SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE
p-FILIFORM NILRADICALS AND MAXIMAL COMPLEMENTED SPACE OF ITS
NILRADICAL
J. Q. ADASHEV1, L.M. CAMACHO2, B. A. OMIROV1,3
1Ins i u e o Ma hema ics, Uzbekis an Academy o Sciences, 100170, Tashken , Uzbekis an,
adashe jq@mail. u
2Dp o. Ma emá ica Aplicada I. Uni e sidad de Se illa. A da. Reina Me cedes, s/n. 41012 Se illa.
(Spain) E-mail add ess: lc[email p o ec ed]
3Na ional Uni e si y o Uzbekis an, 4, Uni e si y s ., 100174, Tashken , Uzbekis an,
omi [email p o ec ed]
Abs ac . The p esen a icle is a pa o he s udy o sol able Leibniz algeb as wi h a gi en
nil adical. In his pape sol able Leibniz algeb as, whose nil adicals is na u ally g aded p-fili o m non-
Lie Leibniz algeb a (n−p≥4) and he complemen ed space o nil adical has maximal dimension, a e
desc ibed up o isomo phism. Mo eo e , among ob ained algeb as we indica e he igid and comple e
algeb as.
1. In oduc ion
Du ing he las decades he heo y o Leibniz algeb as has been ac i ely in es iga ed and many
esul s o he Lie Theo y ha e been ans e ed o Leibniz algeb as.
Le i’s decomposi ion asse s ha e e y ini e-dimensional Lie algeb a is a semidi ec sum o a
semisimple Lie subalgeb a and sol able adical [16], while semisimple Lie algeb as o e he ield o
complex numbe s ha e been classi ied by E. Ca an [11] and o e he ield o eal numbe s by F.
Gan mache [12]. Thus, he p oblem desc ip ion o ini e-dimensional Lie algeb as is educed o he
s udy o sol able Lie algeb as. Till p esen he classi ica ion o sol able Lie algeb as is known o
dimensions no g ea e han six [13], [22]. Also he e a e se e al wo ks de o ed o he classi ica ion o
sol able Lie algeb as in an a bi a y ini e-dimensions [2–4], [20,23,24]. In ac , he e a e sol able Lie
algeb as cons uc ed using he me hod explained in [21].
Fo ini e-dimensional Leibniz algeb as o e a ield o ze o cha ac e is ic, he e is an analogue o
Le i’s decomposi ion: any Leibniz algeb a is decomposed in o a semidi ec sum o a semisimple Lie
algeb a and i s sol able adical [6]. The e o e, simila o Lie case, he main p oblem o he s udy o
Leibniz algeb as educed sol able ones.
In he pape [10], he me hod ha desc ibes sol able Lie algeb as by means o i s adical is adap ed
o Leibniz case.
Since he desc ip ion o ini e-dimensional sol able Leibniz algeb as is a boundless p oblem (e en
o sol able Lie algeb as), new app oaches a e de eloping. Rele an ools o geome ic app oaches
a e p ope ies o Za iski opology and he na u al ac ion o linea educ i e g oup on a ie ies o
algeb as in a such way ha o bi s unde he ac ion consis s o isomo phic algeb as. I is a well-known
esul o algeb aic geome y ha any algeb aic a ie y (e iden ly, algeb as de ined ia iden i ies o m
an algeb aic a ie y) is a union o a ini e numbe o i educible componen s. The mos impo an
algeb as a e hose whose o bi s unde he ac ion a e open se s in sense o Za iski opology (such algeb a
a e called igid algeb as). The algeb as o a a ie y wi h open o bi s a e impo an since he closu es
o o bi s o such algeb as o m i educible componen s o he a ie y. A he same ime he e exis s an
2010 Ma hema ics Subjec Classi ica ion. 17A32, 17A36, 17B30, 17B56.
Key wo ds and ph ases. Leibniz algeb a, na u al g ada ion, p-fili o m algeb a, sol abili y, nil adical, de i a ion, he
second g oup o cohomology.
