Lie Conformal Algebras Research Articles

Universal enveloping Poisson conformal algebras

Lie conformal algebras are useful tools for studying vertex operator algebras and their representations. In this paper, we establish close relations between Poisson conformal algebras and representations of Lie conformal algebras. We also calculate explicitly Poisson conformal brackets on the associated graded conformal algebras of universal associative conformal envelopes of the Virasoro conformal algebra and the Neveu–Schwartz conformal superalgebra.

Lie conformal algebras related to Galilean conformal algebras

Let [Formula: see text] be a Lie conformal algebra related to Galilean conformal algebras, where [Formula: see text] are complex numbers. All the conformal derivations of [Formula: see text] are shown to be inner. The rank one conformal modules and [Formula: see text]-graded free intermediate series modules over [Formula: see text] are completely classified. The corresponding results of the finite conformal subalgebra of [Formula: see text] are also obtained as byproducts.

Structure of a class of Lie conformal algebras of Block type

Let p be a nonzero complex number. Recently, a class of infinite rank Lie conformal algebras was introduced. In this article, we study structure theory of this class of Lie conformal algebras. Specifically, we completely determine conformal derivations, conformal biderivations, and certain second cohomologies of Communicated by K. C. Misra

Conformal classical Yang–Baxter equation, S-equation and $${\mathcal {O}}$$-operators

Conformal classical Yang-Baxter equation and $S$-equation naturally appear in the study of Lie conformal bialgebras and left-symmetric conformal bialgebras. In this paper, they are interpreted in terms of a kind of operators, namely, $\mathcal O$-operators in the conformal sense. Explicitly, the skew-symmetric part of a conformal linear map $T$ where $T_0=T_\lambda\mid_{\lambda=0}$ is an $\mathcal O$-operator in the conformal sense is a skew-symmetric solution of conformal classical Yang-Baxter equation, whereas the symmetric part is a symmetric solution of conformal $S$-equation. One byproduct is that a finite left-symmetric conformal algebra which is a free $\mathbb{C}[\partial]$-module gives a natural $\mathcal O$-operator and hence there is a construction of solutions of conformal classical Yang-Baxter equation and conformal $S$-equation from the former. Another byproduct is that the non-degenerate solutions of these two equations correspond to 2-cocycles of Lie conformal algebras and left-symmetric conformal algebras respectively. We also give a further study on a special class of $\mathcal{O}$-operators called Rota-Baxter operators on Lie conformal algebras and some explicit examples are presented.

Extensions of conformal modules over Lie conformal algebras of Block type

We classify extensions between finite irreducible conformal modules over a class of infinite Lie conformal algebras B(p) of Block type, where p is a nonzero complex number. We find that although certain finite irreducible conformal modules over B(p) are simply conformal modules over its Virasoro conformal subalgebra Vir, there exist more nontrivial extensions between these conformal B(p)-modules. For extensions between other conformal modules, the situation becomes rather different. As an application, we also solve the extension problem for a series of finite Lie conformal algebras b(n) for n≥1.

Quadratic Leibniz conformal algebras

In this paper, we study a class of Leibniz conformal algebras called quadratic Leibniz conformal algebras. An equivalent characterization of a Leibniz conformal algebra [Formula: see text] through three algebraic operations on [Formula: see text] is given. By this characterization, several constructions of quadratic Leibniz conformal algebras are presented. Moreover, one-dimensional central extensions of quadratic Leibniz conformal algebras are considered using some bilinear forms on [Formula: see text]. In particular, we also study one-dimensional Leibniz central extensions of quadratic Lie conformal algebras.

Infinite-dimensional meta-conformal Lie algebras in one and two spatial dimensions

Meta-conformal transformations are constructed as sets of time-space transformations which are not angle-preserving but contain time- and space translations, time-space dilatations with dynamical exponent and whose Lie algebras contain conformal Lie algebras as sub-algebras. They act as dynamical symmetries of the linear transport equation in d spatial dimensions. For d = 1 spatial dimensions, meta-conformal transformations constitute new representations of the conformal Lie algebras, while for their algebraic structure is different. Infinite-dimensional Lie algebras of meta-conformal transformations are explicitly constructed for d = 1 and d = 2 and they are shown to be isomorphic to the direct sum of either two or three centre-less Virasoro algebras, respectively. The form of co-variant two-point correlators is derived. An application to the directed Glauber–Ising chain with spatially long-ranged initial conditions is described.

Generalized conformal derivations of Lie conformal algebras

Let [Formula: see text] be a finite Lie conformal algebra. The purpose of this paper is to investigate the conformal derivation algebra [Formula: see text], the conformal quasiderivation algebra [Formula: see text] and the generalized conformal derivation algebra [Formula: see text]. The generalized conformal derivation algebra is a natural generalization of the conformal derivation algebra. Obviously, we have the following tower [Formula: see text], where [Formula: see text] is the general Lie conformal algebra. Furthermore, we mainly research the connection of these generalized conformal derivations. Finally, the conformal [Formula: see text]-derivations of Lie conformal algebras are studied. Moreover, we obtain some connections between several specific generalized conformal derivations and the conformal [Formula: see text]-derivations. In addition, all conformal [Formula: see text]-derivations of finite simple Lie conformal algebras are characterized.

