Journal of Algebra 386 (2013) 105–112

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Journal of Algebra

www.elsevier.com/locate/jalgebra

The dimension of the c-nilpotent multiplier

Mehdi Araskhan

Department of Mathematics, Yazd Branch, Islamic Azad University, Yazd, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 February 2013Available online 25 April 2013Communicated by Volodymyr Mazorchuk

MSC:17B3017B6017B99

Keywords:Schur multiplierc-Nilpotent multiplierNilpotent Lie algebra

The purpose of this paper is to obtain some inequalities for thedimension of the c-nilpotent multiplier of finite dimensional nilpo-tent Lie algebras and their factor Lie algebras. Finally, we compareour results to upper bound given in Salemkar et al. (2009) [10].

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

All Lie algebras referred to in this article are (of finite or infinite dimension) over a fixed field Λ

and the square brackets [ , ] denotes the Lie product. Let 0 −→ R −→ F −→ L −→ 0 be a free presen-tation of a Lie algebra L, where F is a free Lie algebra. Then we define the c-nilpotent multiplier of L,c � 1, to be

M(c)(L) = (R ∩ γc+1(F )

)/γc+1(R, F ),

where γc+1(F ) is the (c + 1)-th term of the lower central series of F , γ1(R, F ) = R 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 thevariety of nilpotent groups of class at most c given by Baer [1] (see [5,9] for more information on theBaer invariant of groups). The Lie algebra M(1)(L) = M(L) = (R ∩ F 2)/[R, F ] is the most studied Schurmultiplier of L (see for instance [2,3]). It is readily verified that the Lie algebra M(c)(L) is abelian andindependent of the choice of the free presentation of L (see [10]).

E-mail address: m.araskhan@iauyazd.ac.ir.

0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jalgebra.2013.04.010

106 M. Araskhan / Journal of Algebra 386 (2013) 105–112

Notation. Let L be an arbitrary Lie algebra and Ln denote the n-th term of the lower central seriesof L defined inductively by L1 = L and Ln+1 = [Ln, L], for n � 1. Let Zn(L) denote the n-th term ofthe upper central series of L defined inductively by Z0(L) = {0} and Zn+1(L)/Zn(L) be the center ofL/Zn(L) for n � 0.

Let F be a free Lie algebra on the set X = {x1, x2, . . . , xd}. Then the basic commutators in F areinductively defined as follows: The generators x1, x2, . . . , xd are basic commutators of length one andordered by setting xi < x j if i < j. If all the basic commutators ci of length less than k, where k > 1is integer, have been defined and ordered, then we define the basic commutators of length k to beall commutators of the form [ci, c j] such that the sum of lengths of ci and c j is k, ci > c j and ifci = [cs, ct], then c j � ct . The basic commutators of length k follow those of lengths less than k in anyorder with respect to each other. Basic commutators will be numbered so that they will be orderedby their subscripts.

A.I. Shirshov [11] indicated that the set of all basic commutators on X is a basis of F . Also, it isshown in [8] that the number of basic commutators on X of length n, denoted by ld(n), is obtainedby the following formula

ld(n) = 1

n

∑m|n

μ(m)dnm , (∗)

where μ(m) is the Möbius function, defined by μ(1) = 1, μ(k) = 0 if k is divisible by a square, andμ(p1 · · · ps) = (−1)s if p1, . . . , ps are distinct prime numbers.

By the above notations and assumptions, the factor γn(F )/γn+1(F ) is an abelian Lie algebra ofdimension ld(n), and any element of the form [xi1 , xi2 , . . . , xin ] with xi j ∈ X (1 � i � j) is precisely alinear combination of basic commutators on X of length n.

Form the above explanations, we have the following result.

Lemma 1.1. Let L be an abelian Lie algebra of dimension n. Then dim M(c)(L) = ln(c + 1). In particular,dim M(L) = 1

2 n(n − 1).

Proof. Consider a free Lie algebra F freely generated by n elements. By formula (∗), F/F 2 is anabelian Lie algebra of dimension n, and so it is isomorphic to L. Hence dim M(c)(L) = γc+1(F )/γc+2(F ),which gives the result. �

First, we show that for each ideal N in L, there is a close relationship between the M(c)(L) andM(c)(L/N).

