L. A. Bokut, Y. Fong and W.-F. Ke, Composition-Diamond Lemma for associative conformal algebras, J. Algebra 272 (2004) 739–774] which claims that if (i) [Formula: see text] is a Gröbner–Shirshov basis in [Formula: see text], then (ii) the set of [Formula: see text]-irreducible words is a linear basis of the quotient conformal algebra [Formula: see text], but not conversely. In this paper, by introducing some new definitions of normal [Formula: see text]-words, compositions and compositions to be trivial, we give a new Composition-Diamond lemma for associative conformal algebras, which makes the conditions (i) and (ii) equivalent. We show that for each ideal [Formula: see text] of [Formula: see text], [Formula: see text] has a unique reduced Gröbner–Shirshov basis. As applications, we show that Loop Virasoro Lie conformal algebra and Loop Heisenberg–Virasoro Lie conformal algebra are embeddable into their universal enveloping associative conformal algebras.

Classification of finite irreducible conformal modules over some Lie conformal algebras related to the Virasoro conformal algebra

In this paper, we classify all finite irreducible conformal modules over a class of Lie conformal algebras W(b) with b∈ℂ related to the Virasoro conformal algebra. Explicitly, any finite irreducible conformal module over W(b) is proved to be isomorphic to MΔ,α,β with Δ≠0 or β≠0 if b = 0, or MΔ,α with Δ≠0 if b≠0. As a byproduct, all finite irreducible conformal modules over the Heisenberg-Virasoro conformal algebra and the W(2, 2) Lie conformal algebra are classified. Finally, the same thing is done for the Schrödinger-Virasoro conformal algebra.

Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by B\^ocher contractions of the conformal Lie algebra ${\mathfrak{so}}(4,\mathbb {C})$ to itself. In this paper we give a precise definition of B\^ocher contractions and show how they can be classified. They subsume well known contractions of ${\mathfrak{e}}(2,\mathbb {C})$ and ${\mathfrak{so}}(3,\mathbb {C})$ and have important physical and geometric meanings, such as the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. We also classify abstract nondegenerate quadratic algebras in terms of an invariant that we call a canonical form. We describe an algorithm for finding the canonical form of such algebras. We calculate explicitly all canonical forms arising from quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems.

Toward a classification of semidegenerate 3D superintegrable systems

Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intriguing objects. Next to the nondegenerate 4-parameter potential systems they admit the maximum number of symmetry operators, but their symmetry algebras do not close under commutation and not enough is known about their structure to give a complete classification. Some examples are known for which the 3-parameter system can be extended to a 4th order superintegrable system with a 4-parameter potential and 6 linearly independent symmetry generators. In this paper we use Bôcher contractions of the conformal Lie algebra to itself to generate a large family of 3-parameter systems with 4th order extensions, on a variety of manifolds, all from Bôcher contractions of a single ‘generic’ system on the 3-sphere. We give a contraction scheme relating these systems. The results have myriad applications for finding explicit solutions for both quantum and classical systems.

Finite irreducible representations of map Lie conformal algebras

In this paper, we study the finite representation theory of the map Lie conformal algebra [Formula: see text], where G is a finite simple Lie conformal algebra and A is a commutative associative algebra with unity over [Formula: see text]. In particular, we give a complete classification of nontrivial finite irreducible conformal modules of [Formula: see text] provided A is finite-dimensional.

Simplicity of quadratic Lie conformal algebras

ABSTRACTIn this paper, simplicity of quadratic Lie conformal algebras is investigated. From the view point of the corresponding Gel’fand–Dorfman bialgebras, some sufficient 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

ABSTRACTIn 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∈ℂ. Moreover, central extensions, conformal derivations and free conformal modules of rank 1 of TSV(a,b) and TSV(c) are determined.

Extending Structures for Lie Conformal Algebras

The \(\mathbb {C}[\partial ]\)-split extending structures problem for Lie conformal algebras is studied. In this paper, we introduce the definition of unified product of a given Lie conformal algebra R and a given \(\mathbb {C}[\partial ]\)-module Q. This product includes some other interesting products of Lie conformal algebras such as twisted product, crossed product, and bicrossed product. Using this product, a cohomological type object is constructed to provide a theoretical answer to the \(\mathbb {C}[\partial ]\)-split extending structures problem. Moreover, using this general theory, we investigate crossed product and bicrossed product in detail, which give the answers for the \(\mathbb {C}[\partial ]\)-split extension problem and the \(\mathbb {C}[\partial ]\)-split factorization problem respectively.

