Lie Superalgebra Of Vector Fields Research Articles

Homogeneous superstrings with retract $CP^{1|4}$

Any complex-analytic supermanifold whose retract is diffeomorphic to the complex projective superline (superstring) $CP^{1|4}$ is, up to a diffeomorphism, either a member of a 1-parameter family or one of 9 exceptional supermanifolds. I singled out the homogeneous of these supermanifolds and described Lie superalgebras of vector fields on them.

Transitive irreducible Lie superalgebras of vector fields

Let $\mathfrak{d}$ be the Lie superalgebra of superderivations of the sheaf of sections of the exterior algebra of the homogeneous vector bundle $E$ over the flag variety $G/P$, where $G$ is a simple finite-dimensional complex Lie group and $P$ its parabolic subgroup. Then, $\mathfrak{d}$ is transitive and irreducible whenever $E$ is defined by an irreducible $P$-module $V$ such that the highest weight of $V^*$ is dominant. Moreover, $\mathfrak{d}$ is simple; it is isomorphic to the Lie superalgebra of vector fields on the superpoint, i.e., on a $0|n$-dimensional supervariety.

Analogs of Bol operators on superstrings

The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the [Formula: see text]-dimensional supermanifold (superstring) [Formula: see text], we classify analogs of Bol operators invariant under the simple maximal subalgebra [Formula: see text] of the same rank as its simple ambient superalgebra [Formula: see text] of vector fields on [Formula: see text] and containing all elements of negative degree of [Formula: see text] in a [Formula: see text]-grading. We also consider the Lie superalgebras of vector fields [Formula: see text] preserving a contact structure on the superstring [Formula: see text]. We have discovered many new operators.

Generators of simple infinite-dimensional Lie superalgebras of vector fields

In this paper we prove that simple infinite-dimensional Lie superalgebras of vector fields can be generated by 1 element over the field C of complex numbers.

Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields

Abstract According to the classification by Kac, there are eight Cartan series and five exceptional Lie superalgebras in infinite-dimensional simple linearly compact Lie superalgebras of vector fields. In this paper, the Hom-Lie superalgebra structures on the five exceptional Lie superalgebras of vector fields are studied. By making use of the ℤ-grading structures and the transitivity, we prove that there is only the trivial Hom-Lie superalgebra structures on exceptional simple Lie superalgebras. This is achieved by studying the Hom-Lie superalgebra structures only on their 0-th and (−1)-th ℤ-components.

Bicovariant differential calculus on the quantum superspace ℝq(1|2)

Super-Hopf algebra structure on the function algebra on the extended quantum superspace has been defined. It is given a bicovariant differential calculus on the superspace. The corresponding (quantum) Lie superalgebra of vector fields and its Hopf algebra structure are obtained. The dual Hopf algebra is explicitly constructed. A new quantum supergroup that is the symmetry group of the differential calculus is found.

Hom–Lie superalgebra structures on infinite-dimensional simple Lie superalgebras of vector fields

This paper considers the multiplicative Hom-Lie superalgebra structures on infinite dimensional simple Lie superalgebras of vector fields with characteristic zero. The main result is that there is only the multiplicative Hom-Lie superalgebra structure on these Lie superalgebras.

About the Cohomology of the Lie Superalgebra of Vector Fields on ℝ n|n

In this article, we compute the first space of cohomology of Vect (ℝ n|n ), the Lie superalgebra of vector fields on the supermanifold ℝ n|n with coefficients in 𝒻 (ℝ n|n ), the space of smooth functions on ℝ n|n . We give a super analog of the cohomologies of vector fields that where studied for instance by Fuchs [2]. This work allows us to classify the deformations of the action of Vect(ℝ n|n ) on 𝒻 (ℝ n|n ).

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.

Structures of G(2) Type and Nonintegrable Distributions in Characteristic p

Lately we observe: (1) an upsurge of interest (in particular, triggered by a paper by Atiyah and Witten) to manifolds with G(2)-type structure; (2) classifications are obtained of simple (finite dimensional and graded vectorial) Lie superalgebras over fields of complex and real numbers and of simple finite dimensional Lie algebras over algebraically closed fields of characteristic p>3; (3) importance of nonintegrable distributions in (1) -- (2). We add to interrelation of (1)--(3) an explicit description of several exceptional simple Lie algebras for p=2, 3 (Melikyan algebras; Brown, Ermolaev, Frank, and Skryabin algebras) as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by G(2), O(7), Sp(4) and Sp(10). The description is performed in terms of Cartan-Tanaka-Shchepochkina prolongs and is similar to descriptions of simple Lie superalgebras of vector fields with polynomial coefficients. Our results illustrate usefulness of Shchepochkina's algorithm and SuperLie package; two families of simple Lie algebras found in the process might be new.

