Dimensional Lie Superalgebras Research Articles

Characterization of contact Lie superalgebras

We present the concept of finite-dimensional complex homogeneous contact Lie superalgebra. The Z2-graded homogeneous cases are studied in detail producing some relevant examples. We characterize homogeneous contact Lie superalgebras in terms of their Berezinian and their structure matrix. These Lie superalgebras are also characterized by means of deformation theory, and as an application we obtain the complete classification of low dimensional Lie superalgebras of this type.

Simple binary Lie and non-Lie superalgebra has solvable even part

We prove that every simple finite dimensional binary Lie superalgebra over the complex numbers field C with non-zero odd part is either a Lie superalgebra or has a solvable even part.

The Hurwitz-Hopf map and harmonic wave functions for integer and half-integer angular momentum

Harmonic wave functions for integer and half-integer angular momentum are given in terms of the Euler angles (θ, ϕ, ψ) that define a rotation in SO(3), and the Euclidean norm r in , keeping the usual meaning of the spherical coordinates (r, θ, ϕ). They form a Hilbert (super)-space decomposed in the form . Following a classical work by Schwinger, 2-dimensional harmonic oscillators are used to produce raising and lowering operators that change the total angular momentum eigenvalue of the wave functions in half units. The nature of the representation space is approached from the double covering group homomorphism SU(2) → SO(3) and the topology involved is taken care of by using the Hurwitz-Hopf map . It is shown how to reconsider H as a 2-to-1 group map, , translating it into an assignment (z 1, z 2) ↦ (r, θ, ϕ, ψ) whose domain consists of pairs (z 1, z 2) of complex variables, under the appropriate identification of with . It is shown how the Lie algebra of G 0 is coupled with two Heisenberg Lie algebras of 2-dimensional (Schwinger’s) harmonic oscillators generated by the operators and their adjoints. The whole set of operators gets algebraically closed either into a 13-dimensional Lie algebra or into a (4∣8)-dimensional Lie superalgebra. The wave functions in can be written in terms of polynomials in the complex coordinates (z 1, z 2) and their complex conjugates and the representations are explicitly constructed via the various highest weight (or lowest weight) vector representations of G 0. Finally, a new nonrelativistic quantum (Schrödinger-like) equation for the hydrogen atom that takes into account the electron spin is introduced and expressed in terms of (r, θ, ϕ, ψ) and the time t. The equation is succeptible to be solved exactly in terms of the harmonic wave functions hereby introduced.

Simple restricted modules over the N = 1 Ramond algebra as weak modules for vertex operator superalgebras

In the present paper, a class of new simple modules over the N=1 Ramond algebra are constructed, which are induced from simple modules over some finite dimensional solvable Lie superalgebras. These new modules are simple restricted modules over the N=1 Ramond algebra. Combined with the result in [22], a classification of simple weak ψ-twisted W¯(0,c)-modules under certain conditions is also given. At last, some examples of simple restricted N=1 Ramond modules as various versions of Whittaker modules are presented (classical Whittaker modules were studied in [17]).

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.

On multipliers of pairs of Lie superalgebras

We study the notion of the Schur multiplier of a pair (N, L) of Lie superalgebras and obtain some upper bounds concerning dimensions. Moreover, we characterize the pairs of finite dimensional (nilpotent) Lie superalgebras for which for t = 0; 1, where dim N = (m|n).

Superderivations of direct and semidirect sum of Lie superalgebras

It is well known that superderivation of a Lie superalgebra is certain generalization of derivation of a Lie algebra. This paper is devoted to investigate the structure and dimension of superderivation algebra Der(G) of G where G is a direct sum of two finite dimensional Lie superalgebras L and K having no non-trivial common direct factor. We also introduce some of its subsuperalgebras. Moreover, we create a condition which shows the isomorphism between superderivation of direct sum and direct sum of superderivations of two Lie superalgebras. Later on, we take G as a semidirect sum of two Lie superalgebras and obtain the structure of which is a subsuperalgebra of Der(G) that contains those superderivations mapping K to itself. Finally, we give some conditions under which is also a semidirect sum.

2-Local superderivations on the Lie superalgebra of Block type

In this paper, we study 2-local superderivations on the Lie superalgebra of Block type , which is an infinite dimensional Lie superalgebra with an outer derivation. We prove that all 2-local superderivations on are superderivations.

LD-stability for Goldie rings

The lower transcendence degree , introduced by J. J Zhang, is an important non-commutative invariant in ring theory and non-commutative geometry strongly connected to the classical Gelfand-Kirillov transcendence degree. For LD-stable algebras , the lower transcendence degree coincides with the Gelfand-Kirillov dimension. We show that the following algebras are LD -stable and compute their lower transcendence degrees: rings of differential operators of affine domains, universal enveloping algebras of finite dimensional Lie superalgebras, symplectic reflection algebras and their spherical subalgebras , finite W -algebras of type A , generalized Weyl algebras over Noetherian domain (under a mild condition), some quantum groups . We show that the lower transcendence degree behaves well with respect to the invariants by finite groups, and with respect to the Morita equivalence. Applications of these results are given.

