Capability of nilpotent Lie algebras of small dimension
F. Pazandeh Shanbehbazari, P. Niroomand, F.G. Russo, A. Shamsaki
aa r X i v : . [ m a t h . R A ] J u l CAPABILITY OF NILPOTENT LIE ALGEBRAS OF SMALL DIMENSION
FATEMEH PAZANDEH SHANBEHBAZARI, PEYMAN NIROOMAND, FRANCESCO G. RUSSO, AND AFSANEH SHAMSAKI
Abstract.
Given a nilpotent Lie algebra L of dimension ≤ = 2, we show adirect method which allows us to detect the capability of L via computations on the size of its nonabelian exteriorsquare L ∧ L . For dimensions higher than 6, we show a result of general nature, based on the evidences of the lowdimensional case, focusing on generalized Heisenberg algebras. Indeed we detect the capability of L ∧ L via the size ofthe Schur multiplier M ( L/Z ∧ ( L )) of L/Z ∧ ( L ), where Z ∧ ( L ) denotes the exterior center of L . Statement of the results and terminology
Throughout this paper all Lie algebras are considered over a prescribed field F and [ , ] denotes the Lie bracket.Let L and K be two Lie algebras. Following [6, 7], an action of L on K is an F -bilinear map L × K → K given by( l, k ) → l k satisfying [ l,l ′ ] k = l ( l ′ k ) − l ′ ( l k ) and l [ k, k ′ ] = [ l k, k ′ ] + [ k, l k ′ ] , for all l, l ′ ∈ L and k, k ′ ∈ K . These actions are compatible , if k l k ′ = k ′ l k and l k l ′ = l ′ k l for all l, l ′ ∈ L and k, k ′ ∈ K .Now L ⊗ K is the Lie algebra generated by the symbols l ⊗ k subject to the 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 ′ , where c ∈ F , l, l ′ ∈ L and k, k ′ ∈ K .In case L = K , one can find the nonabelian tensor square L ⊗ L as special case, and, if in addition L is abelian, weget the well known abelian tensor product L ⊗ L of L . The map κ : l ⊗ l ′ ∈ L ⊗ L [ l, l ′ ] ∈ L := [ L, L ] = h [ l, l ′ ] | l, l ′ ∈ L i turns out to be epimorphism from L ⊗ L to the derived subalgebra L . It is well known (see [6]) that both J ( L ) = ker κ and the diagonal ideal L (cid:3) L = h l ⊗ l | l ∈ L i are central in L ⊗ L . The quotient Lie algebra L ∧ L = ( L ⊗ L ) / ( L (cid:3) L ) = h l ⊗ l + ( L (cid:3) L ) | l ∈ L i = h l ∧ l | l ∈ L i is called nonabelian exterior square of L and it is easy to check that κ ′ : l ∧ l ′ ∈ L ∧ L [ l, l ′ ] ∈ L is epimorphism as well. Classical results from [7] show that ker κ ′ ≃ M ( L ), where M ( L ) denotes the Schur multiplier of L . There are various ways to define M ( L ), and one is to put M ( L ) = H ( L, Z ), that is, as the second dimensionalLie algebra of homology on L with coefficients in the ring Z of the integers.Of course, L/L is abelian, so L/L ⊗ L/L , and in particular L/L (cid:3) L/L , is well known, since we have a de-composition in the direct sum of ideals of dimension one for abelian Lie algebras (see [20]). In fact if L is of finitedimension dim L , then LL (cid:3) LL ≃ A (1) ⊕ A (1) ⊕ . . . ⊕ A (1) | {z } k − times , where k ≤ dim L and the abelian Lie algebra of dimension n is denoted by A ( n ). Therefore one could ask whethera similar decomposition is known more generally for L (cid:3) L and this has been recently investigated in [12, Proposition3.3, Theorems 3.6 and 3.12 ], where [ L, L ] is replaced by [
L, N ] with N suitable ideal of L .Originally, results of splitting of the diagonal ideal L (cid:3) L are related to the categorical properties of the universalquadratic Whitehead functor Γ, introduced by Whitehead [21] for groups and by Ellis [6] for Lie algebras. Among theproperties of Γ, one can note the existence of a natural epimorphism of Lie algebras γ ( l + L ) ∈ Γ( L/L ) l ⊗ l ∈ L (cid:3) L ,which has been recently studied in [12] in connection with L (cid:3) L . In fact the size of L (cid:3) L may be related to the propertyof a Lie algbra of being isomorphic to the central quotient of another Lie algebra, that is, to the notion of capability;i.e.: we say that a Lie algebra L is capable if L ≃ E/Z ( E ) for some other Lie algebra E . The exterior center Z ∧ ( L )turns out to be a central ideal of L and is defined by Z ∧ ( L ) = { l ∈ L | l ∧ l ′ = 0 , ∀ l ′ ∈ L } Date : July 20, 2020.