Lemma 1.2. Let L be a Lie algebra with a free presentation 0 −→ R −→ F −→ L −→ 0. If S is an ideal in Fwith N ∼= S/R, then the following sequences are exact:

(i) 0 −→ R∩γc+1(S,F )

γc+1(R,F )−→ M(c)(L) −→ M(c)(L/N) −→ N∩γc+1(L)

γc+1(N,L)−→ 0,

(ii) (L/L2)abc⊗ Nab −→ M(c)(L) −→ M(c)(L/N) −→ N ∩ γc+1(L) −→ 0,

under the condition that N is central, Nab = N/[N, N] and M c⊗ N = M ⊗ · · · ⊗ M︸ ︷︷ ︸c-times

⊗ N.

Proof. We prove only part (ii). Since N is central, [S, F ] ⊆ R and

γc+1(

S, F 2 + R) ⊆ γc+1(S, R) + γc+1

(S, F 2) ⊆ γc+1(R, F ) + γc+1

(S, [F , F ]) ⊆ γc+1(R, F ).

Now, we observe that the bilinear map

M. Araskhan / Journal of Algebra 386 (2013) 105–112 107

α : F

F 2 + R× · · · × F

F 2 + R︸ ︷︷ ︸c-times

× S

R−→ R ∩ γc+1(F )

γc+1(R, F ),

induces the following homomorphism

α∗ : F/(

F 2 + R) c⊗ S/R −→ R ∩ γc+1(F )

γc+1(R, F ),

such that Imα∗ = γc+1(S, F )/γc+1(R, F ). Now the result holds by part (i). �Corollary 1.3. Let N be an ideal of Lie algebra L, then

(i) dim M(c)(L) + dim(N ∩ γc+1(L)) = dim M(c)(L/N) + dim(γc+1(N, L)) + dim(R ∩ γc+1(S, F )/

γc+1(R, F )),

(ii) dim M(c)(L) + dim(N ∩ γc+1(L)) = dim M(c)(L/N) + dim(γc+1(S,F )

γc+1(R,F )), where F , S, R are defined in

Lemma 1.1,(iii) dim M(c)(L) + dim(N ∩ γc+1(L))� dim M(c)(L/N) + dim((L/L2)ab

c⊗ Nab).

Proof. We prove only part (ii). Since

(γc+1(S, F )/γc+1(R, F ))

(R ∩ γc+1(S, F )/γc+1(R, F ))∼= γc+1(S, F )

R ∩ γc+1(S, F )∼= γc+1(N, L).

Now the result holds by part (i). �We also obtain a result on the central series of a nilpotent Lie algebra.Let L = L1 ⊃ L2 ⊃ · · · ⊃ Lc ⊃ Lc+1 = 0 be the lower central series of a nilpotent Lie algebra, L. L is

said to have class c if c is the least integer for which Lc+1 = 0. Furthermore, if dim L j/L j+1 = 1 forj = 2,3, . . . , c and dim L/L2 = 2, then L is said to be of maximal class c. Additionally, let 0 = Z0(L) ⊂Z1(L) ⊂ Z2(L) ⊂ · · · ⊂ Zc(L) = L be the upper central series of nilpotent L. If L is of maximal class,then Zi(L) = Lc−i+1 for 0 � i � c (see [4]).

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) and dim L = n.Then

dim M(c)(L) � ln−1(c + 1) + 2c − 1.

Proof. By Corollary 2.8(i) of [10] and using Corollary 1.3(iii) with N = Z(L), we get

dim M(c)(L) + dim(

Z(L))� dim M(c)(L/Z(L)

) + (1)(2)c,

dim M(c)(L) + 1 � ln−1(c + 1) + (1)(2)c . �It is interesting to know the connection between the c-nilpotent multiplier and their factor Lie

algebras. Jones [6,7] gave some inequalities for the Schur multiplier of a finite group G and its factorgroup. The purpose of this paper is to obtain some inequalities for the new concept of the c-nilpotentmultiplier of finite dimensional nilpotent Lie algebras and their factor Lie algebras (Corollary 2.4).Finally, we compare our results to upper bound given in [10].

