Lie Conformal Algebras Research Articles

Atiyah sequences of braided Lie algebras and their splittings

Given an equivariant noncommutative principal bundle, we construct an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. In the case of the principal \mathrm{SU}(2) -bundle over the sphere S^{4}_{\theta} an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra \mathrm{so}_{\theta}(5,1) yields a five parameter family of splittings. On the principal \mathrm{SO}_\theta(2n,\mathbb{R}) -bundle of orthonormal frames over the sphere S^{2n}_\theta , a splitting of the sequence leads to the Levi-Civita connection for the ‘round’ metric on S^{2n}_\theta . The corresponding Riemannian geometry of S^{2n}_{\theta} is worked out.

Sl(2,C)×D symmetry and conformal primary basis for massless fields

Alternative to the embedding formalism, we provide a group theoretic approach to the conformal primary basis for the massless field with arbitrary helicity. To this end, we first point out that sl(2,C) isometry gets enhanced to sl(2,C)×D symmetry for the solution space of the massless field with D the bulk dilatation. Then, associated with sl(2,C)×D symmetry, we introduce the novel quadratic Casimirs and relevant tensor/spinor fields to derive two explicit constraints on the bulk dilatation and sl(2,C) Casimirs. With this, we further argue that the candidate conformal primary basis can be constructed out of the infinite tower of the descendants of the left and right highest (lowest) conformal primary wave function of sl(2,C) Lie algebra, and the corresponding celestial conformal weights are determined by the bulk scaling dimension through solving out the exact on-shell conformal primary wave functions, where on top of the two kinds of familiar-looking on-shell conformal primary wave functions, we also obtain another set of independent on-shell conformal primary wave functions for the massless field with helicity |s|≥1. In passing, we also develop the relationship between the four-dimensional Lorentz Lie algebra and two-dimensional conformal Lie algebra from scratch, and present an explicit derivation for the two important properties associated with the conformal primary wave functions. Published by the American Physical Society 2024

Cohomology of the Lie conformal algebra 𝒲(a,b)

Cohomology of the Lie conformal algebra 𝒲(a,b)

Faithful Representations of Finite Type for Conformal Lie Algebras

Faithful Representations of Finite Type for Conformal Lie Algebras

Canonical Construction of Invariant Differential Operators: A Review

In the present paper, we review the progress of the project of the classification and construction of invariant differential operators for non-compact, semisimple Lie groups. Our starting point is the class of algebras which we called earlier ‘conformal Lie algebras’ (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this purpose, we introduced recently the new notion of a parabolic relation between two non-compact, semi-simple Lie algebras G and G′ that have the same complexification and possess maximal parabolic subalgebras with the same complexification.

Nijenhuis-operator on Hom-Lie conformal algebras

The aim of this paper is to study the Nijenhuis operators on Hom-Lie conformal algebras. We construct a graded Lie algebra whose Maurer-Cartan elements are given by Nijenhuis operators with the aid of Frölicher-Nijenhuis bracket. After constructing graded Lie algebra, we define the cohomology associated with a Nijenhuis operator. We further introduce Hom-NS Lie conformal algebra as an algebraic structure behind Nijenhuis operators on Hom-Lie conformal algebra. We provide various examples of Hom-NS Lie conformal algebras. Finally, as an application to cohomology, we study formal deformations of Nijenhuis operators.

Conformal (σ, τ)-derivations on lie conformal algebras

In this paper, we focus on the conformal (?, ?)-derivation theory of Lie conformal algebras. Firstly, we study the fundamental properties of conformal (?, ?)-derivations. Secondly, we mainly research the interiors of conformal G-derivations. Finally, we discuss the relationships between the conformal (?, ?)-derivations and some generalized conformal derivations of Lie conformal algebras.

Extensions of finite irreducible conformal modules over Lie conformal algebras 𝒲(a,b,r)

In this article, we study extensions of conformal modules over Lie conformal algebras with three parameters a, b, , which contain the Virasoro Lie conformal algebra Vir, and as subalgebras. We obtain the existence conditions of extensions of nontrivial conformal modules over and give a complete classification of extensions of finite irreducible conformal modules of .

Twisting theory, relative Rota-Baxter type operators and L∞-algebras on Lie conformal algebras

Based on Nijenhuis-Richardson bracket and bidegree on the cohomology complex for a Lie conformal algebra, we develop a twisting theory of Lie conformal algebras. By using derived bracket constructions, we construct L∞-algebras from (quasi-)twilled Lie conformal algebras. And we show that the result of the twisting by a C[∂]-module homomorphism on a (quasi-)twilled Lie conformal algebra is also a (quasi-)twilled Lie conformal algebra if and only if the C[∂]-module homomorphism is a Maurer-Cartan element of the L∞-algebra. In particular, we show that relative Rota-Baxter type operators on Lie conformal algebras are Maurer-Cartan elements. Besides, we propose a new algebraic structure, called NS-Lie conformal algebras, that is closely related to twisted relative Rota-Baxter operators and Nijenhuis operators on Lie conformal algebras. As an application of twisting theory, we give the cohomology of twisted relative Rota-Baxter operators and study their deformations.