Key words and phrases.
Tensor square, exterior square, capability, Schur multiplier.
Mathematics Subject Classification 2010.
Primary 17B30; Secondary 17B05, 17B99. and [6, 7] show that L is capable if and only if Z ∧ ( L ) = 0. A series of contributions focused on the property of beingcapable via the size of the exterior centre after [6, 7].Our first main result deals with those nilpotent Lie algebras of finite dimension, classified in [3, 8] and reportedbelow in Theorem 2.6. Note that there are no assumptions on the nature of the ground field of the Lie algebra here. Theorem 1.1.
The only noncapable nilpotent Lie algebras of dimension at most are L , , L , , L , , L , , L , , L , , L , and A (1) . The notations, introduced in Theorem 1.1, are recalled in Section 2 and follow [3, 8]. A computational aspect isdiscussed in Sections 2 and 3, in order to find the nonabelian exterior square, the nonabelian tensor square and theSchur multipliers of all nilpotent Lie algebras up to dimension six. The main results appear in Section 3.Our second result is motivated by an evidence which we noted up to the dimension six. We can basically controlsome extremal situations but we cannot control the capability of L ∧ L completely via that of L . Theorem 1.2.
Assume that L is a nonabelian nilpotent Lie algebra of finite dimension n ≥ and that L /Z ∧ ( L ) iscapable. Then Z ∧ ( L ∧ L ) is isomorphic to a subalgebra of M ( L/Z ∧ ( L )) . In other words, Theorem 1.2 shows that the capability of L ∧ L can be controlled by that of L /Z ∧ ( L ) for highdimensions, provided L /Z ∧ ( L ) is capable. Notations and terminology are standard and follow [1, 4, 5, 15, 16, 20].2. Preliminaries up to dimension five
Some well known properties of the Ganea map are summarised here. This is done just for convenience of the reader.
Proposition 2.1 (See Corollary 2.3 in [17] and Proposition 4.1 in [1]) . Let L be a Lie algebra and N be a central ideal of L . (i) N ⊆ Z ∧ ( L ) if and only if M ( L ) → M ( L/N ) is a monomorphism, (ii) N ⊆ Z ∧ ( L ) if and only if L ∧ L → L/N ∧ L/N is a monomorphism, (iii) L ∧ N → L ∧ L → L/N ∧ L/N → is exact, (iv) M ( L ) → M ( L/N ) → N ∩ L → is exact. While Proposition 2.1 is very general and does not involve any additional property on the Lie algebras, the capabilityof certain families of nilpotent Lie algebras has been recently studied in [10, 11, 17], because they play a fundamentalrole when we want to extend the arguments of classification via dimension.From [17], we recall that a finite dimensional Lie algebra L is Heisenberg provided that L = Z ( L ) and dim L = 1. Such algebras are odd dimensional with basis x , · · · , x m , x and the only nonzero multiplication betweenbasis elements is x = [ x i − , x i ] = − [ x i , x i − ], for i = 1 , · · · , m . The Heisenberg Lie algebra of dimension 2 m + 1 isdenoted by H ( m ).In order to look at Schur multipliers of these Lie algebras we recall a technical result, called K¨unneth Formula (see[20]). This result has categorical nature, but we report it in the way in which we will use it.