1
2 SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE P-FILIFORM NILRADICALS
i educible componen which is no he closu e o o bi o any algeb a. This ac does no de ac he
impo ance o algeb as wi h open o bi s.
This was a mo i a ion o many wo ks ocused o disco e ing o algeb as wi h open o bi s and o
desc ip ion o su icien p ope ies o such algeb as [7,14,15].
The aim o his a icle is o desc ibe sol able Leibniz algeb as wi h na u ally g aded non-Lie p-
ili o m nil adicals and wi h maximal dimension o complemen ed space o i s nil adical. Namely, in
a bi a y ini e dimension, we go h ee ypes o such algeb as (R(µ1, k), R(µ2, k)and R(µ3, k + 2))
and we es ablished ha he algeb a R(µ3, k + 2) is comple e and cohomologically igid.
Th oughou he pape we shall conside ini e-dimensional ec o spaces and complex algeb as.
Mo eo e , in he mul iplica ion able he omi ed p oduc s a e assumed o be ze o and we shall
conside non-nilpo en sol able algeb as (unless s a ed o he wise).
2. P elimina ies
We ecall he necessa y backg ound in o de o make he comp ehensi e pape .
De ini ion 2.1. [18] A Leibniz algeb a Lis a ec o space o e Fequipped wi h a bilinea map
(mul iplica ion) sa is ying he Leibniz iden i y
x, [y, z]=[x, y], z−[x, z], y
o all x, y, z ∈L.
We e e eade s o wo ks [18] and [19] o examples o Leibniz algeb as.
Fu he we will use he ollowing no a ion
L(x, y, z) = [x, [y, z]] −[[x, y], z] + [[x, z], y].
I is ob ious ha he iden i y L(x, y, z) = 0 de e mines he Leibniz algeb as.
Fo a gi en Leibniz algeb a Lwe can de ine he ollowing wo-sided ideals
Ann (L) = {x∈L|[y, x] = 0, o all y∈L},
Cen e (L) = {x∈L|[x, y] = [y, x] = 0, o all y∈L}
called he igh annihila o and he cen e o L, espec i ely. F om he Leibniz iden i y we conclude
ha he ollowing elemen s [x, x],[x, y] + [y, x]in Ann (L) o any x, y ∈L.
A linea map d:L→Lo a Leibniz algeb a Lis said o be a de i a ion i o all x, y ∈L, he
ollowing condi ion holds:
d([x, y]) = [d(x), y] + [x, d(y)].(2.1)
The se o all de i a ions o L(deno ed by De (L)) o ms a Lie algeb a wi h espec o he commu a o .
No e ha he ope a o o igh mul iplica ion on elemen s x∈L( u he deno ed by Rx) is a
de i a ion, which is called inne de i a ion.
De ini ion 2.2. A Leibniz algeb a Lis called comple e i Cen e (L) = 0 and all de i a ions o La e
inne .
Analogously o Lie algeb as, we de ine he ollowing sequences:
L1=L, Lk+1 = [Lk, L], k ≥1, L[1] =L, L[s+1] = [L[s], L[s]], s ≥1,
so-called he lowe cen al and he de i ed se ies o L, espec i ely.
De ini ion 2.3. A Leibniz algeb a Lis nilpo en ( espec i ely, sol able), i he e exis s n∈N(m∈N)
such ha Ln= 0 ( espec i ely, L[m]= 0).
The maximal nilpo en ideal o a Leibniz algeb a is said o be he nil adical o he algeb a.
An analogue o Muba akzjano ’s me hods has been applied o sol able Leibniz algeb as which
shows he impo ance o he conside a ion o non-cha ac e is ically nilpo en Leibniz algeb a [10].
