arXiv:1610.05581v2 [math.RA] 8 Nov 2016

2-CAPABILITY AND 2-NILPOTENT MULTIPLIER OF FINITE
DIMENSIONAL NILPOTENT LIE ALGEBRAS
PEYMAN NIROOMAND AND MOHSEN PARVIZI
Abstract. In the present context, we investigate to obtain some more results
about 2-nilpotent multiplier M(2) (L) of a finite dimensional nilpotent Lie algebra L. For instance, we characterize the structure of M(2) (H) when H is a
Heisenberg Lie algebra. Moreover, we give some inequalities on dim M(2) (L)
to reduce a well known upper bound on 2-nilpotent multiplier as much as
possible. Finally, we show that H(m) is 2-capable if and only if m=1.

1. Introduction
For a finite group G, let G be the quotient of a free group F by a normal subgroup
R, then the c-nilpotent multiplier M(c) (G) is defined as
R ∩ γc+1 (F )/γc+1 [R, F ],
in which γc+1 [R, F ] = [γc [R, F ], F ] for c ≥ 1. It is an especial case of the Baer
invariant [3] with respect to the variety of nilpotent groups of class at most c.
When c = 1, the abelian group M(G) = M(1) (G) is more known as the Schur
multiplier of G and it is much more studied, for instance in [11, 14, 18].
Since determining the c-nilpotent multiplier of groups can be used for the classification of group into isoclinism classes(see [2]), there are multiple papers concerning
this subject.
Recently, several authors investigated to develop some results on the group theory case to Lie algebra. In [22], analogues to the c-nilpotent multiplier of groups,
for a given Lie algebra L, the c-nilpotent multiplier of L is defined as
M(c) (L) = R ∩ F c+1 /[R, F ]c+1 ,
in which L presented as the quotient of a free Lie algebra F by an ideal R, F c+1 =
γc+1 (F ) and [R, F ]c+1 = γc+1 [R, F ]. Similarly, for the case c = 1, the abelian Lie
algebra M(L) = M(1) (L) is more studied by the first author and the others (see
for instance [4, 5, 6, 8, 9, 10, 15, 16, 17, 24]).
The c-nilpotent multiplier of a finite dimensional nilpotent Lie algebra L is a
new field of interest in literature. The present context is involving the 2-nilpotent
multiplier of a finite dimensional nilpotent Lie algebra L. The aim of the current
paper is divided into several steps. In [22, Corollary 2.8], by a parallel result to the
group theory result, showed for every finite nilpotent Lie algebra L, we have
1
(1.1)
dim(M(2) (L)) + dim(L3 ) ≤ n(n − 1)(n + 1).
3
Date: April 9, 2018.
Key words and phrases. 2-nilpotent multiplier; Schur multiplier; Heisenberg algebras; derived
subalgebra; 2-capable Lie algebra,
Mathematics Subject Classification 2010. Primary 17B30 Secondary 17B60, 17B99.
1

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P. NIROOMAND AND M. PARVIZI

Here we prove that abelian Lie algebras just attain the bound 1.1. It shows that
always Ker θ = 0 in [22, Corollary 2.8 (ii)a].
Since Heisenberg algebras H(m) (a Lie algebra of dimension 2m + 1 with L2 =
Z(L) and dim (L2 ) = 1) have interest in several areas of Lie algebra, similar
to the result of [5, Example 3] and [13, Theorem 24], by a quite different way,
we give explicit structure of 2-nilpotent multiplier of these algebras. Among the
other results since the Lie algebra which attained the upper bound 1.1 completely
described in Lemma 2.6 (they are just abelian Lie algebras), by obtaining some
new inequalities on dimension M(2) (L), we reduce bound 1.1 for non abelian Lie
algebras as much as possible.
Finally, among the class of Heisenberg algebras, we show that which of them
is 2-capable. It means which of them is isomorphic to H/Z2 (H) for a Lie algebra
H. For more information about the capability of Lie algebras see [18, 21]. These
generalized the recently results for the group theory case in [19].
2. Further investigation on 2-nilpotent multiplier of finite
dimensional nilpotent Lie algebra
The present section illustrates to obtain further results on 2-nilpotent multiplier
of finite dimensional nilpotent Lie algebra. At first we give basic definitions and
known results for the seek of convenience the reader.
Let F be a free Lie algebra on an arbitrary totaly ordered set X. Recall from
[25], the basic commutator on the set X, which is defined as follows, is a basis of
F.
The elements of X are basic commutators of length one and ordered relative to
the total order previously chosen. Suppose all the basic commutators ai of length
less than k ≥ 1 have been defined and ordered. Then the basic commutators of
length k to be defined as all commutators of the form [ai , aj ] such that the sum
of lengths of ai and aj is k, ai > aj , and if ai = [as , at ], then aj ≥ at . Also the
number of basic commutators on X of length n, namely ld (n), is
n
1X
µ(m)d m ,
n
m|n