Lemma 2.2 (See Theorem 1 in [2] and page 107 in [6] ) . For two Lie algebras H and K we have : (i) ( H ⊕ K ) ⊗ ( H ⊕ K ) ∼ = ( H ⊗ H ) ⊕ ( K ⊗ K ) ⊕ ( H ⊗ K ) ⊕ ( K ⊗ H ) , (ii) ( H ⊕ K ) ∧ ( H ⊕ K ) ∼ = ( H ∧ H ) ⊕ ( K ∧ K ) ⊕ ( H/H ⊗ K/K ) . It is also useful to recall here Schur multipliers of H ( m ) and A ( n ). Lemma 2.3 (See Corollary 2.3, [17]) . (i) dim M ( A ( n )) = n ( n − , (ii) dim M ( H (1)) = 2 , (iii) dim M ( H ( m )) = 2 m − m − for all m ≥ . Capable Lie algebras of dimension n with dim L = 1 are determined below and involve Heisenberg and abelianLie algebras only. This is a first evidence of the role of Heisenberg algebras in the general theory. Lemma 2.4 (See Theorem 3.6, [17]) . Let L be a nilpotent Lie algebra of dimension n such that dim L = 1 . Then L ∼ = H ( m ) ⊕ A ( n − m − for some m > and L is capable if and only if m = 1 . Removing the assumption that the derived subalgebra has dimension one in Lemma 2.4, we may ask whether wehave alternative conditions to determine capable Heisenberg algebras and capable abelian Lie algebras or not. Theanswer is positive and reported below.
Lemma 2.5 (See Theorems 3.3, 3.4 and 3.5, [17]) . A ( n ) is capable if and only if n ≥ , and H ( m ) is capable if andonly if m = 1 . Moreover, for all k ≥ , the nilpotent Lie algebra H ( m ) ⊕ A ( k ) is capable if and only if m = 1 . Recall from [11] that L is a semidirect sum of an ideal I by a subalgebra K if L = I + K and I ∩ K = 0. Wewrite L = K ⋉ I , when these circumstances occur. Of course, if both K and I are ideals of L and K ∩ I = 0,then the decomposition (in direct sum) can be seen as a special case of the notion of semidirect sum. Some importantexamples of semidirect sums are offered by the generalized Heisenberg algebras ; i.e.: a Lie algebra L is called generalizedHeisenberg of rank n ≥
1, if H = Z ( H ) and dim H = n . These have been studied in [11] and allow us to generalizeLemma 2.5. APABILITY OF NILPOTENT LIE ALGEBRAS OF SMALL DIMENSION 3
Nevertheless the relevance of Lemma 2.5 appears in the classification of finite dimensional nilpotent Lie algebras viadimension only. In fact Lemma 2.5 offers a first elementary criterion to detect capable Lie algebras among those whichare nilpotent and of finite dimension. The classification below is reported from [3, 8] and we enrich the statement witha description in terms of semidirect sum.
Theorem 2.6 (Classification of Nilpotent Lie Algebras of Dimension ≤ . Let L be a finite dimensional nilpotentLie algebra (over F of characteristic = 2 ). Then (1) dim L = 3 if and only if L is isomorphic to one of the following Lie algebras: - L , = A (3) , - L , = H (1) ≃ A (1) ⋉ A (2) . (2) dim L = 4 if and only if L is isomorphic to one of the following Lie algebras: - L , = A (4) , - L , = H (1) ⊕ A (1) , - L , = h x , x , x , x | [ x , x ] = x , [ x , x ] = x i ≃ A (1) ⋉ A (3) . (3) dim L = 5 if and only if L is isomorphic to one of the following Lie algebras: - L , = A (5) , - L , = H (1) ⊕ A (2) , - L , = L , ⊕ A (1) , - L , = H (2) , - L , = h x , x , x , x , x | [ x , x ] = x , [ x , x ] = [ x , x ] = x i ≃ A (1) ⋉ L , , - L , = h x , x , x , x , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = [ x , , x ] = x i , - L , = h x , x , x , x , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i , - L , = h x , x , x , x , x | [ x , x ] = x , [ x , x ] = x i , - L , = h x , x , x , x , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i . (4) dim L = 6 if and only if L is isomorphic to one of the following Lie algebras - L ,k = L ,k ⊕ A (1) for k = 1 , , . . . , , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i ≃ A (1) ⋉ L , , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i ≃ A (1) ⋉ L , , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i ≃ A (1) ⋉ L , , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = − x i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = − x