Recognition of a Lie algebra given by its structure constants

2014 ◽  
pp. 37-37
Keyword(s):  
Lie Algebra ◽  
Structure Constants

scholarly journals Pseudo-metric 2-step nilpotent Lie algebras

2018 ◽  
Vol 18 (2) ◽  
pp. 237-263 ◽  
Author(s):  
Christian Autenried ◽  
Kenro Furutani ◽  
Irina Markina ◽  
Alexander Vasiľev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nilpotent Lie Algebra ◽  
Rational Structure ◽  
Lie Triple System ◽  
Nilpotent Lie Algebras ◽  
Lie Bracket ◽  
Orthogonal Lie Algebra ◽  
Structure Constants ◽  
Scalar Products

Abstract The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that a 2-step nilpotent Lie algebra is isomorphic to its standard pseudo-metric form, that is a 2-step nilpotent Lie algebra endowed with some standard non-degenerate scalar product compatible with the Lie bracket. This choice of the standard pseudo-metric form allows us to study the isomorphism properties. If the elements of the centre of the standard pseudo-metric form constitute a Lie triple system of the pseudo-orthogonal Lie algebra, then the original 2-step nilpotent Lie algebra admits integer structure constants. Among particular applications we prove that pseudo H-type algebras have bases with rational structure constants, which implies that the corresponding pseudo H-type groups admit lattices.

scholarly journals On Some Structural Components of Nilsolitons

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Hulya Kadioglu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Identity ◽  
Sufficient Conditions ◽  
Structural Components ◽  
Nilpotent Lie Algebras ◽  
Structure Constants ◽  
Nilsoliton Metric

In this paper, we study nilpotent Lie algebras that admit nilsoliton metric with simple pre-Einstein derivation. Given a Lie algebra η , we would like to compute as much of its structure as possible. The structural components we consider in this study are the structure constants, the index, and the rank of the nilsoliton derivations. For this purpose, we prove necessary or sufficient conditions for an algebra to admit such metrics. Particularly, we prove theorems for the computation of the Jacobi identity for a given algebra so that we can solve the system of the equation(s) and find the structure constants of the nilsoliton.

scholarly journals Exploring exceptional Drinfeld geometries

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Chris D. A. Blair ◽  
Daniel C. Thompson ◽  
Sofia Zhidkova
Keyword(s):  
Field Theory ◽  
Lie Algebra ◽  
Algebraic Structure ◽  
Leibniz Algebra ◽  
Maximal Supergravity ◽  
Structure Constants

Abstract We explore geometries that give rise to a novel algebraic structure, the Exceptional Drinfeld Algebra, which has recently been proposed as an approach to study generalised U-dualities, similar to the non-Abelian and Poisson-Lie generalisations of T-duality. This algebra is generically not a Lie algebra but a Leibniz algebra, and can be realised in exceptional generalised geometry or exceptional field theory through a set of frame fields giving a generalised parallelisation. We provide examples including “three-algebra geometries”, which encode the structure constants for three-algebras and in some cases give novel uplifts for CSO(p, q, r) gaugings of seven-dimensional maximal supergravity. We also discuss the M-theoretic embedding of both non-Abelian and Poisson-Lie T-duality.

scholarly journals On the Monodromy of Meromorphic Cyclic Opers on the Riemann Sphere

2020 ◽  
Author(s):  
Charles LeBarron Alley
Keyword(s):  
Lie Algebra ◽  
Riemann Sphere ◽  
Irreducible Representations ◽  
System Of Equations ◽  
Isomonodromic Deformations ◽  
Direct Sum ◽  
Structure Constants ◽  
Monodromy Map

Abstract We study the monodromy of meromorphic cyclic $\textrm{SL}(n,{\mathbb{C}})$-opers on the Riemann sphere with a single pole. We prove that the monodromy map, sending such an oper to its Stokes data, is an immersion in the case where the order of the pole is a multiple of $n$. To do this, we develop a method based on the work of Jimbo, Miwa, and Ueno from the theory of isomonodromic deformations. Specifically, we introduce a system of equations that is equivalent to the isomonodromy equations of Jimbo–Miwa–Ueno, but which is adapted to the decomposition of the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$ as a direct sum of irreducible representations of $\mathfrak{sl}(2,\mathbb{C})$. Using properties of some structure constants for $\mathfrak{sl}(n,\mathbb{C})$ to analyze this system of equations, we show that deformations of certain families of cyclic $\textrm{SL}(n,\mathbb{C})$-opers on the Riemann sphere with a single pole are never infinitesimally isomonodromic.

