Sadeghieh A, Araskhan M*
Yazd Branch, Islamic Azad University, Iran
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.
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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.
C-nilpotent multiplier; Nilpotent lie algebra; Lie algebra
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
where is 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 is the most studied Schur multiplier of L [4,5]. It is readily verified that the Lie algebra is abelian and independent of the choice of the free presentation of L . 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 . First, we show that for each ideal N in L, there is a close relationship between the
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 , then the following sequences are exact:
under the condition that N is central, and
Proof. We prove only part (ii). Since N is central, and
Now, we have the following homomorphism
such that Imα= . 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. .
Corollary 1.2. Let N be an ideal of Lie algebra L. Then
Where F,S,R are defined in Lemma 1.1.
Suppose that L is generated by n elements. Let F be a free Lie algebra generated by n elements and Witt,s formula from Bahturin et al.  gives us
where μ(m) is the Mobius function, defined by if K is divisible by a square, and if p1,….,ps are distinct prime numbers.
Lemma 1.3. Let L be an abelian Lie algebra of dimension n. Then
dim . In particular, dim
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 , which gives the result.
Let 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 . Furthermore, if dim for j=2,3,…,c and dim , then L is said to be of maximal class c. Additionally, let be the upper central series of nilpotent Lie algebra L. If L is of maximal class, then for .
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
Proof. Using Corollary 1.2(ii) with N= Z(L), we get
dim +dim Z(L) = dim
2 Bounds on dim
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 and (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 a chain of ideals of N such that for all I = 1,2,…. Then
for all i, j.
Proof. We have
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 is a homomorphic image of
Proof. Put for . Now consider the following chain
Since then by Lemma 2.1,
The latter inclusion gives the following epimorphism
Corollary 2.3. Under the assumptions and notation of the above Lemma, we have
Proof. In Corollary 1.2(i), taking Now by Lemma 2.2, we have
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
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 and For convenience, let and Since A is a homomorphic image of B, it follows that By induction,
By Corollary 2.3,
Since , we obtain:
Corollary 2.5. Under the assumptions and notation of the above Theorem, we have
Now, we compare our results to upper bound given , when c = 1.
Theorem 2.6. Let L be a finite dimensional nilpotent Lie algebra of class m and d=d(m). Then
Example 2.7. Let F be a free Lie algebra on 2 generators and . Then L is a Lie algebra of 2 generators and class 2. Thus and dim L=3. By Theorem 2.6,
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,
Also, by Corollary 2.5,
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 and . By Theorem 2.6
Also, by Corollary 2.5,
In this two examples, we see that our results give two better upper bounds for dim(L) than the previously known result.
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.
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