Finite weight modules over twisted affine Lie superalgebras

This work provides the first step toward the classification of irreducible finite weight modules over twisted affine Lie superalgebras. We divide the class of such modules into two subclasses called hybrid and tight. We reduce the classification of hybrid irreducible finite weight modules to the classification of cuspidal modules of finite dimensional cuspidal Lie superalgebras which is discussed in a work of Dimitrov, Mathieu and Penkov.

Simple restricted modules for Neveu-Schwarz algebra

In this paper, we give a construction of simple modules generalizing both highest weight and Whittaker modules for the Neveu-Schwarz algebra, in the spirit of the work of Mazorchuk and Zhao on simple Virasoro modules. We establish a 1-1 correspondence between simple restricted Neveu-Schwarz modules and simple modules of a family of finite dimensional solvable Lie superalgebras associated to the Neveu-Schwarz algebra. Moreover, for two of these superalgebras all simple modules are classified.

Universal central extensions of current Lie superalgebras

We identify the universal central extension of g=A⊗k, where k is a finite dimensional perfect Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form κ which is invariant under all derivations and A is a unital supercommutative associative (super)algebra.

Some properties of isoclinism in Lie superalgebras

Isoclinism of Lie superalgebras has been defined and studied currently. In this article, it is shown that for finite dimensional Lie superalgebras of same dimension, the notation of isoclinism and isomorphism are equivalent. Furthermore, we show that covers of finite dimensional Lie superalgebras are isomorphic using isoclinism concept.

Extended affine Lie superalgebras and their affinizations

Extended affine Lie superalgebras are super versions of the defining axioms of extended affine Lie algebras or more generally invariant affine reflection algebras. This class includes finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras. In this paper, an affinization process is introduced for the class of extended affine Lie superalgebras, and the necessary conditions for an extended affine Lie superalgebra to be invariant under this process are presented. Moreover, new extended affine Lie superalgebras are constructed by means of the affinization process.

On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of Lie algebras and their simple relatives: of special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for ${\mathfrak{o}}_\Pi(8)$). The classification of gradings is new, but all of the corresponding superizations are known. For the simple derived Zassenhaus algebras of height $n>1$, there is an $(n-2)$-parametric family of modulo 2 gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves non-triviality of a deformation of a simple $3|2$-dimensional Lie superalgebra (new result).

Pauli gradings on Lie superalgebras and graded codimension growth

We introduce grading on certain finite dimensional simple Lie superalgebras of type P(t) by elementary abelian 2-group. This grading gives rise to Pauli matrices and is a far generalization of (Z2×Z2)-grading on Lie algebra of (2×2)-traceless matrices.We use this grading for studying numerical invariants of polynomial identities of Lie superalgebras. In particular, we compute graded PI-exponent corresponding to Pauli grading.

Vertex operator superalgebras associated with affine Lie superalgebras

ABSTRACTIt is proved that the induced module of level k associated with a finite dimensional simple Lie superalgebra 𝔤 with a nondegenerate even supersymmetric invariant bilinear form is a ℤ-graded vertex operator superalgebra, and the explicit expression for the virasoro algebra on any twisted -module is given.

A Class of Finite Dimensional Modular Lie Superalgebras of Special Type

This paper is concerned with the Lie superalgebra S(n, m) of special type over a field of prime characteristic. We construct the modular Lie superalgebra S(n, m) and discuss some properties of this algebra. Then the derivation superalgebra of S(n, m) is determined.

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 g0¯-module) over fields of characteristic zero into ideals.

Infinitesimal supersymmetries over Lie groups and their classification for gl(1,1) and sl(1,1)

Infinitesimal supersymmetries over classical Lie groups that do not necessarily integrate to Lie supergroups are described. They yield a notion of supersymmetry that is less rigid than the assumption of a Lie supergroup action but still implies an underlying action of a Lie group. In contrast to Lie supergroups, the arising representation-theoretical Lie supergroups (RTLSG) occur as families associated to Harish-Chandra superpairs. However morphisms of RTLSGs directly correspond to morphisms of Harish-Chandra superpairs. Particular RTLSGs can be derived from the explicit constructions of Lie supergroups given by Berezin and Kostant. The Lie superalgebras $\mathfrak{gl}(1,1)$ or $\mathfrak{sl}(1,1)$ appearing also in higher dimensional classical Lie superalgebras, provide interesting first examples of RTLSGs. A classification of RTLSGs associated to real and complex $\mathfrak{gl}(1,1)$- and $\mathfrak{sl}(1,1)$-Harish-Chandra superpairs is given by parameter spaces and complete sets of invariants. The underlying Lie group is assumed to be connected but possibly not simply connected.