Conformal Triple Derivations and Triple Homomorphisms of Lie Conformal Algebras

Let [Formula: see text] be a finite Lie conformal algebra. We investigate the conformal derivation algebra [Formula: see text], conformal triple derivation algebra [Formula: see text] and generalized conformal triple derivation algebra [Formula: see text], focusing mainly on the connections among these derivation algebras. We also give a complete classification of (generalized) conformal triple derivation algebras on all finite simple Lie conformal algebras. In particular, [Formula: see text], where [Formula: see text] is a finite simple Lie conformal algebra. But for [Formula: see text], we obtain a conclusion that is closely related to [Formula: see text]. Finally, we introduce the definition of a triple homomorphism of Lie conformal algebras. Triple homomorphisms of all finite simple Lie conformal algebras are also characterized.

Hochschild cohomology of the Weyl conformal algebra with coefficients in finite modules

In this work, we find Hochschild cohomology groups of the Weyl associative conformal algebra with coefficients in all finite modules. The Weyl conformal algebra is the universal associative conformal envelope of the Virasoro Lie conformal algebra relative to the locality N = 2. In order to obtain this result, we adjust the algebraic discrete Morse theory to the case of differential algebras.

Representations of the associated Lie conformal algebra of the [formula omitted] algebra and beyond

We introduce the so-called infinite contact Lie conformal superalgebras KN(p) from finite Lie conformal superalgebras of Cartan type K. The annihilation algebra of the first member L in degenerate cases is exactly the associated graded Lie algebra of the filtered W-infinity algebra W1+∞. Using the conformal version of Lie's theorem, we first classify finite conformal modules over KN(p) and its derived subalgebra. Then we classify finite irreducible conformal modules over positive subalgebras of L and its derived subalgebra, which are shown to be weight modules, parameterized by polynomials, in the sense of D'Andrea and Kac [13]. Finally, we conceptually classify extensions between these conformal modules.

Rota---Baxter operators on $Cur(sl_2(\mathbb{C}))$

We classify all Rota---Baxter operators on the simple Lie conformal algebra $\Cur(\sl_2(\mathbb{C}))$ and clarify which of them arise from the solutions to the conformal classical Yang---Baxter equation due to the connection discovered by Y. Hong and C. Bai in 2020.

Conformal Yang–Baxter equation on Cur(sl2(C))

In 2008, Liberati [J. Algebra 319, 2295–2318 (2008)] defined what a conformal Lie bialgebra is and introduced the conformal classical Yang–Baxter equation (CCYBE). An L-invariant solution to the weak version of CCYBE provides a conformal Lie bialgebra structure. We describe all solutions to the CCYBE on the current Lie conformal algebra Cur(sl2(C)) and to the weak version of it.

A class of graded conformal algebras which is induced by Heisenberg-Virasoro conformal algebra

In this paper, we obtain a class of Z-graded conformal algebras which is induced by Heisenberg-Virasoro conformal algebra. More precisely, we classify Z-graded conformal algebras L=⊕i=−1∞Li satisfying the following conditions,(C1)L0 is the Heisenberg-Virasoro conformal algebra;(C2)Each Li for i∈Z≥−1⁎ is an L0-module of rank one;(C3)[X−1λXi]≠0 for i≥0, where Xi is any one of C[∂]-generators of Li for i∈Z≥−1. Further, we prove that all finite nontrivial irreducible modules of these algebras under some special conditions are free of rank one as a C[∂]-module. The conformal derivations of this class of graded Lie conformal algebras are also determined.

Representations of super deformation of Heisenberg-Virasoro type Lie conformal algebras

We introduce a class of Lie conformal superalgebra S(a, b), where a, b are complex parameters. These Lie conformal superalgebras can be viewed as the super deformation of Heisenberg-Virasoro type Lie conformal algebras. We classify the free conformal modules of rank over S(a, b) for all . Using a Cheng-Kac’s lemma, we also classify the finite irreducible conformal modules over S(a, b).

Conformal r-matrix-Nijenhuis structures, symplectic-Nijenhuis structures, and ON-structures

In this paper, we first study infinitesimal deformations of a Lie conformal algebra and a Lie conformal algebra with a module (called an LCMod pair), which lead to the notions of the Nijenhuis operator on the Lie conformal algebra and the Nijenhuis structure on the LCMod pair, respectively. Then, by adding compatibility conditions between Nijenhuis structures and O-operators, we introduce the notion of an ON-structure on an LCMod pair and show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. In particular, we show that compatible O-operators on a Lie conformal algebra can be characterized by Nijenhuis operators on Lie conformal algebras. Finally, we introduce the notions of the conformal r-matrix-Nijenhuis structure and symplectic-Nijenhuis structure on Lie conformal algebras and study their relations.

Schrödinger-Virasoro Lie conformal algebras

Schrödinger-Virasoro Lie algebras spanned by are annihilation algebras of Lie conformal algebras of rank three, which are widely studied by mathematicians and physicists. In this article, we introduce the notion of Schrödinger-Virasoro Lie conformal algebras. Using the pseudoalgebra language, we study the Schrödinger-Virasoro Lie conformal algebras and classify all Schrödinger-Virasoro Lie conformal algebras.

Compatible left-symmetric conformal algebras on W(a, b)

We study the compatible left-symmetric conformal algebraic structures on the Lie conformal algebra W(a, b), which by definition, is a semidirect sum of the Virasoro Lie conformal algebra and its nontrivial rank one module All compatible left-symmetric conformal algebraic structures on W(a, b), under the condition that is an ideal, are completely determined.

Quadratic Lie conformal superalgebras related to Novikov superalgebras

We study quadratic Lie conformal superalgebras associated with Novikov 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 Gelfand–Dorfman superalgebra has a finite faithful conformal representation. This statement is a step towards a solution of the following open problem: whether a finite Lie conformal (super)algebra has a finite faithful conformal representation.