Conside a sol able Leibniz algeb a R=N⊕Qwi h he nil adical Nand complemen a y ec o
space Qo Nwi h a basis {x1,...,xm}. I is known ha o an elemen x∈Q he ope a o Rx|N
is a non-nilpo en de i a ion o N. Mo eo e , o any scala s {α1,...,αm} ∈ C {0}, he ope a o
α1Rx1|N+··· +αmRxm|Nis non-nilpo en , which means ha he elemen s {x1,...,xm}a e nil-
independen . The e o e, he dimension o complemen a y ec o space o Nis no g ea e han he
maximal numbe o nil-independen de i a ions o N( [10, Theo em 3.2]).
SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE p-FILIFORM NILRADICALS 3
Fo a nilpo en Leibniz algeb a Land x∈L L2we conside he dec easing sequence C(x) =
(n1, n2,...,nk)as he dimensions o he Jo dan blocks o he ope a o Rx. On he se o such sequences
we conside lexicog aphic o de .
De ini ion 2.4. The sequence C(L) = max
x∈L L2C(x)is called he cha ac e is ic sequence o he Leibniz
algeb a L.
Simila o he Lie algeb as, we ha e he ollowing de ini ion.
De ini ion 2.5. A Leibniz algeb a Lis called p- ili o m i C(L) = (n−p, 1,...,1
|{z }
p
), whe e p≥0.
No e ha abo e de ini ion, when p > 0ag ees wi h he de ini ion o p- ili o m Lie algeb as [8].
Since in he case o Lie algeb as he e is no singly-gene a ed algeb a, he no ion o 0- ili o m algeb a
o Lie algeb as has no sense, while o he Leibniz algeb as case in each dimension he e exis s up o
isomo phism a unique null- ili o m algeb a [5].
De ini ion 2.6. Gi en an n-dimensional p- ili o m Leibniz algeb a L, pu Li=Li/Li+1,1≤i≤n−p,
and g L=L1⊕L2⊕ · · · ⊕ Ln−p. Then [Li, Lj]⊆Li+jand we ob ain he g aded algeb a g L. I g L
and La e isomo phic, g L∼
=L, we say ha Lis na u ally g aded.
In his pape , we conside na u ally g aded p- ili o m non-Lie Leibniz algeb as. Thei classi ica ion
is gi en in he nex heo em.
Theo em 2.7. [9] An a bi a y n-dimensional na u ally g aded non-spli non-Lie p- ili o m Leibniz
algeb a (n−p≥4) is isomo phic o one o he ollowing non-isomo phic algeb as:
p= 2kis e en
µ1:([ei, e1] = ei+1,1≤i≤n−2k−1,
[e1, j] = k+j,1≤j≤k, µ2:










[ei, e1] = ei+1,1≤i≤n−2k−1,
[e1, 1] = e2+ k+1,
[ei, 1] = ei+1,2≤i≤n−2k−1,
[e1, j] = k+j,2≤j≤k,
p= 2k+ 1 is odd
µ3:




[ei, e1] = ei+1,1≤i≤n−2k−2,
[e1, j] = k+1+j,1≤j≤k,
[ei, k+1] = ei+1,1≤i≤n−2k−2,
whe e {e1, e2,...,en−p, 1, 2,..., p}is a basis o he algeb a.
In o de o simpli y ou nex calcula ions, he ollowing change o basis in µ3:
e′
1= k+1, e′
2=e1− k+1, e′
i+1 =ei,2≤i≤n−2k−1, ′
j= j, ′
k+j= k+1+j,1≤j≤k,
allows o ob ain a mo e con enien o m o µ3:
µ3:([ei, e1] = ei+1,2≤i≤n−2k−1,
[e2, j] = k+j,1≤j≤k.