Extensions of modules over a class of Lie conformal algebras 𝒲(b)

Let [Formula: see text] be a class of free Lie conformal algebras of rank two with [Formula: see text]-basis [Formula: see text] and relations [Formula: see text] where [Formula: see text] is a nonzero complex number. In this paper, we classify extensions between two finite irreducible conformal modules over the Lie conformal algebras [Formula: see text].

Finite irreducible modules of Lie conformal algebras 𝒲(a,b) and some Schrödinger–Virasoro type Lie conformal algebras

Lie conformal algebras [Formula: see text] are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we first give a complete classification of all finite nontrivial irreducible conformal modules of [Formula: see text]. It is shown that all such modules are of rank one. Moreover, with a similar method, all finite nontrivial irreducible conformal modules of Schrödinger–Virasoro type Lie conformal algebras [Formula: see text] and [Formula: see text] are characterized.

Modules for map Virasoro conformal algebras

Any nontrivial finite irreducible module for the map Lie conformal algebra $$\hbox {Vir}\bigotimes {\mathcal {A}}$$ , where $$\hbox {Vir}$$ is the Virasoro conformal algebra and $${\mathcal {A}}$$ is an arbitrary finitely generated commutative associative algebra with unity over $${\mathbb {C}}$$ , is proved to be an evaluation module.

Local and Non-local Multiplicative Poisson Vertex Algebras and Differential-Difference Equations

We develop the notions of multiplicative Lie conformal and Poisson vertex algebras, local and non-local, and their connections to the theory of integrable differential-difference Hamiltonian equations. We establish relations of these notions to $q$-deformed $W$-algebras and lattice Poisson algebras. We introduce the notion of Adler type pseudodifference operators and apply them to integrability of differential-difference Hamiltonian equations.

Extending structures of lie conformal superalgebras

In this article, we generalize the -split extending structures problem of Lie conformal algebras to the super case. First, we introduce a unified product of a given Lie conformal superalgebra and a given -graded -module Q. This product includes some other products such as the twisted product, crossed product, and bicrossed product. Second, using these products, we describe and classify all Lie conformal superalgebra structures on when and Q satisfy some conditions.

Conformal Lie algebras via deformation theory

We discuss possible notions of conformal Lie algebras, paying particular attention to graded conformal Lie algebras with d-dimensional space isotropy: namely, those with a co(d) subalgebra acting in a prescribed way on the additional generators. We classify those Lie algebras up to isomorphism for all d ⩾ 2 following the same methodology used recently to classify kinematical Lie algebras, as deformations of the “most abelian” graded conformal algebra. We find 17 isomorphism classes of Lie algebras for d ≠ 3 and 23 classes for d = 3. Lie algebra contractions define a partial order in the set of isomorphism classes, and this is illustrated via the corresponding Hasse diagram. The only metric graded conformal Lie algebras are the simple Lie algebras, isomorphic to either so(d+1,2) or so(d+2,1). We also work out the central extensions of the graded conformal algebras and study their invariant inner products. We find that central extensions of a Lie algebra in d = 3 and two Lie algebras in d = 2 are metric. We then discuss several other notions of conformal Lie algebras (generalised conformal, Schrödinger, and Lifshitz Lie algebras), and we present some partial results on their classification. We end with some open problems suggested by our results.

On the Hochschild Cohomologies of Associative Conformal Algebras with a Finite Faithful Representation

Associative conformal algebras of conformal endomorphisms are of essential importance for the study of finite representations of conformal Lie algebras (Lie vertex algebras). We describe all semisimple algebras of conformal endomorphisms which have the trivial second Hochschild cohomology group with coefficients in every conformal bimodule. As a consequence, we state a complete solution of the radical splitting problem in the class of associative conformal algebras with a finite faithful representation.

Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra

In this article, under some natural condition, a complete classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra is presented. Moreover, applying this result, we obtain a class of compatible left-symmetric algebraic structures on the coefficient algebra of .

The construction of Hom-left-symmetric conformal bialgebras

ABSTRACT In this paper, we first introduce the notion of Hom-left-symmetric conformal bialgebras and show some nontrivial examples. Also, we present construction methods of matched pairs of Hom-Lie conformal algebras and Hom-left-symmetric conformal algebras. Finally, we prove that a finite Hom-left-symmetric conformal bialgebra is free as a -module is equivalent to a Hom-parakähler Lie conformal algebra. In particular, we investigate the coboundary Hom-left-symmetric conformal bialgebras.

On conformal pseudo-subriemannian fundamental graded Lie algebras of semisimple type

We introduce the notion of a conformal pseudo-subriemannian fundamental graded Lie algebra. Moreover we give a classification of conformal pseudo-subriemannian fundamental graded Lie algebras of semisimple type and their prolongations. Also we apply these results to a conformal pseudo-subriemannian geometry.

Contractions of Degenerate Quadratic Algebras, Abstract and Geometric

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 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between B\^ocher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all B\^ocher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric B\^ocher contractions.

Classification of finite irreducible conformal modules over a class of Lie conformal algebras of Block type

We classify finite irreducible conformal modules over a class of infinite Lie conformal algebras B(p) of Block type, where p is a nonzero complex number. In particular, we obtain that a finite irreducible conformal module over B(p) may be a nontrivial extension of a finite conformal module over Vir if p=−1, where Vir is a Virasoro conformal subalgebra of B(p). As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal algebras b(n) for n≥1.