alexa Verification of Some Properties of the C-nilpotent Multiplier in Lie Algebras | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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

Verification of Some Properties of the C-nilpotent Multiplier in Lie Algebras

Sadeghieh A, Araskhan M*

Yazd Branch, Islamic Azad University, Iran

Corresponding Author:
Araskhan M
Yazd Branch, Islamic Azad University, Iran
Tel: +982147911
E-mail: [email protected]

Received date: April 21, 2015 Accepted date: July 20, 2015 Published date: July 29, 2015

Citation: Sadeghieh A, Araskhan M (2015) Verification of Some Properties of the C-nilpotent Multiplier in Lie Algebras. J Generalized Lie Theory Appl 9:228. doi:10.4172/1736-4337.1000228

Copyright: © 2015 Sadeghieh A, 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 Generalized Lie Theory and Applications

Abstract

The purpose of this paper is to obtain some inequalities and certain bounds for the dimension of the c-nilpotent multiplier of finite dimensional nilpotent Lie algebras and their factor Lie algebras. Also, we give an inequality for the dimension of the c-nilpotent multiplier of L connected with dimension of the Lie algebras γd (L) and L / Zd−1 (L) . Finally, we compare our results with the previously known result.

Keywords

C-nilpotent multiplier; Nilpotent lie algebra; Lie algebra

Introduction

All Lie algebras referred to in this article are (of finite or infinite dimension) over a fixed field F and the square brackets [ , ] denotes the Lie product. Let 0→R→F→L→0 be a free presentation of a Lie algebra L, where F is a free Lie algebra. Then we define the, c-nilpotent multiplier c ≥ 1, to be

image

where imageis the (c+1)-th term of the lower central series of F, γ1(R, F) = 1 and γc+1(R,F) = [γc(R,F),F]. This is analogous to the definition of the Baer-invariant of a group with respect to the variety of nilpotent groups of class at most c given by Baer [1-3] (for more information on the Baer invariant of groups). The Lie algebra image is the most studied Schur multiplier of L [4,5]. It is readily verified that the Lie algebra image is abelian and independent of the choice of the free presentation of L [6]. The purpose of this paper is to obtain some inequalities for the dimension of the c-nilpotent multiplier of finite dimensional nilpotent Lie algebras and their factor Lie algebras (Corollary 2.3 and Corollary 2.5). Finally, we compare our results to upper bound given [6]. First, we show that for each ideal N in L, there is a close relationship between the image

Lemma 1.1. Let L be Lie algebra with a free presentation 0→R→F→L→0. If S is an ideal in F with image , then the following sequences are exact:

image

image

under the condition that N is central, image and image

Proof. We prove only part (ii). Since N is central, image and

image

Now, we have the following homomorphism

image

such that Imα= image. Now the result holds by part (i).

The following corollary is an immediate consequence of Lemma 1.1, which gives some elementary results about dimension of the c-nilpotent multiplier of finite dimensional Lie algebras see corollary 2.2 of Salemkar et al. [6].

Corollary 1.2. Let N be an ideal of Lie algebra L. Then

image

Where F,S,R are defined in Lemma 1.1.

(ii) image

Suppose that L is generated by n elements. Let F be a free Lie algebra generated by n elements and image Witt,s formula from Bahturin et al. [7] gives us

Dim image

where μ(m) is the Mobius function, defined by image if K is divisible by a square, and image if p1,….,ps are distinct prime numbers.

Lemma 1.3. Let L be an abelian Lie algebra of dimension n. Then

dimimage . In particular, dim image

Proof. Consider a free Lie algebra F freely generated by n elements. By Witt’s formula, F/F2 is an abelian Lie algebra of dimension n, and so it is isomorphic to L. Hence dim image, which gives the result.

Let image be the lower central series of nilpotent Lie algebra, L. L is said to have class c if c is the least integer for which image . Furthermore, if dim image for j=2,3,…,c and dim image , then L is said to be of maximal class c. Additionally, let image be the upper central series of nilpotent Lie algebra L. If L is of maximal class, then image for image.

By the above notation we have the following corollary.

Corollary 1.4. Let L be a finite dimensional nilpotent Lie algebra of maximal class (c+1), then

dimimage

Proof. Using Corollary 1.2(ii) with N= Z(L), we get

dim image +dim Z(L) = dim image

Discussion and Results

2 Bounds on dim image

Let L be a finite dimensional nilpotent Lie algebra of class d > 2. First, we give an inequality for the dimension of the c-nilpotent multiplier of L connected with dimension of the Lie algebras image and image (Corollary 2.3) and some inequalities for the dimension of the c-nilpotent multiplier of finite dimensional nilpotent Lie algebras will be given. For this purpose, we need the following two lemmas.