2.1. Cohomology Leibniz algeb as. Since in he las sec ion o his pape we s udy he cohomo-
logical igidi y o ob ained algeb as, we need some concep s o he second cohomology g oup o Leibniz
algeb as. Fo mo e de ails, we e e o [18], [19] and e e ences he ein. The second cohomology g oup
o a Leibniz algeb a Lwi h coe icien i sel is he quo ien space
HL2(L, L) := ZL2(L, L)/BL2(L, L),
whe e he elemen s ψ∈BL2(L, L)and ϕ∈ZL2(L, L)a e de ined by:
ψ(x, y) = [d(x), y] + [x, d(y)] −d([x, y]), o some linea map d∈Hom(L, L),
[x, ϕ(y, z)] −[ϕ(x, y), z] + [ϕ(x, z), y] + ϕ(x, [y, z]) −ϕ([x, y], z) + ϕ([x, z], y) = 0,(2.2)
espec i ely.
I is ob ious ha a Leibniz 2-cocycle ϕo a Leibniz algeb a Lis de e mined by he iden i y
Φ(ϕ)(x, y, z) = 0,whe e
Φ(ϕ)(x, y, z) = [x, ϕ(y, z)] −[ϕ(x, y), z] + [ϕ(x, z), y] + ϕ(x, [y, z]) −ϕ([x, y], z) + ϕ([x, z], y).
4 SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE P-FILIFORM NILRADICALS
De ini ion 2.8. A Leibniz algeb a Lis called cohomologically igid i HL2(L, L) = 0.
Due o esul s o he pape [5], we ha e ha a Leibniz algeb a is igid i he second cohomology
g oup wi h coe icien s in i sel is i ial.
3. Sol able Leibniz algeb as wi h abelian nil adical and maximal dimension o
complemen ed space Q.
In his sec ion we ecall some esul s o he pape [1], which will be used below.
We deno e by ak he k-dimensional abelian algeb as and by R(ak, s) he sol able Leibniz algeb a
wi h akas nil adical and sas he dimension o complemen ed space o ak.
Theo em 3.1. [1] The maximal possible dimension o algeb as o he amily R(ak, s)is equal o 2k,
ha is, s=k. Mo eo e , an a bi a y algeb a o he amily R(ak, k)is decomposed in o a di ec sum
o copies o wo-dimensional non- i ial sol able Leibniz algeb as.
Conside he sol able Leibniz algeb as L(γi)wi h nil adical akunde he condi ion ha
he complemen ed space o he nil adical ha e maximal dimension. Then he e exis s a basis
{ 1, 2,..., k, x1, x2,...,xk}o L(γi)such ha he mul iplica ion able has he o m:
L(γi) : [ i, xi] = i,[xi, i] = γi i,1≤i≤k,
whe e γi∈ {−1,0}.
The algeb a L(γi)is a igid algeb a o any γi∈ {−1,0},1≤i≤k, [1].
Lemma 3.2. Any au omo phism ϕo he algeb a L(γi)has he ollowing o m:
ϕ( i) = αi i, ϕ(xi) = βi i+xi,1≤i≤k,
whe e (1 + γi)βi= 0 o 1≤i≤k.
P oo . Le ϕbe an au omo phism o L(γi).Since he au omo phism o algeb a maps nil adical o
nil adical we can assume
ϕ( i) =
k
X
j=1
Di,j j, ϕ(xi) =
k
X
j=1
Fi,j j+
k
X
j=1
Hi,jxj,1≤i≤k.
F om he ollowing equali ies [ϕ( i), ϕ(xj)] = ϕ([ i, xj]) and [ϕ(xi), ϕ(xj)] = ϕ([xi, xj]) wi h 1≤
i≤k, we de i e 




Di,mHj,m = 0,1≤i6=j, m ≤k,
Di,mHi,m =Di,m,1≤i, m ≤k,
Fi,mHj,m +γmFj,mHi,m = 0,1≤i, j, m ≤k.
So, o a gi en alue o jwe ha e a linea sys em wi h espec o Hj,1, Hj,2, . . . , Hj,k.
Le us p o e ha o a ixed j, 1≤j≤k he e exis s only m0such ha Hj,m0= 1 and Hj,m = 0
wi h 1≤m6=m0≤k.
