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 2-Local derivations of infinite-dimensional Lie algebras

2019 ◽  
Vol 19 (05) ◽  
pp. 2050100 ◽  
Author(s):  
Shavkat Ayupov ◽  
Baxtiyor Yusupov
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Witt Algebra ◽  
Infinite Dimensional ◽  
Local Derivation ◽  
Infinite Dimensional Lie Algebra

In the present paper, we study 2-local derivations of infinite-dimensional Lie algebras over a field of characteristic zero. We prove that all 2-local derivations of the Witt algebra as well as of the positive Witt algebra and the classical one-sided Witt algebra are (global) derivations. We also give an example of an infinite-dimensional Lie algebra with a 2-local derivation which is not a derivation.

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.

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.

Infinite-Dimensional Lie Algebra

2004 ◽  
pp. 202-202
Author(s):  
John Howie ◽  
Steven Duplij ◽  
Ali Mostafazadeh ◽  
Masaki Yasue ◽  
Vladimir Ivashchuk ◽  
...  
Keyword(s):  
Lie Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Algebra

scholarly journals Golod–Shafarevich-Type Theorems and Potential Algebras

2018 ◽  
Vol 2019 (15) ◽  
pp. 4822-4844 ◽  
Author(s):  
Natalia Iyudu ◽  
Agata Smoktunowicz
Keyword(s):  
Linear Growth ◽  
Complete Classification ◽  
Koszul Complex ◽  
Model Program ◽  
Minimal Model Program ◽  
Infinite Dimensional ◽  
Homogeneous Potential ◽  
Finite Dimensional ◽  
The Difference

Abstract Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree $n\geqslant 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class $\mathcal {P}_{n}$ of potential algebras with homogeneous potential of degree $n+1\geqslant 4$, the minimal Hilbert series is $H_{n}=\frac {1}{1-2t+2t^{n}-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

Sign in / Sign up

Export Citation Format

Share Document