i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i , - L , ( ε ) = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = εx i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x = x , [ x , x ] = x i , - L , ( ε ) = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = εx i , - L , ( ε ) = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = εx , [ x , x ] = x i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i , - L , ( ε ) = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = εx , [ x , x ] = x , [ x , x ] = x i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i , - L , = h x , · · · , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i . Note that the presence of the parameter ε , which is a scalar of the ground field F , determines L , ( ε ) ≃ L , ( δ )if and only if δε − is a square. Similarly, L , ( ε ) ≃ L , ( δ ) if and only if δε − is a square, and the same conditiongives criteria of isomorphisms for L , ( ε ) ≃ L , ( δ ) and L , ( ε ) ≃ L , ( δ ).The main idea of the present paper is to describe the nonabelian exterior square and the capability of the aboveLie algebras, but complications arise when the dimension is higher than six. Remark . There are only finitely many ismorphism classes of nilpotent Lie algebras of dimension n <
7. Indimension n = 7 there exist 1-parameter families of mutually non-isomorphic nilpotent Lie algebras. A classificationhas been achieved for n = 7 over the real and complex numbers, by many different authors [9, 18].The above remark shows our motivation to focus up to the case of dimension six. On the other hand, we will usea numerical argument for higher dimensions, loosing information in terms of generators and relations. Unfortunately,this difficulty is related to the nature of the topic. For sure, the abelian case and that of Heisenberg algebras are clearby Lemmas 2.3, 2.5 and the following result. Lemma 2.8 (See Lemma 3.2, [17] ) . For all n ≥ , we have A ( n ) ⊗ A ( n ) ≃ A ( n ) and A ( n ) ∧ A ( n ) ≃ A ( n ( n − .For all m ≥ , we have H ( m ) ⊗ H ( m ) ≃ A (4 m ) and H ( m ) ∧ H ( m ) ≃ A ( m (2 m − ≥
4. Moreover Lemmas 2.5 and 2.8 allow us to reduce
F. PAZANDEH SHANBEHBAZARI, P. NIROOMAND, F.G. RUSSO, AND A. SHAMSAKI to the nonabelian case of ≥
4, to the case of ≥ Lemma 2.9.
We have (i) M ( L , ) ≃ A (6) , M ( L , ) ≃ A (4) and M ( L , ) ≃ A (2) ; (ii) L , ∧ L , ≃ A (6) , L , ∧ L , ≃ A (5) , L , ∧ L , ≃ A (4) ; (iii) L , ⊗ L , ≃ A (16) , L , ⊗ L , ≃ A (11) , L , ⊗ L , ≃ A (7) . (iv) L , (cid:3) L , ≃ A (10) , L , (cid:3) L , ≃ A (6) , L , (cid:3) L , ≃ A (3) ;Proof. (i). These Schur multipliers can be found in [2]. (ii), (iii) and (iv) follow from Lemmas 2.8, 2.2 and [10,Corollary 2.14 (e)] (cid:3) Let’s analyse the case of dimension five.
Lemma 2.10.
The Schur multipliers of the nilpotent Lie algebras L ,k (with k = 1 , · · · , ) are the following: M ( L ,k ) ∼ = A (10) if k = 1 ,A (7) if k = 2 ,A (4) if k = 3 , ,A (5) if k = 4 ,A (3) if k = 6 , , ,A (6) if k = 8 . Proof.
The case L , follows by Lemma 2.3. The case L , by Lemmas 2.2 and 2.3. Similarly we have the case of L , .Now consider L , and, instead of applying Lemma 2.8, we use method of Hardy and Stitzinger in [2]. We use[ x , x ] = x + s , [ x , x ] = s , [ x , x ] = s , [ x , x ] = s [ x , x ] = s , [ x , x ] = s , [ x , x ] = s , [ x , x ] = x + s , [ x , x ] = s , [ x , x ] = s , Where { s , · · · s } generate M ( L ). A change of variables s = s = 0. Next we use of the Jacobi identity on allpossible triples. Shows [ x , [ x , x ]] + [ x , [ x , x ]] + [ x , [ x , x ]] =0 , [ x , s ] + [ x , s ] + [ x , x ] =0 ,s = 0 . By a similar technique, we have s = s = 0. Thus dim M ( L ) = 5.By the same argument, we can see that dim M ( L , ) = 4, dim M ( L , ) = 3, dim M ( L , ) = 3, dim M ( L , ) = 6and dim M ( L , ) = 3. (cid:3) The following lemma summarises some well known facts when 1 ≤ k ≤
5, but there are additional information for6 ≤ k ≤ Lemma 2.11.
