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.

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.

Generalized Heisenberg Algebras and Toroidal Lie Algebras

2010 ◽  
Vol 17 (03) ◽  
pp. 375-388 ◽  
Author(s):  
Yingjue Fang ◽  
Liangang Peng
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Heisenberg Algebra ◽  
Simple Lie Algebras ◽  
Simple Lie Algebra ◽  
Heisenberg Algebras ◽  
Toroidal Lie Algebra

In this article we provide two kinds of infinite presentations of toroidal Lie algebras. At first we define generalized Heisenberg algebras and prove that each toroidal Lie algebra is an amalgamation of a simple Lie algebra and a generalized Heisenberg algebra in the sense of Saito and Yoshii. This is one kind of presentations of toroidal Lie algebras given by the generators of generalized Heisenberg algebras and the Chevalley generators of simple Lie algebras with certain amalgamation relations. Secondly by using the generalized Chevalley generators, we give another kind of presentations. These two kinds of presentations are different from those given by Moody, Eswara Rao and Yokonuma.

Automorphism Group of a Class of Gradation Shifting Toroidal Lie Algebras

2010 ◽  
Vol 17 (04) ◽  
pp. 705-720 ◽  
Author(s):  
Zhangsheng Xia ◽  
Shaobin Tan ◽  
Haifeng Lian
Keyword(s):  
Automorphism Group ◽  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Groups ◽  
Type B ◽  
Laurent Polynomials ◽  
Lie Bracket ◽  
Toroidal Algebra ◽  
Toroidal Lie Algebra

Let [Formula: see text] be the ring of Laurent polynomials in commuting variables. As a generalization of the toroidal Lie algebra, the gradation shifting toroidal Lie algebra [Formula: see text] is isomorphic to the corresponding (centerless) toroidal Lie algebra so(n, ℂ) ⨂ A of type B or D as a vector space, with the Lie bracket twisted by n fixed elements E1,…,En from A. In this paper, we study the automorphisms of the gradation shifting toroidal algebra [Formula: see text], which is proved to be closely related to a class of subgroups of GL(n,ℤ), called the linear groups over semilattices. We use the linear group over a special semilattice to determine the automorphism group of the gradation shifting toroidal algebra [Formula: see text], which extends our earlier work.

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.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

scholarly journals Structure of n-Lie Algebras with Involutive Derivations

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Ruipu Bai ◽  
Shuai Hou ◽  
Yansha Gao
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Two Dimensional

We study the structure of n-Lie algebras with involutive derivations for n≥2. We obtain that a 3-Lie algebra A is a two-dimensional extension of Lie algebras if and only if there is an involutive derivation D on A=A1  ∔  A-1 such that dim A1=2 or dim A-1=2, where A1 and A-1 are subspaces of A with eigenvalues 1 and -1, respectively. We show that there does not exist involutive derivations on nonabelian n-Lie algebras with n=2s for s≥1. We also prove that if A is a (2s+2)-dimensional (2s+1)-Lie algebra with dim A1=r, then there are involutive derivations on A if and only if r is even, or r satisfies 1≤r≤s+2. We discuss also the existence of involutive derivations on (2s+3)-dimensional (2s+1)-Lie algebras.

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