108 M. Araskhan / Journal of Algebra 386 (2013) 105–112

2. Bounds on dim M (c)(L)

First, we give some inequalities for the dimension of the c-nilpotent multiplier of finite dimen-sional nilpotent Lie algebras. For this purpose, we need the following two lemmas, Lemma 2.1 issimilar to the work of Jones [6] for the group case.

Lemma 2.1. Let L be a finite dimensional Lie algebra with an ideal N. Let 0 −→ R −→ F −→ L −→ 0 be afree presentation of L and N ∼= S/R for some ideal S of the free Lie algebra F . Then γc+1(S, F )/(γc+1(R, F ) +γc+1([F , S], F ) + γc+1(S)) is a homomorphic image of (L/N)ab

c⊗ Nab.

Proof. Define

θ : L/(L2 + N

) × · · · × L/(L2 + N

)︸ ︷︷ ︸

c-times

× N/N2 −→ γc+1(S, F )

γc+1(R, F ) + γc+1([F , S], F ) + γc+1(S),

θ(

f1 + (F 2 + S

), . . . , fc + (

F 2 + S), x + (

S2 + R)) �−→ [x, f1, . . . , fc] + T ,

where T = (γc+1(R, F ) + γc+1([F , S], F ) + γc+1(S)), f1, . . . , fc ∈ F and x ∈ S . Note that L/(L2 + N) ×N/N2 ∼= F/(F 2 + S)× S/(S2 + R). We claim that θ is well defined. Let f ′

1 ≡ f1 (mod F 2 + S), . . . , f ′c ≡

fc (mod F 2 + S) and x′ ≡ x (mod S2 + R). This implies that f ′1 = f1 + g1 + s1, . . . , f ′

c = fc + gc + sc

and x′ = x + s′ + r for some gi ∈ F 2, si ∈ S , r ∈ R and s′ ∈ S2 (1 � i � c). Simple calculations show that[x′, f ′

1, . . . , f ′c] ≡ [x, f1, . . . , fc] mod (γc+1(R, F ) + γc+1([F , S], F ) + γc+1(S)). Hence θ is well defined.

Therefore, there exists a unique homomorphism

θ∗ : L/(L2 + N

) ⊗ · · · ⊗ L/(L2 + N

)︸ ︷︷ ︸

c-times

× N/N2 −→(

γc+1(S, F )

γc+1(R, F ) + γc+1([F , S], F ) + γc+1(S)

)

such that Im(θ∗) = γc+1(S, F )/(γc+1(R, F ) + γc+1([F , S], F ) + γc+1(S)) and the lemma is proved. �Lemma 2.2. Let H and N be ideals of Lie algebra L and N = N0 ⊇ N1 ⊇ · · · , a chain of ideals of N such that[Ni, L] ⊆ Ni+1 for all i = 1,2, . . . . Then

[Ni, [H, j L]] ⊆ Ni+ j+1 for all i, j.

Proof. We have

[Ni, [H, j+1 L]] = [

Ni,[[H, j L], L

]]⊆ [[

Ni, [H, j L]], L] + [[Ni, L], [H, j L]]

⊆ [Ni+ j+1, L] + [Ni+1, [H, j L]]

⊆ Ni+ j+2 + Ni+ j+2

= Ni+ j+2.

Now, the assertion follows by induction on j. �Theorem 2.3. Let L be a finite dimensional nilpotent Lie algebra of class d � 2. Let 0 −→ R −→ F −→ L −→ 0be a free presentation of L and N ∼= S/R for some ideal S of the free Lie algebra F . Then

M. Araskhan / Journal of Algebra 386 (2013) 105–112 109

(i) γc+1(γi+1(F )+R,F )

γc+1(γi+2(F )+R,F )is a homomorphic image of (L/γi+1(L))ab

c⊗ (γi+1(L))ab , for 1 � i � c − 1,

(ii) γc+1(γd(F )+R,F )

γc+1(R,F )is a homomorphic image of γd(L) ⊗ L

Zd−1(L)⊗ · · · ⊗ L

Zd−1(L)︸ ︷︷ ︸c-times

.

Proof. (i) For 1 � i � c − 1, we have

γc+1(γi+2(F ) + R, F

) = γc+1(γi+2(F ), F

) + γc+1(R, F )

= γc+1([

γi+1(F ), F], F

) + γc+1(R, F )

= γc+1(γi+1(F ) + R

) + γc+1([

F , γi+1(F ) + R], F

) + γc+1(R, F ).