Infinite rank Schrödinger-Virasoro type Lie conformal algebras

Motivated by the structure of certain modules over the loop Virasoro Lie conformal algebra and the Lie structures of Schrödinger-Virasoro algebras, we construct a class of infinite rank Lie conformal algebras CSV(a, b), where a, b are complex numbers. The conformal derivations of CSV(a, b) are uniformly determined. The rank one conformal modules and ℤ-graded free intermediate series modules over CSV(a, b) are classified. Corresponding results of the conformal subalgebra CHV(a, b) of CSV(a, b) are also presented.

The Ado theorem for finite Lie conformal algebras with Levi decomposition

We prove that a finite torsion-free conformal Lie algebra with a splitting solvable radical has a finite faithful conformal representation.

Loop Schrödinger–Virasoro Lie conformal algebra

In this paper, we introduce two kinds of Lie conformal algebras, associated with the loop Schrödinger–Virasoro Lie algebra and the extended loop Schrödinger–Virasoro Lie algebra, respectively. The conformal derivations, the second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and [Formula: see text]-graded free intermediate series modules over these two conformal algebras are also classified in the present paper.

Structures of W(2.2) Lie conformal algebra

Abstract The purpose of this paper is to study W(2, 2) Lie conformal algebra, which has a free ℂ[∂]-basis {L, M} such that [ L λ L ] = ( ∂ + 2 λ ) L , [ L λ M ] = ( ∂ + 2 λ ) M , [ M λ M ] = 0 $\begin{equation}[{L_\lambda }L] = (\partial + 2\lambda )L,[{L_\lambda }M] = (\partial + 2\lambda )M,[{M_\lambda }M] = 0]\end{equation}$ . In this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.

On certain generalizations of the Schrödinger-Virasoro algebra

We study certain generalizations of the Schrödinger-Virasoro algebra, introduced by Roger and Unterberger, in the context of vertex algebras. As the main result, we associate these Lie algebras with vertex algebras and their twisted modules with respect to semi-simple (not necessarily finite order) automorphisms, and we give a connection among these Lie algebras in terms of twisted current algebras of conformal Lie algebras.

The Lie Conformal Algebra of a Block Type Lie Algebra

Let L be a Lie algebra of Block type over ℂ with basis {Lα,i | α,i ∈ ℤ} and brackets [Lα,i,Lβ,j]=(β(i+1)-α(j+1)) Lα+β,i+j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with ℂ[∂]-basis {Lα(w) | α ∈ ℤ} and λ-brackets [Lα(w)λ Lβ(w)]= (α∂+(α+β)λ) Lα+β(w). Finally, we give a classification of free intermediate series B-modules.

Homogeneous averaging operators on simple finite conformal Lie algebras

We describe all homogeneous averaging operators on finite simple conformal Lie algebras. In the case of current algebras, these operators are closely related to rational solutions of the classical Yang–Baxter equation.

A class of Schrödinger–Virasoro type Lie conformal algebras

In this paper, a class of Lie conformal algebras associated to a Schrödinger–Virasoro type Lie algebra is constructed, which is nonsimple and can be regarded as an extension of the Virasoro conformal algebra. Then conformal derivations, second cohomology group with trivial coefficients and conformal modules of rank 1 of this Lie conformal algebra are investigated.

On conformal subriemannian fundamental graded Lie algebras and Cartan connections

Under some condition we classify the prolongations of conformal subriemannian fundamental graded Lie algebras. Also we show the existence of a normal Cartan connection associated with a bracket generating regular conformal subriemannian manifold.

Left-symmetric conformal algebras and vertex algebras

A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. By an equivalent characterization of vertex algebra using Lie conformal algebra and left-symmetric algebra given by Bakalov and Kac in [3], in studying vertex algebra, we have to deal with such a question: Do there exist compatible left-symmetric algebra structures on a class of special Lie algebras named formal distribution Lie algebras? In this paper, we study this question. We introduce the definitions of left-symmetric conformal algebra and Novikov conformal algebra. Many examples of these algebras are obtained. As an application, we present a construction of vertex algebra using left-symmetric conformal algebra.