scholarly journals Loop Schrödinger–Virasoro Lie conformal algebra

2016 ◽  
Vol 27 (06) ◽  
pp. 1650057 ◽  
Author(s):  
Haibo Chen ◽  
Jianzhi Han ◽  
Yucai Su ◽  
Ying Xu
Keyword(s):  
Lie Algebra ◽  
Conformal Algebra ◽  
Cohomology Groups ◽  
Conformal Algebras ◽  
Rank One ◽  
Intermediate Series

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.

Lie conformal algebras related to Galilean conformal algebras

2020 ◽  
pp. 2150075
Author(s):  
Xiu Han ◽  
Dengyin Wang ◽  
Chunguang Xia
Keyword(s):  
Conformal Algebra ◽  
Complex Numbers ◽  
Conformal Algebras ◽  
Rank One ◽  
Intermediate Series

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.

Cohomology of Lie Conformal Algebra Vir⋉Cur g

2021 ◽  
Vol 28 (03) ◽  
pp. 507-520
Author(s):  
Maosen Xu ◽  
Yan Tan ◽  
Zhixiang Wu
Keyword(s):  
Lie Algebra ◽  
Conformal Algebra ◽  
Cohomology Groups ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Irreducible Modules

In this article, we compute cohomology groups of the semisimple Lie conformal algebra [Formula: see text] with coefficients in its irreducible modules for a finite-dimensional simple Lie algebra [Formula: see text].

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

2019 ◽  
Vol 30 (06) ◽  
pp. 1950026 ◽  
Author(s):  
Lipeng Luo ◽  
Yanyong Hong ◽  
Zhixiang Wu
Keyword(s):  
Complete Classification ◽  
Conformal Algebra ◽  
Direct Sums ◽  
Conformal Algebras ◽  
Irreducible Modules ◽  
Rank One ◽  
Virasoro Type

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.

scholarly journals Finite irreducible conformal modules of rank two Lie conformal algebras

2020 ◽  
pp. 2150145
Author(s):  
Maosen Xu ◽  
Yanyong Hong ◽  
Zhixiang Wu
Keyword(s):  
Irreducible Module ◽  
Conformal Algebra ◽  
Conformal Algebras ◽  
Irreducible Modules ◽  
Rank One

In the present paper, we prove that any finite nontrivial irreducible module over a rank two Lie conformal algebra [Formula: see text] is of rank one. We also describe the actions of [Formula: see text] on its finite irreducible modules explicitly. Moreover, we show that all finite nontrivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one.

scholarly journals Loop W(a,b) Lie conformal algebra

2016 ◽  
Vol 27 (02) ◽  
pp. 1650016 ◽  
Author(s):  
Guangzhe Fan ◽  
Henan Wu ◽  
Bo Yu
Keyword(s):  
Lie Algebra ◽  
Conformal Algebra ◽  
Central Extensions ◽  
Rank One ◽  
Formal Distribution

Fix [Formula: see text], let [Formula: see text] be the loop [Formula: see text] Lie algebra over [Formula: see text] with basis [Formula: see text] and relations [Formula: see text], where [Formula: see text]. In this paper, a formal distribution Lie algebra of [Formula: see text] is constructed. Then the associated conformal algebra [Formula: see text] is studied, where [Formula: see text] has a [Formula: see text]-basis [Formula: see text] with [Formula: see text]-brackets [Formula: see text] and [Formula: see text]. In particular, we determine the conformal derivations and rank one conformal modules of this conformal algebra. Finally, we study the central extensions and extensions of conformal modules.

scholarly journals The Lie Conformal Algebra of a Block Type Lie Algebra

2015 ◽  
Vol 22 (03) ◽  
pp. 367-382 ◽  
Author(s):  
Ming Gao ◽  
Ying Xu ◽  
Xiaoqing Yue
Keyword(s):  
Lie Algebra ◽  
Conformal Algebra ◽  
Block Type ◽  
Intermediate Series ◽  
Formal Distribution

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.

On left-symmetric conformal bialgebras

2014 ◽  
Vol 14 (01) ◽  
pp. 1450079 ◽  
Author(s):  
Yanyong Hong ◽  
Fang Li
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Conformal Algebra ◽  
Baxter Equation ◽  
Matched Pairs ◽  
Conformal Algebras

The notion of left-symmetric bialgebra was introduced in [C. M. Bai, Left-symmetric bialgebra and an analogue of the classical Yang–Baxter equation, Commun. Contemp. Math. 10(2) (2008) 221–260] which is equivalent to a parakähler Lie algebra which is the Lie algebra of a Lie group G with a G-invariant parakähler structure. In this paper, we study a conformal analog of left-symmetric bialgebras. The notions of left-symmetric conformal coalgebra and bialgebra are introduced. Moreover, the constructions of matched pairs of Lie conformal algebras and left-symmetric conformal algebras are presented. We show that a finite left-symmetric conformal bialgebra which is free as a ℂ[∂]-module is equivalent to a parakähler Lie conformal algebra. We also obtain a conformal analog of the S-equation (see [C. M. Bai, Left-symmetric bialgebras and an analogue of the classical Yang–Baxter equation, Commun. Contemp. Math. 10(2) (2008) 221–260]), and give a construction of the conformal symplectic double.

$\mathcal{W}(a,b)$ Lie Conformal Algebra and Its Conformal Module of Rank One

2015 ◽  
Vol 22 (03) ◽  
pp. 405-412 ◽  
Author(s):  
Ying Xu ◽  
Xiaoqing Yue
Keyword(s):  
Lie Algebra ◽  
Conformal Algebra ◽  
Conformal Module ◽  
Rank One

For any complex parameters a, b, let [Formula: see text] be the Lie algebra with basis {Li,Hi | i ∈ ℤ} and relations [Li,Lj]=(j-i)Li+j, [Li,Hj]=(a+j+bi)Hi+j and [Hi,Hj]=0. In this paper, we construct the [Formula: see text] conformal algebra for some a, b and its conformal module of rank one.

A class of Schrödinger–Virasoro type Lie conformal algebras

2015 ◽  
Vol 26 (08) ◽  
pp. 1550058 ◽  
Author(s):  
Wei Wang ◽  
Ying Xu ◽  
Chunguang Xia
Keyword(s):  
Lie Algebra ◽  
Cohomology Group ◽  
Conformal Algebra ◽  
Conformal Algebras ◽  
Virasoro Type

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.

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

2017 ◽  
pp. 1-31
Author(s):  
María Julia Redondo ◽  
Lucrecia Román
Keyword(s):  
Lie Algebra ◽  
Hochschild Cohomology ◽  
Cohomology Groups ◽  
Efficient Tool ◽  
Simple Description ◽  
Monomial Algebra ◽  
Cup Product ◽  
Lie Bracket ◽  
Projective Resolutions ◽  
Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

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