Carina Boyallian, Victor G. Kac, Jose I. Liberati
We classify all finite growth representations of all infinite rank subalgebras of the Lie conformal algebra gc_1 that contain a Virasoro subalgebra.
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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 first give a complete classification of all finite nontrivial irreducible conformal modules of $\mathcal{W}(a,b)$. 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 $TSV(a,b)$ and $TSV(c)$ are characterized.
I give a short proof of the following algebraic statement: if a vertex algebra is simple, then its underlying Lie conformal algebra is either abelian, or it is an irreducible central extension of a simple Lie conformal algebra.
In this paper, we introduce the notion of BiHom-Lie conformal superalgebras. We develop its representation theory and define the cohomology group with coefficients in a module. Finally, we introduce conformal derivations of BiHom-Lie conformal superalgebras and study some of their properties.
We prove that a Lie conformal algebra L with bounded locality function is embeddable into an associative conformal algebra A with the same bound on the locality function. If L is nilpotent, then so is A, and the nilpotency index remains the same. We also give a list of open questions concerning the embedding of Lie conformal algebras into associative conformal.
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 Z-graded free intermediate series modules over these two conformal algebras are also classified in the present paper
Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this paper we prove a rank-level duality for type so(2r…
The aim of this paper is to study a Lie conformal algebra of Block type. In this paper, conformal derivation, conformal module of rank 1 and low-dimensional comohology of the Lie conformal algebra of Block type are studied. Also, the vertex Poisson algebra structure associated with the Lie conformal algebra of Block type is constructed.
In this paper, we construct six families of infinite simple conformal superalgebra of finite growth based on our earlier work on constructing vertex operator superalgebras from graded assocaitive algebras. Three subfamilies of these conformal superalgebras are generated by simple Jordan algebras of types A, B and C in a certain sense.
Conformal Lie 2-algebras and conformal omni-Lie algebras
本文引入并研究了共形李2-代数与共形全李代数的概念，证明了共形李2-代数范畴与2项共形$L_{\infty}$-代数范畴等价，并从共形全李代数和莱布尼茨共形代数构造了共形李2-代数。
We classify unital associative conformal algebras of linear growth and provide new examples of such.
Let $R$ be a Lie conformal algebra. The purpose of this paper is to investigate the conformal derivation algebra $CDer(R)$, the conformal quasiderivation algebra $QDer(R)$ and the generalized conformal derivation algebra $GDer(R)$. The generalized conformal derivation algebra is a natural generalization of the conformal derivation algebra. Obviously, we have the following tower $CDer(R)\subseteq QDer(R)\subseteq GDer(R)\subseteq gc(R)$, where $gc(R)$ is the general Lie conformal algebra. Furthermore, we mainly research the connection of these generalized conformal derivations. Finally, the con…
The Lie conformal algebra of loop Virasoro algebra, denoted by $\mathscr{CW}$, is introduced in this paper. Explicitly, $\mathscr{CW}$ is a Lie conformal algebra with $\mathbb{C}[\partial]$-basis $\{L_i\,|\,i\in\mathbb{C}\}$ and $λ$-brackets $[L_i\, {}_λ\, L_j]=(-\partial-2λ) L_{i+j}$. Then conformal derivations of $\mathscr{CW}$ are determined. Finally, rank one conformal modules and $\mathbb{Z}$-graded free intermediate series modules over $\mathscr{CW}$ are classified.
One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra [K]. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which C[\partial] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra [BDK]. The finite (i.e., finitely generated over H) simple Lie pseudoalgebras were classified in [BDK]. In a series of papers, starting with the present one, we classify all irreducible finite modules over finite simple Li…
We find a necessary and sufficient condition for the existence of the tensor product of modules over a Lie conformal algebra. We provide two algebraic constructions of the tensor product. We show the relation between tensor product and conformal linear maps. We prove commutativity of the tensor product.
We prove that a finite torsion-free conformal Lie algebra with a splitting solvable radical has a finite faithful conformal representation.
Fix a positive integer number $r$. A class of $r$-dim Lie conformal superalgebras named $r$-dim $i$-linear Lie conformal superalgebras are studied for $1\leq i \leq r$. We present an equivalent characterization of this class of Lie conformal superalgebras in $r$ dimension. Moreover, by this equivalent characterization, several constructions and examples are given.
Lie conformal algebras appear in the theory of vertex algebras. Their relation is similar to that of Lie algebras and their universal enveloping algebras. Associative conformal algebras play a role in conformal representation theory. We introduce the notions of conformal identity and unital associative conformal algebras and classify finitely generated simple unital associative conformal algebras of linear growth. These are precisely the complete algebras of conformal endomorphisms of finite modules.