REPRESENTATIONS OF THE EXCEPTIONAL LIE SUPERALGEBRA E(3,6) III: CLASSIFICATION OF SINGULAR VECTORS

We continue the study of irreducible representations of the exceptional Lie superalgebra E(3,6). This is one of the two simple infinite-dimensional Lie superalgebras of vector fields which have a Lie algebra sℓ(3) × sℓ(2) × gℓ(1) as the zero degree component of its consistent ℤ-grading. We provide the classification of the singular vectors in the degenerate Verma modules over E(3,6), completing thereby the classification and construction of all irreducible E(3,6)-modules that are L0-locally finite.

Errata to "Structure of Some ℤ-graded Lie Superalgebras of Vector Fields''

N. CANTARINIDipartimento di MatematicaPura ed ApplicataUniversita di Padova,via Belzoni 7,35131 Padova, Italycantarin@math.unipd.itS.{J. CHENGDepartment of Mathematics,National Cheng{Kung University,Tainan, Taiwanchengsj@mail.ncku.edu.twV. KACDepartment of MathematicsMIT, CambridgeMA 02139, USAkac@math.mit.eduWe shall give here the correct description of the Lie superalgebra E(3;8) presentedat the end of [Section 5.3, CK]. The superalgebra E(3;8) has even part E(3;8)

Representations of the exceptional Lie superalgebra E(3,6) I: Degeneracy conditions

Recently one of the authors obtained a classification of simple infinite dimensional Lie superalgebras of vector fields which extends the well known classification of E. Cartan in the Lie algebra case. The list consists of many series defined by simple equations, and of several exceptional superalgebras, among themE(3, 6).In the article we study irreducible representations of the exceptional Lie superalgebraE(3, 6). This superalgebra hassℓ(3)×sℓ(2)×gℓ(1) as the zero degree component of the consistent ℤ-grading which leads us to believe that its representation theory has potential for physical applications.

Representations of the exceptional lie superalgebraE(3, 6) I: Degeneracy conditions

Recently one of the authors obtained a classification of simple infinite dimensional Lie superalgebras of vector fields which extends the well known classification of E. Cartan in the Lie algebra case. The list consists of many series defined by simple equations, and of several exceptional superalgebras, among themE(3, 6).

A Program for Computing the Cohomology of Lie Superalgebras of Vector Fields

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. At present, since the required algebraic computations are very tedious, the cohomology is explicitly computed only in a few cases for different classes of Lie (super)algebras. That is why application of computer algebra methods is important for this problem. We describe an algorithm (and its C implementation) for computing the cohomology of Lie algebras and superalgebras. In elaborating the algorithm, we focused mainly on the cohomology with coefficients in trivial, adjoint, and coadjoint modules for Lie (super)algebras of the formal vector fields. These algebras have many applications to modern supersymmetric models of theoretical and mathematical physics. As an example, we consider the cohomology of the Poisson algebra Po(2) with coefficients in the trivial module and present 3- and 5-cocycles found by a computer. Bibliography: 6 titles.

The Shapovalov Determinant for the Poisson Superalgebras

Among simple ℤ-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C 2: these are the Lie superalgebra of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra of special Hamiltonian fields in 2k odd indeterminates, and the Kac–Moody version of . Using C 2 we compute N. Shapovalov determinant for and , and for the Poisson superalgebras associated with . A. Shapovalov described irreducible finite dimensional representations of and ; we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over and

The five exceptional simple Lie superalgebras of vector fields and their fourteen regradings

The five simple exceptional complex Lie superalgebras of vector fields are described. One of them, f a s \mathfrak {fas} , is new; the other four are explicitly described for the first time. All nonisomorphic maximal subalgebras of finite codimension of these Lie superalgebras, i.e., all other realizations of these Lie superalgebras as Lie superalgebras of vector fields, are also described; there are 14 of them altogether. All of the exceptional Lie superalgebras are obtained with the help of the Cartan prolongation or a generalized prolongation.

Explicit Bracket in an Exceptional Simple Lie Superalgebra

This note is devoted to a more detailed description of one of the five simple exceptional Lie superalgebras of vector fields, [Formula: see text] a subalgebra of [Formula: see text]. We derive differential equations for its elements, and solve these equations. Hence we get an exact form for the elements of [Formula: see text]. Moreover we realize [Formula: see text] by "glued" pairs of generating functions on a (3∣3)-dimensional periplectic (odd symplectic) supermanifold and describe the bracket explicitly.

Conformal Lie superalgebras and moduli spaces

A conformal Lie superalgebra is a superextension of the centerless Virasoro algebra W—the Lie algebra of complex vector fields on the circle. The algebras of Ramond and Neveu-Schwarz are not the only examples of such superalgebras. All known superconformal algebras can be obtained as comlexifications of Lie superalgebras of vector fields on a supercircle with an additional structure. For every such superalgebra ▪ a class of geometric objects—complex ▪— is defined. For the superalgebras of Neveu-Schwarz and Ramond they are super Riemann surfaces with punctures of different kinds. We construct moduli superspaces for compact ▪, and show that the superalgebra ▪ acts infinitesimally on the corresponding moduli space.

Irreducible representations of Lie superalgebras of vector fields and invariant differential operators

Irreducible representations of Lie superalgebras of vector fields and invariant differential operators