scholarly journals Irreducible continuous representations of the simple linearly compact n-Lie superalgebra of type S

2019 ◽  
Vol 18 (02) ◽  
pp. 1950036
Author(s):  
Carina Boyallian ◽  
Vanesa Meinardi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Superalgebra ◽  
Bijective Correspondence ◽  
Cartan Type ◽  
Type Formula ◽  
Continuous Representations

In the present paper, we classify all irreducible continuous representations of the simple linearly compact [Formula: see text]-Lie superalgebra of type [Formula: see text]. The classification is based on a bijective correspondence between the continuous representations of the [Formula: see text]-Lie algebras [Formula: see text] and continuous representations of the Lie algebra of Cartan type [Formula: see text], on which some two-sided ideal acts trivially.

Restricted Envelopes of Lie Superalgebras

2015 ◽  
Vol 22 (02) ◽  
pp. 309-320
Author(s):  
Liping Sun ◽  
Wende Liu ◽  
Xiaocheng Gao ◽  
Boying Wu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Cartan Type ◽  
Modular Lie Algebras ◽  
Modular Lie Superalgebras

Certain important results concerning p-envelopes of modular Lie algebras are generalized to the super-case. In particular, any p-envelope of the Lie algebra of a Lie superalgebra can be naturally extended to a restricted envelope of the Lie superalgebra. As an application, a theorem on the representations of Lie superalgebras is given, which is a super-version of Iwasawa's theorem in Lie algebra case. As an example, the minimal restricted envelopes are computed for three series of modular Lie superalgebras of Cartan type.

Some studies on central derivation of nilpotent Lie superalgebra

2018 ◽  
Vol 13 (04) ◽  
pp. 2050068
Author(s):  
Rudra Narayan Padhan ◽  
K. C. Pati
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Nilpotent Lie Algebra ◽  
New Concepts

Many theorems and formulas of Lie superalgebras run quite parallel to Lie algebras, sometimes giving interesting results. So it is quite natural to extend the new concepts of Lie algebra immediately to Lie superalgebra case as the later type of algebras have wide applications in physics and related theories. Using the concept of isoclinism, Saeedi and Sheikh-Mohseni [A characterization of stem algebras in terms of central derivations, Algebr. Represent. Theory 20 (2017) 1143–1150; On [Formula: see text]-derivations of Filippov algebra, to appear in Asian-Eur. J. Math.; S. Sheikh-Mohseni, F. Saeedi and M. Badrkhani Asl, On special subalgebras of derivations of Lie algebras, Asian-Eur. J. Math. 8(2) (2015) 1550032] recently studied the central derivation of nilpotent Lie algebra with nilindex 2. The purpose of the present paper is to continue and extend the investigation to obtain some similar results for Lie superalgebras, as isoclinism in Lie superalgebra is being recently introduced.

scholarly journals Fermionic realization of twisted toroidal Lie algebras

2020 ◽  
pp. 2150143
Author(s):  
Naihuan Jing ◽  
Chad R. Mangum ◽  
Kailash C. Misra
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Type Formula ◽  
Toroidal Lie Algebra

In this paper, we construct a fermionic realization of the twisted toroidal Lie algebra of type [Formula: see text] and [Formula: see text] based on the newly found Moody–Rao–Yokonuma-like presentation.

A geometry of the generalized quadrangle (Ω,𝔏) of type O6−(2) and Lie algebras of type E6 with characteristic two

2019 ◽  
Vol 19 (01) ◽  
pp. 2050016
Author(s):  
Yousuf Alkhezi ◽  
Mashhour Bani Ata
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Generalized Quadrangle ◽  
Geometric Properties ◽  
Type Formula ◽  
Characteristic Two

The purpose of this paper is to study certain geometric properties of generalized quadrangle [Formula: see text] of type [Formula: see text]. We define certain root elements which generate a Lie algebra of type [Formula: see text] for fields [Formula: see text] of characteristic two. The construction will be mainly based on the geometric properties of the generalized quadrangle [Formula: see text]. In fact, we will explicity construct a Chevalley base of this Lie algebra.

On root-involutions and root-subgroups of E6(K) for fields K of characteristic two

2019 ◽  
Vol 18 (01) ◽  
pp. 1950017 ◽  
Author(s):  
S. Aldhafeeri ◽  
M. Bani-Ata
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Algebra ◽  
Chevalley Group ◽  
Chevalley Groups ◽  
Type Formula ◽  
Characteristic Two

The purpose of this paper is to investigate the root-involutions and root-subgroups of the Chevalley group [Formula: see text] for fields [Formula: see text] of characteristic two. The approach we follow is elementary and self-contained depends on the notion of [Formula: see text]-sets which we have introduced in [Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type [Formula: see text] for fields of characteristic two, Beit. Algebra Geom. 58 (2017) 529–534]. The approach is elementary on the account that it consists of little more than naive linear algebra. It is remarkable to mention that Chevalley groups over fields of characteristic two have not much been researched. This work may contribute in this regard. This paper is divided into three main sections: the first section is a combinatorial section, the second section is on relations among [Formula: see text]-sets, the last one is on Lie algebra.

