alexa
Reach Us +44-1625-708989
Matrix Representation for Seven-Dimensional Nilpotent Lie Algebras | OMICS International
ISSN: 2090-0902
Journal of Physical Mathematics
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on Medical, Pharma, Engineering, Science, Technology and Business
All submissions of the EM system will be redirected to Online Manuscript Submission System. Authors are requested to submit articles directly to Online Manuscript Submission System of respective journal.

Matrix Representation for Seven-Dimensional Nilpotent Lie Algebras

Ghanam R1*, Basim Mustafa B2, Mustafa MT3 and Thompson G4

1Department of Liberal Arts and Sciences, Virginia Commonwealth University in Qatar, Qatar

2Department of Mathematics, An-Najah National University, Palestine

3Department of Mathematics, Statistics and Physics, Qatar University, Qatar

4Department of Mathematics, University of Toledo, USA

*Corresponding Author:
Ghanam R
Department of Liberal Arts and Sciences
Virginia Commonwealth University in Qatar, Qatar
Tel: +974 4402 0795
E-mail: [email protected]

Received Date: September 15, 2015 Accepted Date: January 12, 2016 Published Date: January 30, 2016

Citation: Ghanam R, Basim Mustafa B, Mustafa MT, Thompson G (2016) Matrix Representation for Seven-Dimensional Nilpotent Lie Algebras. J Phys Math 7:155. doi:10.4172/2090-0902.1000155

Copyright: © 2016 Ghanam R, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Visit for more related articles at Journal of Physical Mathematics

Abstract

This paper is concerned with finding linear representations for seven-dimensional real, indecomposable nilpotent Lie algebras. We consider the first 39 algebras presented in Gong’s classification which was based on the upper central series dimensions. For each algebra, we give a corresponding matrix Lie group, a representation of the Lie algebra in terms of left-invariant vector field and left-invariant one forms.

Keywords

Lie algebra; Lie group; Representation, Nilpotent

Introduction

Given a real Lie algebra g of dimension n a well known theorem due to Ado [1,2] asserts that g has a faithful representation as a subalgebra of gl(p,R) for some p. The theorem does not give much information about the value of p but leads one to believe that p may be very large in relation to the size of n and consequently it seems to be of limited practical value. In previous work we have found linear representation for all indecompsable real Lie algebras in dimesion six or less [3-5] In this paper we are concerned with finding linear representations for seven-dimensional nilpotent Lie algebras. Let g be a nilpotent Lie algebra, we shall assume throughout that g is indecomposable in the sense that g is not isomorphic to a direct sum of two proper nilpotent ideals. If g is decomposable and we have representations for both factors then we can easily find a representation for g. It is well known that a nilpotent Lie algebra has a non-trivial center and so finding representations is difficult because the adjoint representation is not faithful. In this article, we consider the list of seven-dimensional nilpoten Lie algebras over R classified by Gong [6], his list contained 147 indecomposable Lie algebras that are classifed according to the upper cenral series. We consider the first 39 cases for which the dimensions the upper centeral series are (27) (37) (247) (257) and (357). For each algebra g we give a corresponding Lie group that is a subgroup of GL(7,R). The representation for the Lie algebra is then easily obtained by differentiating and evaluating at the identity. The structure of the paper is as follows: in Section 2, we give a brief history and background on the classification of the seven-dimensional nilpotent Lie algebras, In section 3, we give a matrix Lie gorup that corresponds to each Lie algebra. We also give a representation of the Lie algebra in terms of left-invariant one-forms and left-invariant vectros fields. Throught out the paper we will use (p,q,r,x,y,z,w) as local coordinates on Lie groups.

Classifying Nilpotent Lie Algebras in Dimension Seven

Classification of solvable Lie algebras is not an easy problem, this is due to the fact the solvable Lie algebras are unlike semisimple Lie algebras. Semisimple Lie algebras are considered very beautiful since over the complex numbers we have the killing form, Dynkin diagrams, root space decompositions, the Serre representation, the theory of of highest weight represenation, the Weyl character formula and much more [7-10]. On the other hand, we have Lie , Engel and Ado’s theorms for solvable Lie algebras. There has been several attempts to classify the seven-dimensional nilpotent Lie algebras, we will mention the most recent ones that we are aware of. In 1993, Seeley [10] gave a classification of over the filed of comples numbers. His classification was based on the upper central series of the Lie algebras and knowledge of all lower dimensional nilpotent Lie algebras. His list contained 161 Lie algebras; 130 indecomposable and 31 indecomposable. In 1998, Gong [8], presented a new list of seven-dimensional nilpotent Lie algebras. Gong’s classification was based on the Skjelbred-Sund method [11]. Gong provided a classifcation of the seven-dimensional indecomposable algebras over Algebriacally closed fileds (χ≠2) and another classification over the field of real numbers. Once again, he used the same labeling as Seeley; the dimensions of the upper central series.

Representations

In this section, we present a matrix Lie group representation for the first 39 seven-dimensional Lie algebras with upper central series (27) (37) (247) (257) and (357)

Algebras with upper central series dimensions (27)

1. image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

The vector field representation is given by:

image

2. (27B) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

Algebras with upper central series dimensions (37)

1. (37A) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

2. (37B) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

3. (37C) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

4. (37D) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

Algebras with upper central series dimensions (247)

1. (247A) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

2. (247B) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

3. (247C) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

4. (247D) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

A matrix Lie group is given by:

5. (247E) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

6. (247F) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

7. (247G) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

8. (247H) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

9. (247I ) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

10. (247J ) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

11. (247K) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

12. (247L) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

13. (247M) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

14. (247N) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

15. (247O) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

16. (247P) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

17. (247Q) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

18. (247R) :

image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

The Vector field representation is given by:

image

Algebras with upper central series dimensions (257)

1. (257A) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

2. (257B) :

A matrix Lie group is given by:image

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

3. (257C) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

4. (257D) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

5. (257E) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

6. (257F) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

7. (257G) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

8. (257H): image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

9. (257I ) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

10. (257J ) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

11. (257K) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

12. (257L) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

Algebras with upper central series dimensions (357)

1. (357A) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

2. (357B) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

The vector field representation is given by:

image

3. (357C) : image

A matrix Lie group is given by:

image

Left invariant differential forms are given by:

image

Vector field representation is given by:

image

Conclusion

The Explanation regarding finding linear representations for seven-dimensional real, indecomposable nilpotent Lie algebras is done.

References

Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Article Usage

  • Total views: 8691
  • [From(publication date):
    March-2016 - Jul 23, 2019]
  • Breakdown by view type
  • HTML page views : 8502
  • PDF downloads : 189
Top