In this paper, we study the derivations, the central extensions and the automorphism group of the extended Schrodinger-Virasoro Lie algebra, introduced by J. Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that the extended Schrodinger-Virasoro Lie algebra is an infinite-dimensional complete Lie algebra and the universal central extension of the extended Schrodinger-Virasoro Lie algebra in the category of Leibniz algebras is the same as that in the category of Lie algebras.
In this paper, conjugate-linear anti-involutions and unitary Harish-Chandra modules over the Schrödinger-Virasoro algebra are studied. It is proved that there are only two classes conjugate-linear anti-involutions over the Schrödinger-Virasoro algebra. The main result of this paper is that a unitary Harish-Chandra module over the Schrödinger-Virasoro algebra is simply a unitary Harish-Chandra module over the Virasoro algebra.
In a series of papers by Henkel, Roger and Unterberger, Schrödinger-Virasoro algebras and their deformations were introduced and investigated. In the present paper we determine the 2-cocycles of a class of deformative Schrödinger-Virasoro algebras.
In this paper, we investigate the skew-symmertric biderivations of the deformative Schrodinger-Virasoro Lie algebras which contain the twisted and original deformative Schrodinger-Virasoro Lie algebras. As an application, we give the explicit form of each linear commuting map on the deformative Schrodinger-Virasoro Lie algebras. In particular, we obtain that there exist non-inner biderivations and non-standard linear commuting maps for the certain deformative Schrodinger-Virasoro Lie algebras.
Both original and twisted Schrödinger-Virasoro algebras also their deformations were introduced and investigated in a series of papers by Henkel, Roger and Unterberger. In the present paper we aim to determine the 2-cocycles of original deformative Schrödinger-Virasoro algebras.
Both the basic cohomology groups and the reduced cohomology groups of the Schrödinger-Virasoro conformal algebra with trivial coefficients are completely determined.
In this paper we investigate Lie bialgebra structures on the Schrödinger-Virasoro algebra $\LL$. Surprisingly, we find out an interesting fact that not all Lie bialgebra structures on the Schrödinger-Virasoro algebra are triangular coboundary, which is different from the related known results of some Lie algebras related to the Virasoro algebra.
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.
In this paper, we characterize the biderivations of Schrödinger-Virasoro Lie algebra. We get a classes of non-inner and non-skewsymmetric biderivations. As application, we characterize the commutative post-Lie algebra structures on Schrödinger-Virasoro Lie algebra.
In this paper we investigate Lie bialgebra structures on the deformative Schrödinger-Virasoro algebras mainly using the techniques introduced recently by Liu, Pei and Zhu, which indicate that all cases considered in this paper except one behave different from their centerless ones.
In this paper it is proved that an irreducible weight module with finite-dimensional weight spaces over the Schrödinger-Virasoro algebras is a highest/lowest weight module or a uniformly bounded module. Furthermore, indecomposable modules of the intermediate series over these algebras are completely determined.
In this paper we investigate Lie bialgebra structures on a twisted Schrödinger-Virasoro type algebra $\LL$. All Lie bialgebra structures on $\LL$ are triangular coboundary, which is different from the relative result on the original Schrödinger-Virasoro type Lie algebra. In particular, we find for this Lie algebra that there are more hidden inner derivations from itself to $\LL\otimes\LL$ and we develop one method to search them.
All the basic cohomology groups and reduced cohomology groups of the extended Schrödinger-Virasoro conformal algebra with trivial coefficients are completely determined. In particular, we introduce the notion of the relative cohomology of Lie conformal algebra, and then we can express our main results in a more concise manner.
In this paper, we determine the derivation algebra and the automorphism group of the original deformative ${\rm Schr\ddot{o}dinger}$-{\rm Virasoro} algebras which is the semi-direct product Lie algebra of the Witt algebra and its tensor density module ${\rm I^g}(a,b)$.
In this paper, Lie bialgebra structures on the extended Schrodinger-Virasoro Lie algebra are classified. It is obtained that all the Lie bialgebra structures on L are triangular coboundary. As a by-product, it is derived that the first cohomology group is trivial.
It is shown that the support of an irreducible weight module over the Schrödinger-Virasoro Lie algebra with an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite-dimensional. As a side-product, it is obtained that every simple weight module over the Schrödinger-Virasoro Lie algebra with a nontrivial finite-dimensional weight space, is a Harish-Chandra module.
In this paper, two new 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\in \mathbb{C}. Moreover, central extensions, conformal derivations and free conformal modules of rank 1 of TSV(a,b) and TSV(c) are determined.
Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we give a complete classification of extensions of finite irreducible conformal modules of $\mathcal{W}(a,b)$. With a similar method, we characterize all extensions of finite irreducible conformal modules of Schrödinger-Virasoro type Lie conformal algebras $TSV(a,b)$ and $TSV(c)$.