where µ is the Möbius function.
From [8], let F be a fixed field, L, K be two Lie algebras and [ , ] denote the Lie
bracket. By an action of L on K we mean an F -bilinear map
(l, k) ∈ L × K 7→ l k ∈ K satisfying
[l,l′ ]

′

′

k = l ( l k)− l ( l k) and l [k, k ′ ] = [ l k, k ′ ]+[k, l k ′ ], for all c ∈ F, l, l′ ∈ L, k, k ′ ∈ K.

When L is a subalgebra of a Lie algebra P and K is an ideal in P , then L acts
on K by Lie multiplications l k = [l, k]. A crossed module is a Lie homomorphism
σ : K → L together with an action of L on K such that
σ(l k) = [l, σ(k)] and

σ(k) ′

k = [k, k ′ ] for all k, k ′ ∈ K and l ∈ L.

Let σ : L → M and η : K → M be two crossed modules, L and K act on each
other and on themselves by Lie. Then these actions are called compatible provided
that
k
′
l
′
l ′
k = k ( l k) and k l′ = l ( k l).

2-CAPABILITY AND 2- NILPOTENT MULTIPLIER OF FINITE DIMENSIONAL NILPOTENT LIE ALGEBRAS
3

The non-abelian tensor product L ⊗ K of L and K is the Lie algebra generated
by the symbols l ⊗ k with defining relations
c(l ⊗ k) = cl ⊗ k = l ⊗ ck, (l + l′ ) ⊗ k = l ⊗ k + l′ ⊗ k,
l ⊗ (k + k ′ ) = l ⊗ k + l ⊗ k ′ , l l′ ⊗ k = l ⊗

l′

k − l′ ⊗ l k, l ⊗ k k ′ =

k′

l ⊗ k − k l ⊗ k′ ,

′

[l ⊗ k, l′ ⊗ k ′ ] = − k l ⊗ l k ′ , for all c ∈ F, l, l′ ∈ L, k, k ′ ∈ K.
The non-abelian tensor square of L is a special case of tensor product L ⊗ K
when K = L. Note that we denote the usual abelian tensor product L ⊗Z K, when
L and K are abelian and the actions are trivial.
Let LK be the submodule of L ⊗ K generated by the elements l ⊗ k such that
σ(l) = η(k). The factor Lie algebra L ∧ K ∼
= L ⊗ K/LK is called the exterior
product of L and K, and the image of l ⊗ k is denoted by l ∧ k for all l ∈ L, k ∈ K.
Throughout the paper Γ is denoted the universal quadratic functor (see [8]).
Recall from [18], the exterior centre of a Lie algebra Z ∧ (L) = {l ∈ L | l ∧ l′ =
1L∧L , ∀ l′ ∈ L} of L. It is shown that in [18] the exterior centre L is a central
ideal of L which allows us to decide when Lie algebra L is capable, that is, whether
L∼
= H/Z(H) for a Lie algebra H.
The following Lemma is a consequence of [18, Lemma 3.1].
Lemma 2.1. Let L be a finite dimensional Lie algebra, L is capable if and only if
Z ∧ (L) = 0.
The next two lemmas are special cases of [22, Proposition 2.1 (i)] when c = 2
and that is useful for proving the next theorem.
Lemma 2.2. Let I be an ideal in a Lie algebra L. Then the following sequences
are exact.
I∩L3
→ 0.
(i) Ker(µ2I ) → M(2) (L) → M(2) (L/I) → [[I,L],L]
(ii) (I ∧ L/L3) ∧ L/L3 → M(2) (L) → M(2) (L/I) → I ∩ L3 → 0, when
[[I, L], L] = 0.
Lemma 2.3. Let I be an ideal of L, and put K = L/I. Then
I ∩ L3
(i) dim M(2) (K) ≤ dim M(2) (L) + dim
.
[[I, L], L]
(ii) Moreover, if I is a 2-central subalgebra. Then
(a). (I ∧ L) ∧ L → M(2) (L) → M(2) (K) → dim I ∩ L3 → 0.
(b). dim M(2) (L)+ dim I ∩L3 ≤ dim M(2) (K)+ dim (I ⊗ L/L3)⊗ L/L3 .
Proof. (i). Using Lemma 2.2 (i).
(ii)(a). Since [I, L] ⊆ Z(L), Ker µ2I = (I ∧ L) ∧ L and [[I, L], L] = 0 by Lemma
2.2. It follows the result.
(ii)(b). Since there is a natural epimorphism (I ⊗ L/L3) ⊗ L/L3 → (I ∧ L/L3 ) ∧
L/L3, the result deduces from Lemma 2.2 (ii).