Le us suppose ha Hj,m0=Hj,m1= 1, hen we ge Di,m0=Di,m1= 0 o 1≤i6=j≤k. On he
o he hand, de (Di,m)k
i,m=1 = 0, ha is, we a i e a con adic ion. Wi hou loss o gene ali y, we can
assume ha Hj,j = 1 and Hj,i = 0 wi h 1≤j6=i≤k.
Then, we ob ain he ollowing es ic ions:
(Dj,j 6= 0, Dj,m = 0,1≤m6=j≤k,
Fi,j = 0,(1 + γj)Fj,j = 0,1≤i6=j≤k,
which imply
ϕ( i) = Di,i i, ϕ(xi) = Fi,i i+xi,1≤i≤k,
whe e (1 + γi)Fi,i = 0 and γi∈ {−1,0} o 1≤i≤k. 
4. sol able leibniz algeb as wi h n-dimensional na u ally g aded p- ili o m non-Lie
Leibniz algeb a and maximal dimension o Q.
In his sec ion we gi e a desc ip ion o sol able Leibniz algeb as whose nil adical is a na u ally
g aded p- ili o m Leibniz algeb a and he dimension o Qis maximal. Fi s ly, we ecall he de i a ions
o he algeb as µi, i = 1,2,3gi en in [1].
SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE p-FILIFORM NILRADICALS 5
4.1. De i a ions o algeb as µi, i = 1,2,3.
P oposi ion 4.1. Any de i a ion o he algeb a µ1has he ollowing ma ix o m:
D=A B
C D, wi h D =D1D2
0a1E+D1,
whe e
A=
n−2k
X
i=1
ia1ei,i +
n−2k−1
X
i=1
n−2k
X
j=i+1
aj−i+1ei,j, B =
2k
X
i=1
bie1,i +
k
X
i=1
bie2,k+i, C =
k
X
i=1
ciei,n−2k,
A∈Mn−2k,n−2k, B ∈Mn−2k,2k, C ∈M2k,n−2k, D1, D2,E∈Mk,k and ma ix uni s ei,j.
P oposi ion 4.2. Any de i a ion o he algeb a µ2has he ollowing ma ix o m:
D=A B
C D, wi h D =D1D2
0D3,
whe e
A=
n−2k
X
i=1
(ia1+ (i−1)b1)ei,i +
n−2k−1
X
i=1
n−2k
X
j=i+1
aj−i+1ei,j, B =
2k
X
i=1
bie1,i +
k
X
i=1
bie2,k+i,
C=
k
X
i=1
ciei,n−2k, D1=
k
X
i=1
k
X
j=2
di,jei,j + (a1+b1)e1,1, D3=D1+a1E−
k
X
j=1
bje1,j,
wi h A∈Mn−2k,n−2k, B ∈Mn−2k,2k, C ∈M2k,n−2k, D1, D2, D3,E∈Mk,k and ma ix uni s ei,j .
P oposi ion 4.3. Any de i a ion o he algeb a µ3has he ollowing ma ix o m:
D=A B
C D, wi h D =D1D2
0a2E+D1,
whe e
A=a1e1,1+
n−2k
X
i=2
((i−2)a1+a2)ei,i +βe1,n−2k+
n−2k−1
X
i=2
n−2k
X
j=i+1
aj−i+2ei,j,
B=
2k
X
i=1
b1,ie1,i +
k
X
i=1
b2,ie2,k+i+
k
X
i=1
b1,ie3,k+i, C =
k
X
i=1
ciei,n−2k,
wi h A∈Mn−2k,n−2k, B ∈Mn−2k,2k, C ∈M2k,n−2k, D1, D2,E∈Mk,k and ma ix uni s ei,j .
The heo em bellow desc ibes he maximal dimensions o he complemen ed space o µi, i = 1,2,3.