For Lie algebras L ,k and positive integers k = 1 , · · · , we have L ,k ∧ L ,k ∼ = A (6) if k = 3 , , , ,A (8) if k = 2 , ,A (10) if k = 1 ,H (1) ⊕ A (3) if k = 6 , . Proof.
By Lemma 2.10, we have L , ∧ L , ∼ = A (10). Lemmas 2.2 and 2.9 imply( L , ∧ L , ) = ( L , ∧ L , ) ⊕ ( A (1) ∧ A (1)) ⊕ ( L , /L , ⊗ A (1)) ≃ A (8) . Similarly dim ( L , ∧ L , ) = 6 and it is an abelian Lie algebra. Now cosider L , . It is clear that ( L ∧ L ) ≤ L ∧ L .Since L , ≃ A (1), dim( L , ∧ L , ) = dim( L , ∧ L , ) = 0 by Lemma 2.3. Hence L , ∧ L , is abelian. By usingthe fact that dim L , ∧ L , = dim M ( L , ) + dim L , and Lemma 2.10, we have L , ∧ L , ∼ = A (6). The case of L ∼ = L , either can be found directly or refer to [10,Corollary 2.15 (f)]. Let L , = h x , . . . , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i by using [7, (6 ′ )]and the relations of L , we have x ∧ x = − ([ x , x ] ∧ x ) = − ( x ∧ [ x , x ] − x ∧ [ x , x ]) = x ∧ x ,x ∧ x = − ([ x , x ] ∧ x ) = − ( x ∧ [ x , x ] − x ∧ [ x , x ]) = x ∧ x ,x ∧ x = x ∧ x = 0 . APABILITY OF NILPOTENT LIE ALGEBRAS OF SMALL DIMENSION 5
Therefore { x ∧ x , x ∧ x , x ∧ x , x ∧ x , x ∧ x , x ∧ x } is a generating set for L ∧ L . Now we obtain the relationsof L ∧ L . By using [7, (7)] and the relations of L , we have[ x ∧ x , x ∧ x ] = [ x , x ] ∧ [ x , x ] = x ∧ x = − x ∧ x [ x ∧ x , x ∧ x ] = [ x , x ] ∧ [ x , x ] = x ∧ x = 0[ x ∧ x , x ∧ x ] = [ x , x ] ∧ [ x , x ] = x ∧ x = 0[ x ∧ x , x ∧ x ] = [ x , x ] ∧ [ x , x ] = x ∧ x = 0[ x ∧ x , x ∧ x ] = [ x , x ] ∧ [ x , x ] = x ∧ x = 0[ x ∧ x , x ∧ x ] = [ x , x ] ∧ [ x , x ] = x ∧ x = 0Hence L , ∧ L , ∼ = H (1) ⊕ A (3) . By the same argument, we have L , ∧ L , ∼ = H (1) ⊕ A (3), L , ∧ L , ∼ = A (8) and L , ∧ L , ∼ = A (6). (cid:3) Lemma 2.12.
For Lie algebras L ,k and positive integers k = 1 , · · · , we have L ,k (cid:3) L ,k ∼ = A (3) if k = 6 , , ,A (6) if k = 3 , , ,A (10) if k = 2 , ,A (15) if k = 1 . Proof.
Let L be a n -dimensional nilpotent Lie algebra with the derived subalgebra of dimention m . Then L (cid:3) L ∼ = L/L (cid:3) L/L and dim L/L (cid:3) L/L = ( n − m )( n − m + 1) by using [14, Lemma 2.1 and 2.2]. Now by inserting values n and m for all L ,k such that k = 1 , . . . , (cid:3) Corollary 2.13.
For Lie algebras L ,k and positive integers k = 1 , · · · , we have L ,k ⊗ L ,k ∼ = A (9) if k = 9 ,A (12) if k = 3 , ,A (14) if k = 8 ,A (16) if k = 4 ,A (18) if k = 2 ,A (25) if k = 1 ,H (1) ⊕ A (6) if k = 6 , . Proof.
By [14, Theorem 2.5], we have L ⊗ L ∼ = L ∧ L ⊕ L (cid:3) L. Now by using Lemmas 2.11 and 2.12 the result follows. (cid:3) Proofs of the main results
We begin to describe directly the Schur multipliers of L ,k for k = 1 , . . . , Proposition 3.1.