Lemma 2.1. Let H and N be ideals of Lie algebra L and image a chain of ideals of N such that image for all I = 1,2,…. Then

image for all i, j.

Proof. We have

image

Now, the assertion follows by induction on j.

Lemma 2.2. Let L be a finite dimensional nilpotent Lie algebra of class d ≥ 2 . Let 0→R→F→L→0 be a free presentation of L, then image is a homomorphic image of image

Proof. Put image for image. Now consider the following chain

image

Since image then by Lemma 2.1,

image

Therefore,

image

image

The latter inclusion gives the following epimorphism

image

image

Corollary 2.3. Under the assumptions and notation of the above Lemma, we have

image

Proof. In Corollary 1.2(i), taking image Now by Lemma 2.2, we have

image

image

image

image

In following, we give another an inequality for the dimension of the c-nilpotent multiplier of finite dimensional nilpotent Lie algebras.

Theorem 2.4. Let L be a finite dimensional nilpotent Lie algebra of class ≥ 1 , then

image

Proof. We use induction on the class of L. If L is of class 1, then L2 = 0 and the result holds. Assume the result for nilpotent Lie algebras of class to be less than d and let L have class m= d-1. Note that image and image For convenience, let image and image Since A is a homomorphic image of B, it follows that imageBy induction,

image

image

image

By Corollary 2.3,

image

image

Also, image

Therefore,

image

image

image

Since image, we obtain:

Corollary 2.5. Under the assumptions and notation of the above Theorem, we have

image

Now, we compare our results to upper bound given [7], when c = 1.

Theorem 2.6. Let L be a finite dimensional nilpotent Lie algebra of class m and d=d(m). Then

image

Example 2.7. Let F be a free Lie algebra on 2 generators and image. Then L is a Lie algebra of 2 generators and class 2. Thus image and dim L=3. By Theorem 2.6, image

Note that L is a finite dimensional nilpotent Lie algebra of maximal class (1 + 1) and Z(L)=L2. By Corollary 1.4 and Lemma 1.3,

image

Also, by Corollary 2.5,

image

Example 2.8. Let F be a free Lie algebra on 2 generators and L = F / F4 . Then L is a Lie algebra of 2 generators and class 3. Thus image and image . By Theorem 2.6

image

image

image

Also, by Corollary 2.5,

image

In this two examples, we see that our results give two better upper bounds for dim(L) than the previously known result.

Acknowledgements

The authors wish to thank Yazd Branch, Islamic Azad University for its support of research project under the title Verification of some properties of The c-nilpotent multiplier in Lie algebras.

References

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

Share This Article

Relevant Topics

Article Usage

  • Total views: 11987
  • [From(publication date):
    December-2015 - Dec 12, 2017]
  • Breakdown by view type
  • HTML page views : 8052
  • PDF downloads : 3935
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh

Peer Reviewed Journals
 
Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
International Conferences 2017-18
 
Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings

Contact Us

Agri & Aquaculture Journals

Dr. Krish

[email protected]

1-702-714-7001Extn: 9040

Biochemistry Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Business & Management Journals

Ronald

[email protected]

1-702-714-7001Extn: 9042

Chemistry Journals

Gabriel Shaw

[email protected]

1-702-714-7001Extn: 9040

Clinical Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Engineering Journals

James Franklin

[email protected]

1-702-714-7001Extn: 9042

Food & Nutrition Journals

Katie Wilson

[email protected]

1-702-714-7001Extn: 9042

General Science

Andrea Jason

[email protected]

1-702-714-7001Extn: 9043

Genetics & Molecular Biology Journals

Anna Melissa

[email protected]

1-702-714-7001Extn: 9006

Immunology & Microbiology Journals

David Gorantl

[email protected]

1-702-714-7001Extn: 9014

Materials Science Journals

Rachle Green

[email protected]

1-702-714-7001Extn: 9039

Nursing & Health Care Journals

Stephanie Skinner

[email protected]

1-702-714-7001Extn: 9039

Medical Journals

Nimmi Anna

[email protected]

1-702-714-7001Extn: 9038

Neuroscience & Psychology Journals

Nathan T

[email protected]

1-702-714-7001Extn: 9041

Pharmaceutical Sciences Journals

Ann Jose

[email protected]

1-702-714-7001Extn: 9007

Social & Political Science Journals

Steve Harry

[email protected]

1-702-714-7001Extn: 9042

 
© 2008- 2017 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version