Lie Conformal Superalgebras Research Articles

Structures of sympathetic Lie conformal superalgebras

In this paper, we'll introduce the concept of sympathetic Lie conformal superalgebras and show that some classical properties of Lie conformal superalgebras are still valid for sympathetic Lie conformal superalgebras. We prove that the unique decomposition of each sympathetic Lie conformal superalgebra into a direct sum of indecomposable sympathetic ideals. We also show the existence of a greatest sympathetic ideal and a sympathetic decomposition in every perfect Lie conformal superalgebra. In the end, we also study the ideal I of a Lie conformal superalgebra R such that R/I is a sympathetic Lie conformal superalgebra.

A Lie conformal superalgebra and duality of representations for E(4,4)

We construct a duality functor in the category of continuous representations of the Lie superalgebra E(4,4), the only exceptional simple linearly compact Lie superalgebra, for which it wasn't known. This is achieved by constructing a Lie conformal superalgebra of type (4,4) for which E(4,4) is the annihilation algebra. Along the way we obtain an explicit realization of E(4,4) by vector fields on a (4|4)-dimensional supermanifold.

Hom-Gel'fand-Dorfman conformal superbialgebras

Gel'fand Dorfman superbialgebra, which is both a Lie superalgebra and a (left) Novikov superalgebra with some compatibility condition, appears in the study of Hamiltonian pairs in completely integrable systems and a class of special Lie conformal superalgebras called quadratic Lie conformal superalgebras. In the present paper, we generalize this algebraic structure to the Hom-conformal case. We introduce first, Hom-Novikov conformal superalgebras and exihibit several properties. Then we introduce Hom-Gel'fand Dorfman superbialgebra and provide some construction results.

An upper bound on the degree of singular vectors for E(1,6)

The aim of this work is to prove a technical result that had been stated by Boyallian et al. [Classification of finite irreducible modules over the Lie conformal superalgebra [Formula: see text], Comm. Math. Phys. 317(2013) 503–546], on the degree of singular vectors of finite Verma modules over the exceptional Lie superalgebra [Formula: see text] that is isomorphic to the annihilation superalgebra associated with the conformal superalgebra [Formula: see text].

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.

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).

Classification of Finite Irreducible Conformal Modules over N = 2 Lie Conformal Superalgebras of Block Type

We introduce the N = 2 Lie conformal superalgebras \({\frak {K}}(p)\) of Block type, and classify their finite irreducible conformal modules for any nonzero parameter p. In particular, we show that such a conformal module admits a nontrivial extension of a finite conformal module M over K2 if p = − 1 and M has rank (2 + 2), where K2 is an N = 2 conformal subalgebra of \({\frak {K}}(p)\). As a byproduct, we obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal superalgebras \({\frak k}(n)\) for n ≥ 1. Composition factors of all the involved reducible conformal modules are also determined.

The Heisenberg-Virasoro Lie conformal superalgebra

In this paper, we introduce a finite Lie conformal superalgebra called the Heisenberg-Virasoro Lie conformal superalgebra s by using a class of Heisenberg-Virasoro Lie conformal modules. The super Heisenberg-Virasoro algebra of Ramond type S is defined by the formal distribution Lie superalgebra of s. Then we construct a class of simple S-modules, which are induced from simple modules of some finite dimensional solvable Lie superalgebras. These modules are isomorphic to simple restricted S-modules, and include the highest weight modules, Whittaker modules and high order Whittaker modules. As a byproduct, we present a subalgebra of S, which is isomorphic to the super Heisenberg-Virasoro algebra of Neveu-Schwarz type.

Representations of twisted infinite Lie conformal superalgebras

Following the twisting rule from Ramond Lie superalgebra to Neveu-Schwarz Lie superalgebra, we introduce the twisted version T(p) of a class of Z-graded Lie conformal superalgebras, where p is a nonzero parameter. We completely classify their small rank conformal modules and finite irreducible conformal modules, which turn out to be much richer than those over the original version. As applications, we also develop the representation theory of many new finite Lie conformal superalgebras.

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.