By Lemma 2.1, the Lie algebra (γc+1(γi+1(F )+R,F )

γc+1(γi+1(F ))+γc+1([F ,γi+1(F )+R],F )+γc+1(R,F )) is a homomorphic image of

(L/γi+1(L))abc⊗ (γi+1(L))ab , which gives the result.

(ii) Put Zk(L) = Tk/R for 0 � k � d. Now consider the following chain

S = Td ⊇ · · · ⊇ Tk ⊇ Tk−1 ⊇ · · · ⊇ T1 ⊇ T0 = R.

Since [Tk, F ] ⊆ Tk−1, then by Lemma 2.2, [Td−1, [γd−2(F ), F ]] ⊆ Td−1−(d−2+1) = T0 = R .Therefore,

[γd(F ) + R, Td−1, . . . , Td−1︸ ︷︷ ︸

c-times

] ⊆ [γd(F ), Td−1, . . . , Td−1︸ ︷︷ ︸

c-times

] + [R, Td−1, . . . , Td−1︸ ︷︷ ︸c-times

]

⊆ [γd(F ), Td−1, . . . , Td−1︸ ︷︷ ︸

c-times

] + γc+1(R, F )

⊆ [[[Td−1, F ], γd−1(F )], Td−1, . . . , Td−1︸ ︷︷ ︸

(c−1)-times

]

+ [[[Td−1,

[γd−2(F ), F

]], F

], Td−1, . . . , Td−1︸ ︷︷ ︸

(c−1)-times

] + γc+1(R, F )

⊆ [[Td−2, γd−1(F )

], Td−1, . . . , Td−1︸ ︷︷ ︸

(c−1)-times

] + γc+1(R, F )

...

⊆ [[T0, γ1(F )

], Td−1, . . . , Td−1︸ ︷︷ ︸

(c−1)-times

] + γc+1(R, F )

⊆ γc+1(R, F ).

The latter inclusion gives the following epimorphism

γd(F ) + R

R× F

Td−1× · · · × F

Td−1︸ ︷︷ ︸c-times

−→ γc+1(γd(F ) + R, F )

γc+1(R, F ),

(x + R, f1 + Td−1, . . . , fc + Td−1) �−→ [x, f1, . . . , fc] + γc+1(R, F ). �

110 M. Araskhan / Journal of Algebra 386 (2013) 105–112

Corollary 2.4. Under the assumptions and notation of the above lemma, we have

(i) dim M(c)(L) + dimγc+1(L)� dim M(c)(L/L2) + ∑d−1i=1 dim(( L

γi+1(L))ab

c⊗ (γi+1(L))ab),

(ii) dim M(c)(L) + dim(γd(L) ∩ γc+1(L)) � dim M(c)(L/γd(L)) + dimγd(L)[dim( LZd−1(L)

)]c .

Proof. (i) In Corollary 1.3(i), taking N = [L, L], we obtain

dim M(c)(L) + dimγc+1(L) = dim M(c)(L/L2) + dimγc+1(L2, L

) + dim

(R ∩ γc+1(F 2 + R, F )

γc+1(R, F )

).

Moreover,

dimγc+1(L2, L

) + dim

(R ∩ γc+1(F 2 + R, F )

γc+1(R, F )

)

= dim

(γc+1(F 2 + R, F ) + R

R

)+ dim

((R ∩ γc+1(F 2, F )) + γc+1(R, F )

γc+1(R, F )

)

= dim

(γc+1(F 2 + R, F )

(R ∩ γc+1(F 2, F )) + γc+1(R, F )

)+ dim

((R ∩ γc+1(F 2, F )) + γc+1(R, F )

γc+1(R, F )

)

= dim

(γc+1(F 2 + R, F )

γc+1(R, F )

)

= dim

(γc+1(γd+1(F ) + R, F )

γc+1(R, F )

)+

d−1∑i=1

(γc+1(γi+1(F ) + R, F )

γc+1(γi+2(F ) + R, F )

).