On isomorphism criterion for a subclass of complex filiform Leibniz algebras

2017 ◽  
Vol 27 (07) ◽  
pp. 953-972
Author(s):  
I. S. Rakhimov ◽  
A. Kh. Khudoyberdiyev ◽  
B. A. Omirov ◽  
K. A. Mohd Atan
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Central Extension ◽  
Linear Deformation ◽  
Leibniz Algebras ◽  
Structure Constants ◽  
Filiform Lie Algebras ◽  
Filiform Lie Algebra ◽  
Isomorphism Criterion

In this paper, we present an algorithm to give the isomorphism criterion for a subclass of complex filiform Leibniz algebras arising from naturally graded filiform Lie algebras. This subclass appeared as a Leibniz central extension of a linear deformation of filiform Lie algebra. We give the table of multiplication choosing appropriate adapted basis, identify the elementary base changes and describe the behavior of structure constants under these base changes, then combining them the isomorphism criterion is given. The final result of calculations for one particular case also is provided.

scholarly journals Riemannian Geometry Framed as a Generalized Heisenberg Lie Algebra

2019 ◽  
Author(s):  
Joseph E. Johnson
Keyword(s):  
Quantum Theory ◽  
Lie Algebra ◽  
Riemannian Geometry ◽  
Fourier Transforms ◽  
Heisenberg Algebra ◽  
Space Time ◽  
Einstein Equations ◽  
Curved Space ◽  
Mathematical Generalization ◽  
Structure Constants

The Heisenberg Lie algebra (HA) plays an important role in mathematics with Fourier transforms, as well as for the foundations of quantum theory where it expresses the operators of space-time, X, and their commutation rules with the momentum operators, D, that execute infinitesimal translations in X. Yet it is known that space-time is curved and thus the D operators must interfere thus giving “structure constants” that vary with location which suggests a mathematical generalization of the concept of a Lie algebra to allow for “structure constants” that are functions of X. We here investigate the mathematics of such a “generalized Heisenberg algebra” (GHA) which has “structure constants” that are functions of X and thus are in the enveloping algebra rather than constants. As expected, the Jacobi identity no longer holds globally but only in small regions of space-time where the [D, X] commutator can be considered locally constant and thus where one has a true Lie algebra. We show that one is able to reframe Riemannian geometry in this GHA. As an example, it is then shown that one can express the Einstein equations of general relativity as commutation rules. If one requires that the GHA commutator reduces to the HA of quantum theory in the limit of no curvature, then there are observable effects for quantum theory in this curved space time.

scholarly journals Computations in finite-dimensional Lie algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
A. M. Cohen ◽  
W. A. Graaf ◽  
L. Rónyai
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Collaborative Effort ◽  
Finite Dimensional ◽  
Levi Subalgebra ◽  
International Audience ◽  
Structure Constants ◽  
Technology Foundation ◽  
Finite Dimensional Lie Algebras ◽  
Made In

International audience This paper describes progress made in context with the construction of a general library of Lie algebra algorithms, called ELIAS (Eindhoven Lie Algebra System), within the computer algebra package GAP. A first sketch of the package can be found in Cohen and de Graaf[1]. Since then, in a collaborative effort with G. Ivanyos, the authors have continued to develop algorithms which were implemented in ELIAS by the second author. These activities are part of a bigger project, called ACELA and financed by STW, the Dutch Technology Foundation, which aims at an interactive book on Lie algebras (cf. Cohen and Meertens [2]). This paper gives a global description of the main ways in which to present Lie algebras on a computer. We focus on the transition from a Lie algebra abstractly given by an array of structure constants to a Lie algebra presented as a subalgebra of the Lie algebra of n×n matrices. We describe an algorithm typical of the structure analysis of a finite-dimensional Lie algebra: finding a Levi subalgebra of a Lie algebra.

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