DERIVATION ALGEBRAS OF RINGS RELATED TO HEISENBERG ALGEBRAS

2004 ◽  
Vol 03 (02) ◽  
pp. 181-191 ◽  
Author(s):  
JEFFREY BERGEN
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Quotient Ring ◽  
Finite Codimension ◽  
Enveloping Algebras ◽  
Cartan Type ◽  
Direct Sum ◽  
Heisenberg Algebras ◽  
Martindale Quotient Ring ◽  
The Ideal

In this paper, we will determine the Lie algebra of derivations of rings which are generalizations of the enveloping algebras of Heisenberg Lie algebras. First, we will determine which derivations are X-inner and also determine which elements in the Martindale quotient ring induce X-inner derivations. Then, we will show that the Lie algebra of derivations is the direct sum of the ideal of X-inner derivations and a subalgebra which is isomorphic to a subalgebra of finite codimension in a Cartan type Lie algebra.

scholarly journals FUSION RULES AND COMPLETE REDUCIBILITY OF CERTAIN MODULES FOR AFFINE LIE ALGEBRAS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350062 ◽  
Author(s):  
DRAŽEN ADAMOVIĆ ◽  
OZREN PERŠE
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Operator Algebras ◽  
Affine Lie Algebras ◽  
Fusion Rules ◽  
Type Formula ◽  
Complete Reducibility ◽  
Affine Lie Algebra ◽  
Branching Rules ◽  
Level 1

We develop a new method for obtaining branching rules for affine Kac–Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in our previous paper J. Algebra319 (2008) 2434–2450, is closed under fusion. Then, we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type [Formula: see text], obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for ℓ ≥ 3. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type [Formula: see text]. Next, we notice that the category of [Formula: see text] modules at level -2ℓ + 3 has the isomorphic fusion algebra. This enables us to decompose certain [Formula: see text] and [Formula: see text]-modules at negative levels.

scholarly journals The group of automorphisms of the Lie algebra of derivations of a polynomial algebra

2017 ◽  
Vol 16 (05) ◽  
pp. 1750088 ◽  
Author(s):  
V. V. Bavula
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Short Proof ◽  
Polynomial Algebra ◽  
Group Of Automorphisms ◽  
Cartan Type

A short proof is given of Rudakov’s result (announced in [A. N. Rudakov, Subalgebras and automorphisms of Lie algebras of Cartan type, Funktsional. Anal. i Prilozhen. 20(1) (1986) 83–84]), that the group of automorphisms of the Lie algebra [Formula: see text] of derivations of a polynomial algebra [Formula: see text], over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the polynomial algebra [Formula: see text].

Automorphism groups of modular graded Lie superalgebras of Cartan-type

2017 ◽  
Vol 16 (03) ◽  
pp. 1750050
Author(s):  
Wende Liu ◽  
Jixia Yuan
Keyword(s):  
Automorphism Groups ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Infinite Dimensional ◽  
Cartan Type ◽  
Type Formula ◽  
Finite Dimensional ◽  
Normal Series

Suppose the underlying field is of characteristic [Formula: see text]. In this paper, we prove that the automorphisms of the finite-dimensional graded (non-restircited) Lie superalgebras of Cartan-type [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text] can uniquely extend to the ones of the infinite-dimensional Lie superalgebra of Cartan-type [Formula: see text]. Then a concrete group embedding from [Formula: see text] into [Formula: see text] is established, where [Formula: see text] is any finite-dimensional Lie superalgebra of Cartan-type [Formula: see text] or [Formula: see text] and [Formula: see text] is the underlying (associative) superalgebra of [Formula: see text]. The normal series of the automorphism groups of [Formula: see text] are also considered.

scholarly journals Classification of Simple Lie Superalgebras in Characteristic 2

2021 ◽  
Author(s):  
Sofiane Bouarroudj ◽  
Alexei Lebedev ◽  
Dimitry Leites ◽  
Irina Shchepochkina
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Simple Lie Algebra ◽  
Hamiltonian Vector Fields ◽  
Finite Dimensional ◽  
Characteristic 2

Abstract All results concern characteristic 2. We describe two procedures; each of which to every simple Lie algebra assigns a simple Lie superalgebra. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures. For Lie algebras, in addition to the known “classical” restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: a $2|4$-structure, which is a direct analog of the classical restrictedness, and a novel $2|2$-structure—one more analog, a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras known only for non-alternating orthogonal and Hamiltonian series.

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