The Schur multiplier of 6-dimensional nilpotent Lie algebras are abelian algebras L ,k for k =1 , · · · , . In particular, M ( L ,k ) ∼ = A (2) if k = 14 , ,A (3) if k = 15 , , ,A (4) if k = 13 , ∀ ε ) , ,A (5) if k = 6 , , , , , ∀ ε ) , , ∀ ε ) ,A (6) if k = 10 , , , ,A (7) if k = 3 , ,A (8) if k = 22( ∀ ε ) , ,A (9) if k = 4 , ,A (11) if k = 2 ,A (15) if k = 1 . .Proof. By Lemma 2.3, we have dim M ( L , ) = 15. By Lemmas 2.2 and 2.10, the dimension of M ( L ,k ) is obtainedfor k = 1 , · · · ,
9. Now consider L ∼ = L , and apply the argument we have used in Lemma 2.11. Start with[ x , x ] = x + s , [ x , x ] = x + s , [ x , x ] = s , [ x , x ] = s , [ x , x ] = s [ x , x ] = s , [ x , x ] = s , [ x , x ] = s , [ x , x ] = s , [ x , x ] = s [ x , x ] = s , [ x , x ] = s , [ x , x ] = x + s , [ x , x ] = s [ x , x ] = s , F. PAZANDEH SHANBEHBAZARI, P. NIROOMAND, F.G. RUSSO, AND A. SHAMSAKI where { s , · · · s } generated M ( L ). A change of variables allows that s = s = 0. By use of the Jacobi identity onall possible triples, we have s = s = s = s = s = s = 0. Thus dim M ( L ) = 6.Now let L ∼ = L , . Start with[ x , x ] = x + s , [ x , x ] = x + s , [ x , x ] = x + s , [ x , x ] = s , [ x , x ] = s , [ x , x ] = x + s , [ x , x ] = s , [ x , x ] = x + s , [ x , x ] = s , [ x , x ] = s , [ x , x ] = s , [ x , x ] = s , [ x , x ] = x + s [ x , x ] = s [ x , x ] = s , where { s , · · · s } generate M ( L ).A change of variables shows that s = s = s = 0. Use of the Jacobi identity on all possible triples, we have[ x , [ x , x ]] + [ x , [ x , x ]] + [ x , [ x , x ]] =0[ x , x ] − [ x , x ] =0 . Thus s = s . Again use of the Jacobi identity, we have[ x , [ x , x ]] + [ x , [ x , x ]] + [ x , [ x , x ]] =0 − [ x , x ] − [ x , x ] =0 . Therefore s = − s . By a same way we have s = s = s = s = s = 0. Thus dim M ( L ) = 5. The rest ofproof is obtained by a similar technique, using the same idea (or via GAP [19]) when 12 ≤ k ≤ (cid:3) Now we describe the nonabelian exterior square up to dimension 6.
Proposition 3.2.
For Lie algebras L ,k with k = 1 , . . . , , we have L ,k ∧ L ,k ∼ = A (7) if k = 13 ,A (8) if k = 9 , , ∀ ε ) , , ∀ ε ) ,A (9) if k = 3 , , , , ,A (10) if k = 4 , ∀ ε ) ,A (11) if k = 8 , ,A (12) if k = 2 ,A (15) if k = 1 ,H (1) ⊕ A (3) if k = 16 ,H (1) ⊕ A (4) if k = 15 , , ,H (1) ⊕ A (5) if k = 6 , , , , , ε = 0) ,L , ⊕ A (1) if k = 14 ,L , ⊕ A (3) if k = 21( ε = 0) , Proof.
By Lemmas 2.2 and 2.11, the result is clear for k = 1 , . . . ,
9. Let L ∼ = L , = h x , . . . , x | [ x , x ] = x , [ x , x ] = x , [ x , x ] = x i . By using [7, (6 ′ )] and the relations of L , we have x ∧ x = x ∧ [ x , x ] = − ( x ∧ [ x , x ] − x ∧ [ x , x ]) = 0 x ∧ x = x ∧ [ x , x ] = − ( x ∧ [ x , x ] − x ∧ [ x , x ]) = 0 x ∧ x = x ∧ x = x ∧ x = x ∧ x = x ∧ x = 0Therefore { x ∧ x , x ∧ x , x ∧ x , x ∧ x , x ∧ x , x ∧ x , x ∧ x , x ∧ x } is a generating set for L ∧ L . Now weobtain the relations of L ∧ L . By using [7, (7)] and the relations of L , we have[ x ∧ x , x ∧ x ] = [ x , x ] ∧ [ x , x ] = x ∧ x = 0[ x ∧ x , x ∧ x ] = [ x , x ] ∧ [ x , x ] = x ∧ x = 0[ x ∧ x , x ∧ x ] = [ x , x ] ∧ [ x , x ] = x ∧ x = 0Thus L ∧ L ∼ = A (8) . In a similar way, we obtain the exterior square of the remaining Lie algebras of dimension 6. (cid:3)
We have all that we need for the proof of the first main theorem.