The following theorem gives the explicit structure of the Schur multiplier of all
Heisenberg algebra.
Theorem 2.4. [5, Example 3] and [13, Theorem 24] Let H(m) be Heisenberg
algebra of dimension 2m + 1. Then
(i) M(H(1)) ∼
= A(2).
(ii) M(H(m)) = A(2m2 − m − 1) for all m ≥ 2.

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P. NIROOMAND AND M. PARVIZI

The following result comes from [20, Theorem 2.8] and shows the behavior of
2-nilpotent multiplier respect to the direct sum of two Lie algebras.
Theorem 2.5. Let A and B be finite dimensional Lie algebras. Then
M(2) (A ⊕ B))

∼
= M(2) (A) ⊕ M(2) (B) ⊕ (A/A2 ⊗Z A/A2 ) ⊗Z B/B 2

⊕ (B/B 2 ⊗Z B/B 2 ) ⊗Z A/A2 .



The following theorem is proved in [22] and will be used in the next contribution.
At this point, we may give a short proof with a quite different way of [22, Proposition
1.2] as follows.
Theorem 2.6. Let L = A(n) be an abelian Lie algebra of dimension L. Then
M(2) (L) ∼
= A( 13 n(n − 1)(n + 1)).
Proof. We perform induction on n. Assume n = 2. Then Theorem 2.5 allows us to
conclude that

M(2) (L) ∼
= M(2) (A(1)) ⊕ M(2) (A(1)) ⊕ A(1) ⊗Z A(1) ⊗Z A(1)

⊕ A(1) ⊗Z A(1)) ⊗Z A(1) ∼
= A(1) ⊕ A(1) ∼
= A(2).

Now assume that L ∼
= A(n) ∼
= A(n − 1) ⊕ A(1). By using induction hypothesis
and Theorem 2.5, we have

M(2) (A(n − 1) ⊕ A(1)) ∼
= M(2) (A(n − 1)) ⊕ A(n − 1) ⊗Z A(n − 1) ⊗Z A(1)

⊕ A(1) ⊗Z A(1) ⊗Z A(n − 1)
∼
=

A( 31 n(n − 1)(n − 2)) ⊕ A((n − 1)2 ) ⊕ A(n − 1)

∼
=

A( 13 n(n − 1)(n + 1)).


The main strategy, in the next contribution, is to give the similar argument of
Theorem 2.4 for the 2-nilpotent multiplier. In the first theorem, we obtain the
structure of M(2) (L) when L is non-capable Heisenberg algebra.
Theorem 2.7. Let L = H(m) be a non-capable Heisenberg algebra. Then
8m3 − 2m
).
M(2) (H(m)) ∼
= A(
3
Proof. Since L is non-capable, Lemma 2.1 implies Z ∧ (L) = L2 = Z(L). Invoking
Lemma 2.3 by putting I = Z ∧ (L), we have M(2) (H(m)) ∼
= M(2) (H(m)/H(m)2 ).
Now result follows from Theorem 2.6.