Conformal Super-Biderivations on Lie Conformal Superalgebras

In this paper, the conformal super-biderivations of two classes of Lie conformal superalgebras are studied. By proving some general results on conformal super-biderivations, we determine the conformal super-biderivations of the loop super-Virasoro Lie conformal superalgebra and Neveu–Schwarz Lie conformal superalgebra. Especially, any conformal super-biderivation of the Neveu–Schwarz Lie conformal superalgebra is inner.

Finite irreducible conformal modules over the Lie conformal superalgebra [formula omitted

In the present paper, we introduce a class of infinite Lie conformal superalgebras S(p), which are closely related to Lie conformal algebras of extended Block type defined in [6]. Then all finite non-trivial irreducible conformal modules over S(p) for p∈C⁎ are completely classified. As an application, we also present the classifications of finite non-trivial irreducible conformal modules over finite quotient algebras s(n) for n≥1 and sh which is isomorphic to a subalgebra of Lie conformal algebra of N=2 superconformal algebra. Moreover, as a generalized version of S(p), the infinite Lie conformal superalgebras GS(p) are constructed, which have a subalgebra isomorphic to the finite Lie conformal algebra of N=2 superconformal algebra.

Lie conformal superalgebras and duality of modules over linearly compact Lie superalgebras

We construct a duality functor on the category of continuous representations of linearly compact Lie superalgebras, using representation theory of Lie conformal superalgebras. We compute the dual representations of the generalized Verma modules.

On [formula omitted]-Hom-Jordan Lie conformal superalgebras

In this paper, we introduce the representation theory of δ-Hom-Jordan Lie conformal superalgebras and discuss the case of adjoint representations. Furthermore, we develop the cohomology theory of δ-Hom-Jordan Lie conformal superalgebras and discuss some applications to the study of deformations of regular δ-Hom-Jordan Lie conformal superalgebras. Finally, we introduce derivations of multiplicative δ-Hom-Jordan Lie conformal superalgebras and study their properties.

Classification of finite irreducible conformal modules over Lie conformal superalgebras of Block type

We introduce a class of infinite Lie conformal superalgebras S(p) of Block type, and classify their finite irreducible conformal modules for any nonzero parameter p. In particular, we show that such a conformal module admits a nontrivial extension of a finite conformal module over NS if p=−1, where NS is a Neveu-Schwarz conformal subalgebra of S(p). As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal superalgebras s(n) for n≥1.

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.

Deformations and generalized derivations of Lie conformal superalgebras

In this paper, we study deformations and generalized derivations of Lie conformal superalgebras. First, we study conformal derivations of the semidirect product of a Lie conformal superalgebra and its conformal module. Second, we extend the cohomology theory of Lie conformal algebras to the super case and discuss some applications to deformations of Lie conformal superalgebras. Finally, we introduce generalized derivations of Lie conformal superalgebras and study their properties.

On principal realization of modules for the affine Lie algebra 𝐴₁⁽¹⁾ at the critical level

We present complete realization of irreducible A 1 ( 1 ) A_1^{(1)} –modules at the critical level in the principal gradation. Our construction uses vertex algebraic techniques, the theory of twisted modules and representations of Lie conformal superalgebras. We also provide an alternative Z-algebra approach to this construction. All irreducible highest weight A 1 ( 1 ) A_1^{(1)} –modules at the critical level are realized on the vector space M 1 2 + Z ( 1 ) ⊗ 2 M_{\tfrac {1}{2}+\mathbb {Z}}(1)^{\otimes 2} where M 1 2 + Z ( 1 ) M_{\tfrac {1}{2} + \mathbb {Z}}(1) is the polynomial ring C [ α ( − 1 / 2 ) , α ( − 3 / 2 ) , … ] {\mathbb {C}}[\alpha (-1/2), \alpha (-3/2),\dots ] . Explicit combinatorial bases for these modules are also given.

The Automorphism Group Functor of the N = 4 Lie Conformal Superalgebra

In this article, we study the structure and representability of the automorphism group functor of the N = 4 Lie conformal superalgebra over an algebraically closed field k of characteristic zero.

The variational Poisson cohomology

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.