Let $G$ be a filtered Lie conformal algebra whose associated graded conformal algebra is isomorphic to that of general conformal algebra $gc_1$. In this paper, we prove that $G\cong gc_1$ or ${\rm gr\,}gc_1$ (the associated graded conformal algebra of $gc_1$), by making use of some results on the second cohomology groups of the conformal algebra $\fg$ with coefficients in its module $M_{b,0}$ of rank 1, where $\fg=\Vir\ltimes M_{a,0}$ is the semi-direct sum of the Virasoro conformal algebra $\Vir$ with its module $M_{a,0}$. Furthermore, we prove that ${\rm gr\,}gc_1$ does not have a nontrivial…
We study quadratic Lie conformal superalgebras associated with No\-vikov superalgebras. For every Novikov superalgebra $(V,\circ)$, we construct an enveloping differential Poisson superalgebra $U(V)$ with a derivation $d$ such that $u\circ v = ud(v)$ and $\{u,v\} = u\circ v - (-1)^{|u||v|} v\circ u$ for $u,v\in V$. The latter means that the commutator Gelfand--Dorfman superalgebra of $V$ is special. Next, we prove that every quadratic Lie conformal superalgebra constructed on a finite-dimensional special Gel'fand--Dorfman superalgebra has a finite faithful conformal representation. This statem…
We develop structure theory of finite Lie conformal superalgebras.
In this paper we use Lie conformal algebras to realize some moonshine type VOAs, whose Greiss algebras are Jordan algebras. On the other hand, we consider some free fields which realizes the corresponding simple VOAs. As an application, we can calculate the correlation functions of these VOAs in a relatively easy way.
Twisting theory, relative Rota-Baxter type operators and $L_\infty$-algebras on Lie conformal algebras
本文建立 Lie 共形代数的扭理论，利用导出括号构造 $L_\infty$-代数，刻画相对 Rota-Baxter 型算子，并引入 NS-Lie 共形代数及相关上同调与形变。
In this note we show a close relation between the following objects: Classical Yang -- Baxter equation (CYBE), conformal algebras (also known as vertex Lie algebras), and averaging operators on Lie algebras. It turns out that the singular part of a solution of CYBE (in the operator form) on a Lie algebra $\mathfrak g$ determines an averaging operator on the corresponding current conformal algebra $\mathrm{Cur} \mathfrak g$. For a finite-dimensional semisimple Lie algebra $\mathfrak g$, we describe all homogeneous averaging operators on $\mathrm{Cur}\mathfrak g$. It turns out that all these ope…
In this paper, simplicity of quadratic Lie conformal algebras are investigated. From the point view 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 classes of new infinite simple Lie conformal algebras. These results will be useful for classification purposes.
The purpose of this paper is to introduce and study super Hom-Gel'fand-Dorfman bialgebras and Hom-Lie conformal superalgebras. In this paper, we provide different ways for constructing super Hom-Gel'fand-Dorfman bialgebras and obtain some infinite-dimensional Hom-Lie superalgebras from affinization of super Hom-Gel'fand-Dorfman bialgebras. Also, we give a general construction of Hom-Lie conformal superalgebras from Hom-Lie superalgebras and establish equivalence of quadratic Hom-Lie conformal superalgebras and super Hom-Gel'fand-Dorfman bialgebras. Finally, we characterize one-dimensional cent…
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.
The loop super-Virasoro conformal superalgebra $\mathfrak{cls}$ associated with the loop super-Virasoro algebra is constructed in the present paper. The conformal superderivation algebra of $\mathfrak{cls}$ is completely determined, which is shown to consist of inner superderivations. And nontrivial free and free $\mathbb{Z}$-graded $\mathfrak{cls}$-modules of rank two are classified. We also give a classification of irreducible free $\mathfrak{cls}$-modules of rank two and all irreducible submodules of each free $\mathbb{Z}$-graded $\mathfrak{cls}$-module of rank two.
A quadratic Lie conformal algebra corresponds to a Hamiltonian pair in \cite{GD}, which plays fundamental roles in completely integrable systems. Moreover, it also corresponds to certain compatible pairs of a Lie algebra and a Novikov algebra which was called Gel'fand-Dorfman bialgebra by Xu in \cite{X1}. In this paper, central extensions and conformal derivations of quadratic Lie conformal algebras are studied in terms of Gel'fand-Dorfman bialgebras. It is shown that central extensions and conformal derivations of a quadratic Lie conformal algebra are related with some bilinear forms and some…
Non-linear Lie Conformal Algebras and One-Loop Corrections of self-dual Yang-Mills amplitudes
本文基于celestial holography的发展，利用非线性李一致性代数框架重新表述了QCD共线奇异性与二维CFT运算乘积展开的关系，并探讨自对偶杨-米尔斯幅值的一loop修正。
The role of automorphisms of infinite-dimensional Lie algebras in conformal field theory is examined. Two main types of applications are discussed; they are related to the enhancement and reduction of symmetry, respectively. The structures one encounters also appear in other areas of physics and mathematics. In particular, they lead to two conjectures on the sub-bundle structure of chiral blocks, and they are instrumental in the study of conformally invariant boundary conditions.