The following theorem from [18, Theorem 3.4] shows in the class of all Heisenberg
algebras which one is capable.
Theorem 2.8. H(m) is capable if and only if m = 1.
Corollary 2.9. H(m) is not 2-capable for all m ≥ 2.
Proof. Since every 2-capable Lie algebra is capable, the result follows from Theorem
2.8.


2-CAPABILITY AND 2- NILPOTENT MULTIPLIER OF FINITE DIMENSIONAL NILPOTENT LIE ALGEBRAS
5

Since H(m) for all m ≥ 2 is not 2-capable, we just need to discus about the
2-capability of H(1). Here, we obtain 2-nilpotent multiplier of H(1) and in the
next section we show H(1) is 2-capable.
Theorem 2.10. Let L = H(1). Then
M(2) (H(1)) ∼
= A(5).
Proof. We know that H(1) is in fact the free nilpotent Lie algebra of rank 2 and
class 2. That is H(1) ∼
= F/F 3 in which F is the free Lie algebra on 2 letters x, y.
The second nilpotent multiplier of H(1) is F 4 ∩F 3 /[F 3 , F, F ] which is isomorphic to
F 3 /F 5 ant the latter is the abelian Lie algebra on the set of all basic commutators of
weights 3 and 4 which is the set {[y, x, x], [y, x, y], [y, x, x, x], [y, x, x, y], [y, x, y, y]}.
So the result holds.

We summarize our result as below
Theorem 2.11. Let H(m) be Heisenberg algebra of dimension 2m + 1. Then
(i) M(2) (H(1)) ∼
= A(5).
3
(ii) M(2) (H(m)) = A( 8m 3−2m ) for all m ≥ 2.
The following Lemma lets us to obtain the structure of the 2-nilpotent multiplier
of all nilpotent Lie algebras with dim L2 = 1.
Lemma 2.12. [15, Lemma 3.3] Let L be an n-dimensional Lie algebra and dim L2 =
1. Then
L∼
= H(m) ⊕ A(n − 2m − 1).
Theorem 2.13. Let L be an n-dimensional Lie algebra with dim L2 = 1. Then

if m > 1,
 A( 13 n(n − 1)(n − 2))
(2)
M (L) ∼
=

A( 13 n(n − 1)(n − 2) + 3) if m = 1.

Proof. By using Lemma 2.12, we have L ∼
= H(m) ⊕ A(n − 2m − 1). Using the
behavior of 2-nilpotent multiplier respect to direct sum
M(2) (L) ∼
= M(2) (H(m)) ⊕ M(2) (A(n − 2m − 1))
⊕
⊕


(H(m)/H(m)2 ⊗Z H(m)/H(m)2 ) ⊗Z A(n − 2m − 1)

(A(n − 2m − 1) ⊗Z A(n − 2m − 1)) ⊗Z H(m)/H(m)2 .

First assume that m = 1, then by virtue of Theorems 2.6 and 2.11
1
M(2) (H(1)) ∼
= A( (n − 2)(n − 3)(n − 4)).
= A(5) and M(2) (A(n − 3)) ∼
3
Thus
M(2) (L) ∼
= A(5) ⊕ A( 13 (n − 2)(n − 3)(n − 4))

⊕ A(2) ⊗Z A(2)) ⊗Z A(n − 3)

⊕ (A(n − 3) ⊗Z A(n − 3)) ⊗Z A(2)
∼
= A( 13 n(n − 1)(n − 2) + 3).
The case m ≥ 1 is obtained by a similar fashion.



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P. NIROOMAND AND M. PARVIZI

Theorem 2.14. Let L be a n-dimensional nilpotent Lie algebra such that dim L2 =
m(m ≥ 1). Then

1
dim M(2) (L) ≤ (n − m) (n + 2m − 2)(n − m − 1) + 3(m − 1) + 3.
3
In particular, dim M(2) (L) ≤ 13 n(n − 1)(n − 2) + 3. The equality holds in last
inequality if and only if L ∼
= H(1) ⊕ A(n − 3).
Proof. We do induction on m. For m = 1, the result follows from Theorem 2.13.
Let m ≥ 2, and taking I a 1-dimensional central ideal of L. Since I and L/L3 act
L/L3
L/L3
to each other trivially we have (I ⊗ L/L3 ) ⊗ L/L3 ∼
= (I ⊗Z (L/L3 )2 ) ⊗Z (L/L3 )2 ).
Thus by Lemma 2.3 (ii)(b)
dim M(2) (L) + dim I ∩ L3 ≤ dim M(2) (L/I) + dim (I ⊗Z