Proof of Theorem 1.1.
A first distinction comes from the abelian case and from the nonabelian case, looking at The-orem 2.6. Clearly Lemma 2.5 shows that A (1) is noncapable and A ( n ) for n ≥ ≤ A (1): this means that there is no loss of generality in assuming in the restof the proof that the Lie algebras are nonabelian. We look again at Theorem 2.6 in case of dimension 3 and find thatthere are no noncapable Lie algebras by Lemma 2.5. The same happens also in dimension 4 looking at Theorem 2.6:indeed L , is capable by Lemma 2.5. By Proposition 2.1 (ii) and (iv), we can see that for any central ideal K ofdimension one in a finite dimensional nilpotent Lie algebra L ,( † ) dim M ( L ) = dim M ( L/K ) − dim( L ∩ K ) ⇐⇒ K ⊆ Z ∧ ( L ) . APABILITY OF NILPOTENT LIE ALGEBRAS OF SMALL DIMENSION 7
Let’s apply this when L ≃ L , , in order to show that Z ∧ ( L ) = 0. By Lemma 2.9, we have dim M ( L , ) = 2 . Notethat Z ( L , ) = h x i , so either Z ∧ ( L , ) = 0, or Z ∧ ( L , ) = Z ( L , ) . In the first case, the claim follows. In the secondcase L , /Z ( L , ) ≃ L , and dim M ( L , ) = 2 by Lemma 2.3 (i). Therefore ( † ) impliesdim M ( L , ) = dim M ( L , /Z ( L , ) − dim( L , ∩ Z ( L , ))and so Z ( L , ) = Z ∧ ( L , ) cannot happen. We conclude that L , is capable and that there are no noncapable Liealgebras of dimension 4.Now we consider Lie algebras of the dimension 5 and we see that here we have the first examples of noncapable Liealgebras. Of course, L , is capable by Lemma 2.5. Therefore we consider L , with K any ideal of dimension one in Z ( L , ) ≃ A (2). Depending on the possible two choices of K in Z ( L , ), the quotient L , /K is isomorphic either to L , , or to L , . Also dim M ( L , ) = 4 , dim M ( L , ) = 6 and dim M ( L , ) = 4 by Lemmas 2.9 and 2.10. Again weapply ( † ) and note that dim M ( L , ) = dim M ( L , /K ) − dim( L , ∩ K )for all one dimensional central ideals of L , , so we have necessarily Z ∧ ( L , ) = 0, hence L , is capable.Then we pass to consider L , and clearly this is noncapable by Lemma 2.5. Now we examine the case of L ,k , when k = 5 , . . . , L , with help of Lemma 2.10. The same idea applies in the case of dimension six and we find that L ,k (for k = 4 , , , , ,
20) are the only noncapable Lie algebras. Here Proposition 3.1 is used. Then the result follows. (cid:3)
Theorem 1.1 shows an evidence, which we report below.
Proposition 3.3.