Motivated by the structure of certain modules over the loop Virasoro Lie conformal algebra and the Lie structures of Schrodinger-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.
In this paper, by studying the maximal good subspaces, we determine the dual Lie coalgebras of the centerless twisted Heisenberg-Virasoro algebra. Based on this, we construct the dual Lie bialgebras structures of the twisted Heisenberg-Virasoro type. As by-products, four new infinite dimensional Lie algebras are obtained.
In this paper we investigate Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra. With the classifications of Lie bialgebra structures on the Virasoro algebra, we determined such structures on the twisted Heisenberg-Virasoro algebra. Moreover, some general and useful results are obtained. With our methods and results we also can easily to determine such structures on some Lie algebras related to the twisted Heisenberg-Virasoro algebra.
This paper is devoted to investigating the structure theory of a class of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras. In particular, we completely determine the derivation algebras, the automorphism groups and the second cohomology groups of these Lie superalgebras.
In this paper, Whittaker modules for the Schrödinger-Virasoro algebra $\mathfrak{sv}$ are defined. The Whittaker vectors and the irreducibility of the Whittaker modules are studied. $\mathfrak{sv}$ has a triangular decomposition according to the Cartan algebra $\mathfrak{h}:$ $$\mathfrak{sv}=\mathfrak{sv}^{-}\oplus\mathfrak{h}\oplus\mathfrak{sv}^{+}.$$ For any Lie algebra homomorphism $ψ:\mathfrak{sv}^{+}\to\mathbb{C}$, we can define Whittaker modules of type $ψ.$ When $ψ$ is nonsingular, the Whittaker vectors, the irreducibility and the classification of Whittaker modules are completely deter…
We determine all two-dimensional Lie subalgebras of the centreless Virasoro algebra and complete the characterization of all finite dimensional Lie subalgebras of the complex Virasoro algebra.
In this paper, we classify all finite irreducible conformal modules over a class of Lie conformal algebras $\mathcal{W}(b)$ with $b\in\mathbb{C}$ related to the Virasoro conformal algebra. Explicitly, any finite irreducible conformal module over $\mathcal{W}(b)$ is proved to be isomorphic to $M_{Δ,α,β}$ with $Δ\neq 0$ or $β\neq 0$ if $b=0$, or $M_{Δ,α}$ with $Δ\neq 0$ if $b\neq0$. As a byproduct, all finite irreducible conformal modules over the Heisenberg-Virasoro conformal algebra and the Lie conformal algebra of $\mathcal{W}(2,2)$-type are classified. Finally, the same thing is done for the…
In this paper we construct ternary $q$-Virasoro-Witt algebras which $q$-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos using $su(1,1)$ enveloping algebra techniques. The ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a parameter and are not Nambu-Lie algebras for all but finitely many values of this parameter. For the parameter values for which the ternary Virasoro-Witt algebras are Nambu-Lie, the corresponding ternary $q$-Virasoro-Witt algebras constructed in this article are also Hom-Nambu-Lie because they are obt…
In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.
In this paper, we study conformal biderivations of a Lie conformal algebra. First, we give the definition of 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.
In this paper, we provide a uniform method to thoroughly classify all Harish-Chandra modules over some Lie algebras related to the Virasoro algebras. We first classify such modules over the Lie algebra $W(\varrho)[s]$ for $s=0,\frac12$. With this result and method, we can also do such works for some Lie algebras and superconformal algebras related to the Virasoro algebra, including the several kinds of Schrödinger-Virasoro Lie algebras, which are open up to now.
Let $HV$ be the loop Heisenberg-Virasoro Lie algebra over $\C$ with basis $\{L_{\a,i},H_{\b,j}\,|\,\a,\,\b,i,j\in\Z\}$ and brackets $[L_{\a,i},L_{\b,j}]=(\a-\b)L_{\a+\b,i+j}, [L_{\a,i},H_{\b,j}]=-\b H_{\a+\b,i+j},[H_{\a,i},H_{\b,j}]=0$. In this paper, a formal distribution Lie algebra of $HV$ is constructed. Then the associated conformal algebra $CHV$ is studied, where $CHV$ has a $\C[\partial]$-basis $\{L_i,H_i\,|\,i\in\Z\}$ with $λ$-brackets $[L_i\, {}_λ\, L_j]=(\partial+2λ) L_{i+j}, [L_i\, {}_λ\, H_j]=(\partial+λ) H_{i+j}, [H_i\, {}_λ\, L_j]=λL_{i+j}$ and $[H_i\, {}_λ\, H_j]=0$. In particul…
2-local derivation is a generalized derivation for a Lie algebra, which plays an important role to the study of local properties of the structure of the Lie algebra. In this paper, we prove that every 2-local derivation on the twisted Heisenberg-Virasoro algebra is a derivation.
