scholarly journals Classification of simple Gelfand–Tsetlin modules of 𝔰𝔩(3)

2021 ◽  
pp. 1-109
Author(s):  
Vyacheslav Futorny ◽  
Dimitar Grantcharov ◽  
Luis Enrique Ramirez
Keyword(s):  
Lie Algebra ◽  
Infinite Dimensional ◽  
Injective Modules ◽  
Explicit Realization

We provide a classification and an explicit realization of all simple Gelfand–Tsetlin modules of the complex Lie algebra [Formula: see text]. The realization of these modules, including those with infinite-dimensional weight spaces, is given via regular and derivative Gelfand–Tsetlin tableaux. Also, we show that all simple Gelfand–Tsetlin [Formula: see text]-modules can be obtained as subquotients of localized Gelfand–Tsetlin [Formula: see text]-injective modules.

KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank

1982 ◽  
Vol 34 (6) ◽  
pp. 1215-1239 ◽  
Author(s):  
L. J. Santharoubane
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Maximal Rank ◽  
Killing Form ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Infinite Dimensional ◽  
Dimensional Version

Introduction. The natural problem of determining all the Lie algebras of finite dimension was broken in two parts by Levi's theorem:1) the classification of semi-simple Lie algebras (achieved by Killing and Cartan around 1890)2) the classification of solvable Lie algebras (reduced to the classification of nilpotent Lie algebras by Malcev in 1945 (see [10])).The Killing form is identically equal to zero for a nilpotent Lie algebra but it is non-degenerate for a semi-simple Lie algebra. Therefore there was a huge gap between those two extreme cases. But this gap is only illusory because, as we will prove in this work, a large class of nilpotent Lie algebras is closely related to the Kac-Moody Lie algebras. These last algebras could be viewed as infinite dimensional version of the semisimple Lie algebras.

scholarly journals 2-Local derivations on the W-algebra W(2,2)

2020 ◽  
pp. 2150237
Author(s):  
Xiaomin Tang
Keyword(s):  
Lie Algebra ◽  
Complete Classification ◽  
Infinite Dimensional ◽  
Local Derivation ◽  
Infinite Dimensional Lie Algebra

This paper is devoted to study 2-local derivations on [Formula: see text]-algebra [Formula: see text] which is an infinite-dimensional Lie algebra with some outer derivations. We prove that all 2-local derivations on the [Formula: see text]-algebra [Formula: see text] are derivations. We also give a complete classification of the 2-local derivation on the so-called thin Lie algebra and prove that it admits many 2-local derivations which are not derivations.

scholarly journals Classification of Kantowski–Sachs metric via conformal Ricci collineations

2017 ◽  
Vol 15 (01) ◽  
pp. 1850006 ◽  
Author(s):  
Tahir Hussain ◽  
Fawad Khan ◽  
Ashfaque H. Bokhari ◽  
Sumaira Saleem Akhtar
Keyword(s):  
Lie Algebra ◽  
Physical Interpretation ◽  
Ricci Tensor ◽  
Ricci Collineations ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Spacetime Metric

In this paper, we present a classification of the Kantowski–Sachs spacetime metric according to its conformal Ricci collineations (CRCs). Solving the CRC equations, it is shown that the Kantowski–Sachs metric admits 15-dimensional Lie algebra of CRCs when its Ricci tensor is non-degenerate and an infinite dimensional group of CRCs when the Ricci tensor is degenerate. Some examples of Kantowski–Sachs metric admitting nontrivial CRCs are presented and their physical interpretation is provided.

Tensor Product Realizations of Simple Torsion Free Modules

2001 ◽  
Vol 53 (2) ◽  
pp. 225-243 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Complex Numbers ◽  
Infinite Dimensional ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Elementary Construction ◽  
Free Modules ◽  
Torsion Free

AbstractLet be a finite dimensional simple Lie algebra over the complex numbers C. Fernando reduced the classification of infinite dimensional simple -modules with a finite dimensional weight space to determining the simple torsion free -modules for of type A or C. Thesemodules were determined by Mathieu and using his work we provide a more elementary construction realizing each one as a submodule of an easily constructed tensor product module.