L/L3
L/L3
)
⊗
).
Z
(L/L3 )2
(L/L3 )2

Since
dim M(2) (L/I) ≤


1
(n − m) (n + 2m − 5)(n − m − 1) + 3(m − 2) ,
3

we have
dim M(2) (L) ≤
=
as required.

1
3 (n


− m) (n + 2m − 5)(n − m − 1) + 3(m − 2) + 3 + (n − m)2

1
3 (n − m) (n + 2m − 2)(n − m − 1) + 3(m − 1) + 3,


The following corollary shows that the converse of [22, Proposition 1.2] for c = 2
is also true. In fact it proves always Ker θ = 0 in [22, Corollary 2.8 (ii)a].
Corollary 2.15. Let L be a n-dimensional nilpotent Lie algebra. If dim M(2) (L) =
1
∼
3 n(n − 1)(n + 1), then L = A(n).
3. 2-capability of Lie algebras
Following the terminology of [7] for groups, a Lie algebra L is said to be 2-capable
provided that L ∼
= H/Z2 (H) for a Lie algebra H. The concept Z2∗ (L) was defined in
[23] and it was proved that if π : F/[R, F, F ] → F/R be a natural Lie epimorphism
then
Z2∗ (L) = π(Z2 (F/[[R, F ], F ])), for c ≥ 0
The following proposition gives the close relation between 2-capability and Z2∗ (L).
Proposition 3.1. A Lie algebra L is 2-capable if and only if Z2∗ (L) = 0.
Proof. Let L has a free presentation F/R, and Z2∗ (L) = 0. Consider the natural
epimorphism π : F/[[R, F ], F ] ։ F/R. Obviously
Ker π = R/[[R, F ], F ] = Z2 (F/[[R, F ], F ]),
∼
and hence L = F/[[R, F ], F ]/Z2 (F/[[R, F ], F ]), which is a 2-capable.
Conversely, let L is 2-capable and so H/Z2 (H) ∼
= L for a Lie algebra H. Put
F/R ∼
= H and Z2 (H) ∼
= S/R. There is natural epimorphism η : F/[[S, F ], F ] ։
F/S ∼

= L. Since Z2 (F/[[R, F ], F ]) ⊆ Ker η, Z2∗ (L) = 0, as required.
The following Theorem gives an instrumental tools to present the main.

2-CAPABILITY AND 2- NILPOTENT MULTIPLIER OF FINITE DIMENSIONAL NILPOTENT LIE ALGEBRAS
7

Theorem 3.2. Let I be an ideal subalgebra of L such that I ⊆ Z2∗ (L). Then the
natural Lie homomorphism M(2) (L) → M(2) (L/I) is a monomorphism.
Proof. Let S/R and F/R be two free presentations of L and I, respectively. Looking
the natural homomorphism
φ : M(2) (L) ∼
= R ∩ F 2 /[[R, F ], F ] → M(2) (L/I) ∼
= R ∩ S 2 /[[S, F ], F ]
and the fact that S/R ⊆ Z2 (F/R) show φ has trivial kernel. The result follows. 
Theorem 3.3. A Heisenberg Lie algebra H(m) is 2-capable if and only if m = 1.
Proof. Let m ≥ 2, by Corollary 2.9 H(m) is not capable so it is not 2-capable as
well. Hence we may assume that L ∼
= H(1). Let I be an ideal of L od dimension
1. Then L/I is abelian of dimension 2, and hence dim M(2) (L) = 2. On the
other hands, Theorem 2.11 implies dim M(2) (L) = 5, and Theorem 3.2 deduces
M(2) (L) → M(2) (L/I) can not be a monomorphism, as required.

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School of Mathematics and Computer Science, Damghan University, Damghan, Iran
E-mail address: niroomand@du.ac.ir, p niroomand@yahoo.com
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
E-mail address: parvizi@math.um.ac.ir