In this paper we attempt to investigate the super-biderivations of Lie superalgebras. Furthermore, we prove that all super-biderivations on the centerless super-Virasoro algebras are inner super-biderivations. Finally, we study the linear super commuting maps on the centerless super-Virasoro algebras.
In this paper, the biderivations without the skew-symmetric condition of the twisted Heisenberg-Virasoro algebra are presented. We find some non-inner and non-skew-symmetric biderivations. As applications, the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the twisted Heisenberg-Virasoro algebra are given. It also is proved that every biderivation of the graded twisted Heisenberg-Virasoro left-symmetric algebra is trivial.
We show how the recently again discussed $N$-point Witt, Virasoro, and affine Lie algebras are genus zero examples of the multi-point versions of Krichever--Novikov type algebras as introduced and studied by Schlichenmaier. Using this more general point of view, useful structural insights and an easier access to calculations can be obtained. The concept of almost-grading will yield information about triangular decompositions which are of importance in the theory of representations. As examples the algebra of functions, vector fields, differential operators, current algebras, affine Lie algebra…
We prove that any irreducible Harish-Chandra modules for a class of Lie algebras, which we call gap-$p$ Virasoro algebras, must be a highest weight module, a lowest weight module, or a module of intermediate series.These algebras are closely related to the Heisenberg-Virasoro algebra and the algebra of derivations over a quantum torus. They also contain subalgebras which are isomorphic to the Virasoro algebra $Vir$, but graded by $p\mathbb Z$(unlike $Vir$ by $\mathbb Z$).
The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.
We study a class of infinite dimensional Lie algebras called generalized Witt algebras (in one variable). These include the classical Witt algebra and the centerless Virasoro algebra as important examples. We show that any such generalized Witt algebra is a semisimple, indecomposable Lie algebra which does not contain any abelian Lie subalgebras of dimension greater than one. We develop an invariant of these generalized Witt algebras called the spectrum, and use it to show that there exist infinite families of nonisomorphic, simple, generalized Witt algebras and infinite families of nonisomorp…
The loop-Virasoro algebra is the Lie algebra of the tensor product of the Virasoro algebra and the Laurent polynomial algebra. This paper classifies irreducible Harish-Chandra modules over the loop-Virasoro algebra, which turn out to be highest weight modules, lowest weight modules and evaluation modules of the intermediate series (all wight spaces are 1-dimensional). As a by-product, we obtain a classification of irreducible Harish-Chandra modules over truncated Virasoro algebras. We also determine the necessary and sufficient conditions for highest weigh irreducible modules over the loop-Vir…
In this note, we introduce a notion of multi-parameter formal deformations of ternary hom-Nambu-Lie algebras. Within this framework, we construct formal deformations of the three-dimensional Jacobian determinant and of the cross-product in four-dimensional Euclidean space. We also conclude that the previously defined ternary q-Virasoro-Witt algebra is a formal deformation of the ternary Virasoro-Witt algebra.
In this paper, we study a class of non-weight modules over two kinds of algebras related to the Virasoro algebra, i.e., the loop-Virasoro algebras $\mathfrak{L}$ and a class of Block type Lie algebras $\mathfrak{B(q)}$, where $q$ is a nonzero complex number. We determine those modules whose restriction to the Cartan subalgebra (modulo center) are free of rank one. We also provide a sufficient and necessary condition for such modules to be simple, and determine their isomorphism classes. Moreover, we obtain the simplicity of modules over loop-Virasoro algebras by taking tensor products of some …
Realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is analogous to the classical Jordan-Schwinger map. A number of examples of such Lie algebras of linear vector fields is computed. In particular, we obtain examples of the twisted Heisenberg-Virasoro Lie algebra and the Schrödinger-Virasoro Lie algebras among others. More generally, we construct an embedding of an arbitrary locally convex topological algebra into the Cuntz algebra.
In this article we develop an approach to deformations of the Witt and Virasoro algebras based on $σ$-derivations. We show that $σ$-twisted Jacobi type identity holds for generators of such deformations. For the $σ$-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the $q$-deformations of the Witt and Virasoro algebras associated to $q$-difference operators, providing also correspond…
By elementary and direct calculations the vanishing of the (algebraic) second Lie algebra cohomology of the Witt and the Virasoro algebra with values in the adjoint module is shown. This yields infinitesimal and formal rigidity or these algebras. The first (and up to now only) proof of this important result was given 1989 by Fialowski in an unpublished note. It is based on cumbersome calculations. Compared to the original proof the presented one is quite elegant and considerably simpler.