scholarly journals Jet modules for the centerless Virasoro-like algebra

2019 ◽  
Vol 18 (01) ◽  
pp. 1950002 ◽  
Author(s):  
Xiangqian Guo ◽  
Genqiang Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Block Type ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Infinite Dimensional Lie Algebra ◽  
Area Preserving

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a [Formula: see text]-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over [Formula: see text]. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of [Irreducible representations for toroidal Lie algebras, J. Algebras 221 (1999) 188–231; Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004) 23–42]. Consequently, indecomposable jet modules are described using modules over the algebra [Formula: see text], which is the “positive part” of a Block type algebra studied first by [Some infinite-dimensional simple Lie algebras in characteristic [Formula: see text] related to those of Block, J. Pure Appl. Algebra 127(2) (1998) 153–165] and recently by [A [Formula: see text]-graded generalization of the Witt-algebra, preprint; Classification of simple Lie algebras on a lattice, Proc. London Math. Soc. 106(3) (2013) 508–564]).

LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART

2021 ◽  
pp. 1-41
Author(s):  
VOLODYMYR MAZORCHUK ◽  
RAFAEL MRÐEN
Keyword(s):  
Lie Algebra ◽  
Complete Classification ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Direct Sum ◽  
Levi Decomposition ◽  
Finite Dimensional ◽  
Semisimple Part

Abstract For a finite-dimensional Lie algebra $\mathfrak {L}$ over $\mathbb {C}$ with a fixed Levi decomposition $\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$ , where $\mathfrak {g}$ is semisimple, we investigate $\mathfrak {L}$ -modules which decompose, as $\mathfrak {g}$ -modules, into a direct sum of simple finite-dimensional $\mathfrak {g}$ -modules with finite multiplicities. We call such modules $\mathfrak {g}$ -Harish-Chandra modules. We give a complete classification of simple $\mathfrak {g}$ -Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak {g} = \mathfrak {sl}_2$ , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak {sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules.

scholarly journals Diamond distances in Nottingham algebras

2021 ◽  
Author(s):  
M. Avitabile ◽  
S. Mattarei
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Central Series ◽  
Graded Lie Algebras ◽  
Infinite Dimensional ◽  
Graded Lie Algebra ◽  
The Difference ◽  
Dimension One

Nottingham algebras are a class of just-infinite-dimensional, modular, [Formula: see text]-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree [Formula: see text], and the second occurs in degree [Formula: see text], a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to [Formula: see text]. A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper, we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra [Formula: see text] can be assigned a type, in such a way that the degrees and types of the diamonds completely describe [Formula: see text]. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals [Formula: see text]. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type [Formula: see text].

scholarly journals Polynomial ring representations of endomorphisms of exterior powers

2021 ◽  
Author(s):  
Ommolbanin Behzad ◽  
André Contiero ◽  
Letterio Gatto ◽  
Renato Vidal Martins
Keyword(s):  
Lie Algebra ◽  
Polynomial Ring ◽  
Vertex Operators ◽  
Exterior Power ◽  
Infinite Dimensional ◽  
Symmetric Structure ◽  
Rational Polynomials ◽  
Exterior Powers ◽  
Vertex Representation ◽  
Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

Ricci inheritance collineations in Bianchi type II spacetime

2016 ◽  
Vol 31 (17) ◽  
pp. 1650102 ◽  
Author(s):  
Tahir Hussain ◽  
Sumaira Saleem Akhtar ◽  
Ashfaque H. Bokhari ◽  
Suhail Khan
Keyword(s):  
Lie Algebra ◽  
Ricci Tensor ◽  
Bianchi Type ◽  
Complete Classification ◽  
Type Ii

In this paper, we present a complete classification of Bianchi type II spacetime according to Ricci inheritance collineations (RICs). The RICs are classified considering cases when the Ricci tensor is both degenerate as well as non-degenerate. In case of non-degenerate Ricci tensor, it is found that Bianchi type II spacetime admits 4-, 5-, 6- or 7-dimensional Lie algebra of RICs. In the case when the Ricci tensor is degenerate, majority cases give rise to infinitely many RICs, while remaining cases admit finite RICs given by 4, 5 or 6.

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