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**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

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 γ_{d} (L) and L / Z_{d−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, γ_{1}(R, F) = 1 and γ_{c+1}(R,F) = [γ_{c}(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 [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

**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. [6].

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

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

(ii)

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. [7] gives us

Dim

where μ(m) is the Mobius function, defined by if K is divisible by a square, and if p_{1},….,p_{s} 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/F^{2} 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

dim

**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,

Therefore,

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 L^{2} = 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,

Also,

Therefore,

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 [7], 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)=L^{2}. 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 / F^{4} . 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.

- Baer R (1945) Representations of groups as quotient groups, I, II, III. Trans Amer Math Soc 58: 295-419
- Burns J, Ellis G (1998) Inequalities for Baer invariants of finite groups. Canad Math Bull 41: 385-391.
- Leedham-Green CR, McKay S (1976) Baer-invariant, isologism, varietal laws and homology. Acta Math 137: 99-150
- Batten P, Stitzinger E (1996) On covers of Lie algebras. Comm Alg 24: 4301-4317.
- Batten P, Moneyhun K, Stitzinger E (1996) On characterizing nilpotent Lie algebras by their multipliers. Comm Alg 24: 4319-4330.
- Salemkar AR, Edalatzadeh B, Araskhan M (2009) Some inequalities for the dimension of the c-nilpotent multiplier of Lie Algebras. J Algebra 322: 1575-1585.
- Bahturin Y (1987) Identical relations in Lie algebras. VNU Science Press BV.

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