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**Ghanam R ^{1*}, Basim Mustafa B^{2}, Mustafa MT^{3} and Thompson G^{4}**

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

^{2}Department of Mathematics, An-Najah National University, Palestine

^{3}Department of Mathematics, Statistics and Physics, Qatar University, Qatar

^{4}Department 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.

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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.

Lie algebra; Lie group; Representation, Nilpotent

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.

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.

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.

A matrix Lie group is given by:

Left invariant differential forms are given by:

The vector field representation is given by:

2. (27B) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

**Algebras with upper central series dimensions (37)**

1. (37A) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

2. (37B) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

3. (37C) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

4. (37D) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

**Algebras with upper central series dimensions (247)**

1. (247A) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

2. (247B) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

3. (247C) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

4. (247D) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

A matrix Lie group is given by:

5. (247E) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

6. (247F) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

7. (247G) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

8. (247H) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

9. (247I ) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

10. (247J ) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

11. (247K) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

12. (247L) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

13. (247M) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

14. (247N) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

15. (247O) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

16. (247P) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

17. (247Q) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

18. (247R) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

The Vector field representation is given by:

**Algebras with upper central series dimensions (257)**

1. (257A) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

2. (257B) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

3. (257C) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

4. (257D) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

5. (257E) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

6. (257F) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

7. (257G) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

8. (257H):

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

9. (257I ) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

10. (257J ) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

11. (257K) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

12. (257L) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

**Algebras with upper central series dimensions (357)**

1. (357A) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

2. (357B) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

The vector field representation is given by:

3. (357C) :

A matrix Lie group is given by:

Left invariant differential forms are given by:

Vector field representation is given by:

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

- Ado ID (1935) Note on the representation of finite continuous groups by means of linear substitution. Izv Fiz-Mat Obsch (Kazan) 7: 01-43.
- Ado ID (1947)The representation of Lie algebras by matrices (in Russian), Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo.Uspekhi Matematicheskikh Nauk 2: 159-173.
- Ghanam R, Thompson G,Tonon S(2006) Representations for six-dimensional nilpotent Lie algebras. Hadronic J 29: 299-317.
- Ghanam R Strugar I,Thompson G (2005) Matrix representations for low dimensional Lie algebras. Extracta Math 20: 151-184.
- Ghanam R, Thompson G, Miller E (2004) Variationality of four-dimensional Lie group connections. J Lie Theory 14: 395-425.
- Gong MP (1998) Classification of Nilpotent Lie algebras of dimension 7 (Over algebriacally closed Fields and
*R*).University of Waterloo. - Jacobson N(1962) Lie Algebras. Tracts in Pure and Applied Math Interscience Publishers, Newyork.
- Humphreys JE (1972) Introduction to Lie algebras and representation theory.Graduate Text in Math Springer.
- Seeley C.Degeneration of 6-dimensional Nilpotent Lie Algebras over
*C.*Communications in Algebra 18:3493-3505. - Seeley C (1993) 7-dimensional Nilpotent Lie Algebra.Trans Amer Math Soc 335: 479-496.
- Skjelbred T, Sund T(1977).Classification of Nilpotent Lie Algebras in dimension six.Univerisity of Oslo.

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