By the assumption, we have 1 = γd+1(L) = γd+1(F )+RR . Hence γd+1(F ) + R = R . Now the assertion

follows by Theorem 2.3(i).(ii) In Corollary 1.3(ii), take N = γd(L) = γd(F )+R

R . Now by Theorem 2.3(ii), we have

dim M(c)(L) + dim(γd(L) ∩ γc+1(L)

) = dim M(c)(L/γd(L)) + dim

(γc+1(γd(F ) + R, F )

γc+1(R, F )

)

� dim

(γd(L) ⊗ L

Zd−1(L)⊗ · · · ⊗ L

Zd−1(L)︸ ︷︷ ︸c-times

)

+ dim M(c)(L/γd(L))

= dim M(c)(L/γd(L)) + dimγd(L)

[dim

(L

Zd−1(L)

)]c

. �Now, we compare our result to upper bound given in [10].

Theorem 2.5. Let L be a finite dimensional nilpotent Lie algebra of class m and d = d(L). Then

dim M(c)(L) �m∑

k=1

ld(k + c).

M. Araskhan / Journal of Algebra 386 (2013) 105–112 111

Example 2.6. Let F be a free Lie algebra on 2 generators and L = F/F 4. Then L is a Lie algebraof 2 generators and class 3. Thus dim L = 5, dim L2 = 3, dim L3 = 2 and dim L/L2 = l2(1) = 2. ByTheorem 2.5 (for c = 1)

dim M(L) �3∑

j=1

l2( j + 1) = l2(2) + l2(3) + l2(4)

= 1 + 2 + 1

4

(μ(1)24 + μ(2)22 + μ(4)2

)

= 3 + 1

4(16 − 4) = 6.

Also, by Corollary 2.4(i) (for c = 1),

dim M(L) � dim M(L/L2) +

3−1∑i=1

dim

((L

γi+1(L)

)ab

⊗ (γi+1(L)

)ab

)− dim L2

= dim M(L/L2) + dim

((L

L2

)⊗ L2

)+ dim

((L

L3

)⊗ L3

)− dim L2

= l2(2) + 6 + 6 − 3 = 10.

Moreover, by Theorem 2.5 (for c = 2),

dim M(2)(L) �3∑

j=1

l2( j + 2) = l2(3) + l2(4) + l2(5) = 5 + 6 = 11.

Also, by Corollary 2.4(i) (for c = 2),

dim M(2)(L) � dim M(2)(L/L2) +

3−1∑i=1

dim

((L

γi+1(L)

)ab

⊗ (γi+1(L)

)ab

)− dim L3

= dim M(2)(L/L2) + dim

((L

L2

)⊗ L2

)+ dim

((L

L3

)⊗ L3

)− dim L3

= l2(3) + 12 − 2 = 12.

Thus, the result of Theorem 2.5 was proved to be a better upper bound than the one obtained by ournew theorem.

Example 2.7. Let F be a free Lie algebra on 2 generators and L = F/F 3. Then L is a Lie algebra of2 generators and class 2. Thus dim L/L2 = l2(1) = 2, dim L2/L3 = l2(2) = 1 and dim L = 3. By Theo-rem 2.5 (for c = 1),

dim M(L) �2∑

j=1

l2( j + 1) = l2(2) + l2(3) = 1 + 1

3(6) = 3.

Also, by Corollary 2.4(i) (for c = 1),

112 M. Araskhan / Journal of Algebra 386 (2013) 105–112

dim M(L) � dim M(L/L2) +

2−1∑i=1

dim

((L

γi+1(L)

)ab

⊗ (γi+1(L)

)ab

)− dim L2

= dim M(L/L2) + dim

((L

L2

)⊗ L2

)− dim L2 = l2(2) + 2 − 1 = 2.

Moreover, by Theorem 2.5 (for c = 2),

dim M(2)(L) �2∑

j=1

l2( j + 2) = l2(3) + l2(4) = 2 + 1

4(12) = 5.

Also, by Corollary 2.4(i) (for c = 2),

dim M(2)(L) � dim M(2)(L/L2) +

2−1∑i=1

dim

((L

γi+1(L)

)ab

⊗ (γi+1(L)

)ab

)− dim L3

= dim M(2)(L/L2) + dim

((L

L2

)⊗ L2

)− dim L3 = l2(3) + 2 − 0 = 4.

Thus, we see that in this case, our theorem creates a better upper bound for dim M(c)(L) (for c = 1,2)than the previously known result.

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