Any finite dimensional nonabelian nilpotent Lie algebra of dimension ≤ has capable nonabelianexterior square.Proof. We begin to show the thesis up to dimension ≤
5. If L has dim L ≤
5, then Theorem 2.6 shows that dim L ≤ L = 1, hence [10, Theorem 2.15 (i)] and Lemma 2.5 show that L ∧ L is always capable, because we get an abelian Lie algebra A ( n ) of dimension n always bigger than 2. The secondoption is that dim L = 2, hence [10, Theorem 2.15 (ii) and (iii)] and Lemma 2.5 show that L ∧ L is always capable,because we get an abelian Lie algebra A ( n ) of dimension always bigger than 4. Finally, one can see that the fivedimensional nonabelian nilpotent Lie algebras with derived subalgebra of dimension 3 can be the remaining L , , L , , L , and L , and here the result follows by Lemma 2.11.Assume now that dim L ≤ L , ∧ L , and L , ∧ L , . First of all we look at generators and relations in Theorem 2.6 andrecognize easily that L , is generalized Heisenberg of rank two. Then we invoke [11, Proposition 2.4] which shows thatthe capability of L , ⊕ A (1) is equivalent to that of the factor which is generalized Heisenberg. Since L , is capable,we may conclude that L , ⊕ A (1), hence L , ∧ L , , is capable. The same argument applies to L , ∧ L , and theresult follows. (cid:3) Noting that the derived subalgebra of a Lie algebra of dimension ≤ Question . When can we detect the capability of L ∧ L from that of L /Z ∧ ( L ) ? In other words, it is interestingto decide when the following implication is true: L /Z ∧ ( L ) nilpotent capable Lie algebra ⇒ L ∧ L nilpotent capable Lie algebraThere are important evidences which motivate the above question. Example . Consider the Lie algebra L ≃ H (2) ⊕ A (1). Here L ≃ [ H (2) , H (2)] ≃ A (1) which is noncapable byLemma 2.5, but L ∧ L ≃ A ( k ) with k ≥ L /Z ∧ ( L )is the trivial Lie algebra, which is of course capable. Therefore L noncapable may imply that L ∧ L is capable, butdefinitively this example supports Question 3.4 positively. Another evidence is offered by the generalized Heisenbergalgebras, studied in [10, 11, 12, 13]. If K is a d -generated generalized Heisenberg algebra of rank d ( d − K iscapable by [13, Theorem 2.7]. In particular, K ≃ A ( d ( d − d ≥
3. This means that K /Z ∧ ( K ) is capable and in fact K ∧ K ≃ A ( k ) with k ≥ Proof of Theorem 1.2.
Since L is nonabelian and nilpotent, L = L and dim L ≥ L/L ≥
2. Thisimplies Z ∧ ( L/L ) = 0, because L/L is an abelian capable Lie algebra by Lemma 2.5 (i), and so Z ∧ ( L ) ⊆ L concluding that Z ∧ ( L ) is an ideal of L , that is, the Lie algebra quotient L /Z ∧ ( L ) is well defined.Proposition 2.1 (ii) shows that the map π : x ∧ y ∈ L ∧ L ( x ∧ y ) + Z ∧ ( L ) ∈ L/Z ∧ ( L ) ∧ L/Z ∧ ( L )is an isomorphism of Lie algebras, so it will be sufficient to show that L/Z ∧ ( L ) ∧ L/Z ∧ ( L ) is capable, in order toconclude that so is L ∧ L . F. PAZANDEH SHANBEHBAZARI, P. NIROOMAND, F.G. RUSSO, AND A. SHAMSAKI
Note that the central extension0 −→ M (cid:18) LZ ∧ ( L ) (cid:19) −→ LZ ∧ ( L ) ∧ LZ ∧ ( L ) −→ (cid:20) LZ ∧ ( L ) , LZ ∧ ( L ) (cid:21) −→ L/Z ∧ ( L ) ∧ L/Z ∧ ( L ) M ( L/Z ∧ ( L )) ≃ (cid:20) LZ ∧ ( L ) , LZ ∧ ( L ) (cid:21) = L + Z ∧ ( L ) Z ∧ ( L ) ≃ L Z ∧ ( L ) ⇒ Z ∧ (cid:18) L/Z ∧ ( L ) ∧ L/Z ∧ ( L ) M ( L/Z ∧ ( L )) (cid:19) ≃ Z ∧ (cid:18) L Z ∧ ( L ) (cid:19) = 0 , where the last condition is possible because L /Z ∧ ( L ) is capable.Since the following inclusion is true: Z ∧ ( L/Z ∧ ( L ) ∧ L/Z ∧ ( L )) M ( L/Z ∧ ( L )) ⊆ Z ∧ (cid:18) L/Z ∧ ( L ) ∧ L/Z ∧ ( L ) M ( L/Z ∧ ( L )) (cid:19) , we get Z ∧ ( L/Z ∧ ( L ) ∧ L/Z ∧ ( L )) ⊆ M ( L/Z ∧ ( L ))and the result follows. (cid:3) References [1] V. Alamian, H. Mohammadzadeh and A. R. Salemkar, Some properties of the Schur multiplier and covers of Lie algebras,
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