Structure theory of finite Lie conformal superalgebras

Non-degenerate Killing forms on Hom-Lie superalgebras

In this paper we investigate some important basic properties of simple Hom-Lie superalgebras and show that a Hom-Lie superalgebra does not have any left or right nontrivial ideals. Moreover, we classify invariant bilinear forms on a given simple Hom-Lie superalgebra. Then we study the Killing forms on a Hom-Lie algebra which are examples of the invariant bilinear forms. Making use of the Killing forms, we find conditions for a Hom-Lie superalgebra to be classical. Furthermore, we check the conditions in which the Killing form of a Hom-Lie superalgebra is non-degenerate.

Cohomology and conformal derivations of BiHom-Lie conformal superalgebras

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.

Simple Jordan conformal superalgebras

We classify simple finite Jordan conformal superalgebras and establish preliminary results for the classification of simple finite Jordan pseudoalgebras.

A class of Lie conformal superalgebras in higher dimensions

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.

Equivalence of conformal superalgebras to Hamiltonian superoperators

In this paper, we present a formal variational calculus of super functions in one real variable and find the conditions for a "matrix differential operator" to be a Hamiltonian superoperator. Moreover, we prove that conformal superalgebras are equivalent to certain Hamiltonian superoperators.

Hom-Lie superalgebra structures on finite-dimensional simple Lie superalgebras

Hom-Lie superalgebras, which can be considered as a deformation of Lie superalgebras, are $\mathbb{Z}_2$-graded generalization of Hom-Lie algebras. In this paper, we prove that there is only the trivial Hom-Lie superalgebra structure over a finite-dimensional simple Lie superalgebra.

Extended Affine Lie Superalgebras

We introduce the notion of extended affine Lie superalgebras and investigate the properties of their root systems. Extended affine Lie algebras, invariant affine reflection algebras, finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras are examples of extended affine Lie superalgebras.

Loop super-Virasoro Lie conformal superalgebra

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.

Root Graded Lie Superalgebras

We define root graded Lie superalgebras and study their connection with centerless cores of extended affine Lie superalgebras; our definition generalizes the known notions of root graded Lie superalgebras.

Schramm-Loewner evolution with Lie superalgebra symmetry

We propose a generalization of Schramm-Loewner evolution (SLE) that has internal degrees of freedom described by an affine Lie superalgebra. We give a general formulation of SLE corresponding to representation theory of an affine Lie superalgebra whose underlying finite dimensional Lie superalgebra is basic classical type, and write down stochastic differential equations on internal degrees of freedom in case that the corresponding affine Lie superalgebra is $\widehat{\mathfrak{osp}(1|2)}$. We also demonstrate computation of local martingales associated with the solution from a representation …

Universally defined representations of Lie conformal superalgebras

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 …

Finite dimensional compact and unitary Lie superalgebras

Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we describe those Lie superalgebras which have a faithful finite dimensional unitary representation. We call these Lie superalgebras unitary. This is achieved by describing the classification of real finite dimensional compact simple Lie superalgebras, and analyzing, in a rather elementary and direct way, the decomposition of reductive Lie superalgebras ($\g$ is a semisimple $\g_{\bar 0}$-module) over fields of characteristic zero into ideals.

Vertex representations of toroidal special linear Lie superalgebras

Based on the loop-algebraic presentation of 2-toroidal Lie superalgebras, free field representation of toroidal Lie superalgebras of type $A(m, n)$ is constructed using both vertex operators and bosonic fields.

Locally finite basic classical simple Lie superalgebras

In this work, we study direct limits of finite dimensional basic classical simple Lie superalgebras and obtain the conjugacy classes of Cartan subalgebras under the group of automorphisms.

Structure and cohomology of 3-Lie-Rinehart superalgebras

We introduce the concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via the cohomology theory.

Infinite dimensional primitive linearly compact Lie superalgebras

We classify open maximal subalgebras of all infinite-dimensional linearly compact simple Lie superalgebras. This is applied to the classification of infinite-dimensional Lie superalgebras of vector fields, acting transitively and primitively in a formal neighborhood of a point of a finite-dimensional supermanifold.

Mixed cohomology of Lie superalgebras

We investigate a new cohomology of Lie superalgebras, which may be compared to a de Rham cohomology of Lie supergroups involving both differential and integral forms. It is defined by a BRST complex of Lie superalgebra modules, which is formulated in terms of a Weyl superalgebra and incorporates inequivalent representations of the bosonic Weyl subalgebra. The new cohomology includes the standard Lie superalgebra cohomology as a special case. Examples of new cohomology groups are computed.

Toroidal Lie superalgebras and free field representations

A loop-algebraic presentation is given for toroidal Lie superalgebras of classical types. Based on the loop superalgebra presentation free field realizations of toroidal Lie superalgebras are constructed for types $A(m,n)$, $B(m,n)$, C(n) and $D(m,n)$.

On the defining relations of the affine Lie superalgebras and their quantized universal enveloping superalgebras

In this paper, we give defining relations of the affine Lie superalgebras an and defining relations of a super-version of the Drinfeld[D]-Jimbo[J] affine quantized universal enveloping algebras. As a result, we can exactly define the affine quantized universal enveloping superalgebras with generators and relations. Moreover we give a Drinfeld's realizations of $U_h({\hat {sl}}(m|n)^{(1)})$.

Some remarks on quantized Lie superalgebras of classical type

In this paper we use the Etingof-Kazhdan quantization of Lie bi-superalgebras to investigate some interesting questions related to Drinfeld-Jimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: 1) a theorem stating that all highest weight mod…

A Hochschild's 6-term exact sequence for restricted Lie superalgebras

In [5], Hochschild established a 6-term exact sequence for the cohomology of restricted Lie algebras. We generalize this result to restricted Lie superalgebras.

关于仿射李超代数及其模的顶点超代数的积分形式

On integral forms for vertex superalgebras associated with affine Lie superalgebras and their modules

本文研究仿射顶点超代数及其模的积分形式，通过构造仿射李超代数通用包络超代数的积分形式，并利用Chevalley基推广Kostant-Garland积分形式理论，进而构造基于仿射李超代数的顶点（算子）超代数及其模的积分形式，并探讨积分形式包含共形向量以及反梯度模中的积分形式。

Yangian of the general linear Lie superalgebra

We prove several basic properties of the Yangian of the general linear Lie superalgebra.

Lie Superalgebras arising from bosonic representation

A 2-toroidal Lie superalgebra is constructed using bosonic fields and a ghost field. The superalgebra contains $osp(1|2n)^{(1)}$ as a distinguished subalgebra and behaves similarly to the toroidal Lie superalgebra of type $B(0, n)$. Furthermore this algebra is a central extension of the algebra $osp(1|2n)\otimes \mathbb C[s, s^{-1}, t,t^{-1}]$.

Minimal W-superalgebras and modular representations of basic Lie superalgebras

Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1}$ be a basic Lie superalgebra over $\mathbb{C}$, and $e$ a minimal nilpotent element in $\mathfrak{g}_{\bar 0}$. Set $W_χ'$ to be the refined $W$-superalgebra associated with the pair $(\mathfrak{g},e)$, which is called a minimal $W$-superalgebra. In this paper we present a set of explicit generators of minimal $W$-superalgebras and the commutators between them. In virtue of this, we show that over an algebraically closed field $\mathds{k}$ of characteristic $p\gg0$, the lower bounds of dimensions in the modular representations of ba…