Any variety of classical algebras has a so-called conformal counterpart. For example one can consider Lie conformal or associative conformal algebras. Lie conformal algebras are closely related to vertex algebras. We define free objects in the categories of conformal and vertex algebras. In some cases we can explicitly build their Groebner bases.
We distinguish a class of irreducible finite representations of conformal Lie (super)algebras. These representations (called universally defined) are the simplest ones from the computational point of view: a universally defined representation of a conformal Lie (super)algebra $L$ is completely determined by commutation relations of $L$ and by the requirement of associative locality of generators. We describe such representations for conformal superalgebras $W_n$, $n\ge 0$, with respect to a natural set of generators. We also consider the problem for superalgebras $K_n$. In particular, we find …
We classify certain non-linear Lie conformal algebras with three generators, which can be viewed as deformations of the current Lie conformal algebra of sl_2. In doing so we discover an interesting 1-parameter family of non-linear Lie conformal algebras R_{-1}^d and the corresponding freely generated vertex algebras V_{-1}^d, which includes for d=1 the affine vertex algebra of sl_2 at the critical level k=-2. We construct free-field realizations of the algebras V_{-1}^d extending the Wakimoto realization of the affine vertex algebra of sl_2 at the critical level, and we compute their Zhu algeb…
In this paper, we first study infinitesimal deformations of a Lie conformal algebra and a Lie conformal algebra with a module (called an $\mathsf{LCMod}$ pair), which lead to the notions of Nijenhuis operator on the Lie conformal algebra and Nijenhuis structure on the $\mathsf{LCMod}$ pair, respectively. Then by adding compatibility conditions between Nijenhuis structures and $\mathcal{O}$-operators, we introduce the notion of an $\mathcal{O} N$-structure on an $\mathsf{LCMod}$ pair and show that an $\mathcal{O} N$-structure gives rise to a hierarchy of pairwise compatible $\mathcal{O}$-operat…
We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra cohomology complex, and endow it with a structure of a g-complex. On the other hand, we give an explicit construction of the complex of variational calculus in terms of skew-symmetric poly-differential operators.
Conformal Yang-Baxter equation on $\mathrm{Cur}(\mathrm{sl}_2(\mathbb{C}))$
针对J. Liberati 2008年提出的共形经典杨-巴克斯特方程（CCYBE），本文给出当前李共形代数$\text{Cur}(\text{sl}_2(\text{C}))$上CCYBE及其弱版本的所有解，为共形李双代数结构研究提供基础。
We construct a calculus structure on the Lie conformal algebra cochain complex. By restricting to degree one chains, we recover the structure of a g-complex introduced in [DSK]. A special case of this construction is the variational calculus, for which we provide explicit formulas.
We introduce several definitions within the framework of vertex and conformal algebras which are analogous to some important concepts of the classical Lie theory. Most importantly, we define formal vertex laws, which correspond to the notion of formal group law. We prove suitable vertex/conformal versions of a number of classical results such as the Milnor-Moore theorem, Cartier duality, and the equivalence between formal group laws and Lie 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 Virasoro conformal algebra and Neveu--Schwartz conformal superalgebra.
Representations of braid group obtained from rational conformal field theories can be used to obtain explicit representations of Temperley-Lieb-Jones algebras. The method is described in detail for SU(2)$_k$ Wess - Zumino conformal field theories and its generalization to an arbitrary rational conformal field theory outlined. Explicit definition of an associated linear trace operation in terms of a certain matrix element in the space of conformal blocks of such a conformal theory is presented. Further for every primary field of a rational conformal field theory, there is a subfactor of hyperfi…
Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks
该研究针对任意单李代数在一般非临界水平下定义了零亏格不规则真空/共真空空间，利用仿射李代数模余不变量，证明不规则共形块空间构成平坦向量丛，并给出推广现有KZ联络的普适平坦联络。
We prove a finite torsion-free associative conformal algebra to have a finite faithful conformal representation. As a corollary, it is shown that one may join a conformal unit to such an algebra. Some examples are stated to demonstrate that a conformal unit can not be joined to any torsion-free associative conformal algebra. In particular, there exist associative conformal algebras of linear growth and even locally finite ones that have no finite faithful representation. We also consider the problem of existence of a finite faithful representation for a torsion-free finite Lie conformal algebr…
A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras
本文定义 GD 双代数并用以构造 Lie 共形双代数，借助 GDYBE、$\mathcal{O}$-算子、pre-GD 代数、匹配对和 Manin 三元组建立其结构理论。
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