Theo em 4.4. Le Rbe a sol able Leibniz algeb a whose nil adical is µi, i = 1,2,3. Then he
dimension o complemen ed space o nil adical e i ies ha :
dim Q(µi)≤k+ 2 i
3.
P oo . Acco ding o P oposi ions 4.1 and 4.2, we ha e he ollowing exp esions o R(µ1, s)and
R(µ2, s), espec i ely:







[e1, x] =
n−2k
P
i=1
aiei+
2k
P
i=1
bi i,
[e2, x] = 2a1e2+
n−2k
P
i=3
ai−1ei+
k
P
i=1
bi k+i,







[e1, x] =
n−2k
P
i=1
aiei+
2k
P
i=1
bi i,
[e2, x] = (2a1+b1)e2+
n−2k
P
i=3
ai−1ei+
k
P
i=1
bi k+i,
Le us in oduce he ollowing no a ions:
[x, e1] =
n−2k
X
i=1
βiei+
2k
X
i=1
βn−2k+i i,[x, 1] =
n−2k
X
i=1
γiei+
2k
X
i=1
ϕi i.
The equali ies L(x, 1, e1) = L(e1, x, e1) = 0 imply a1= 0.
No e ha { 1, 2,..., k} o m he algeb a akand he space o de i a ions o he algeb a akcoincided
wi h Mk,k.

6 SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE P-FILIFORM NILRADICALS
I is easy o see ha
Rx|ak◦ Ry|ak=Ry|ak◦ Rx|ak
o any x, y ∈Q. This implies ha all ope a o s Rxi|ak,1≤i≤scould be simul aneously ans o med
o hei Jo dan o ms by a basis ans o ma ion. The e o e, he ma ix ope a o Rx|ak(in ou case
Rx|ak=D1) has he ollowing o m:
D1=









d1,1d1,20... 0 0
0d2,2d2,3... 0 0
0 0 d3,3... 0 0
.
.
..
.
..
.
..
.
..
.
..
.
.
0 0 0 . . . dk−1,k−1dk−1,k
0 0 0 ... 0dk,k









,
whe e di,i+1 ∈ {0,1} o 1≤i≤k−1.
Now we a e going o in es iga e he nilpo ency o ma ix D.Due o P oposi ions 4.1-4.3 he nilpo-
ency o Ddepends on he ma ices Aand D1.
Le us conside he ma ix Das ollows
D=A B
C D =A1+A2B
C K1+K2,
whe e A1, K1a e diagonal ma ices and A2, K2a e nilpo en such ha
A1=




diag{0,0,0,...,0}, o µ1,
diag{0, b1,2b1,...,(n−2k−1)b1}, o µ2,
diag{a1, a2, a1+a2,...,(n−2k−2)a1+a2}, o µ3,
K1=




diag{d1,1, d2,2,...,dk,k, d1,1, d2,2,...,dk,k}, o µ1,
diag{b1, d2,2,...,dk,k,0, d2,2,...,dk,k}, o µ2,
diag{d1,1, d2,2,...,dk,k, a2+d1,1, a2+d2,2,...,a2+dk,k}, o µ3.
I is easy o see ha CB = 0 and he ma ices A1A2, A2
2, BC, K1K2, K2
2a e nilpo en .
Mo eo e , ma ices C(A1+A2),(K1+K2)Cha e he ype o Cand ma ices (A1+A2)B, B(K1+K2)
ha e he ype o B.
Acco ding o he abo e a gumen s we ha e he ollowing ecu ence o mula:
D = A
1+e
A2e
B
e
C K
1+e
K2!, ≥1,
whe e e
A2,e
K2−nilpo en ma ices and ma ices e
B, e
Cha e he ypes o Band C, espec i ely.
To sum up, we conclude ha he ma ix Dis nilpo en i and only i A1and K1a e nilpo en s.
The e o e, we ob ain he ollowing conclusions:
•Fo µ1, he nilpo ency o Ddepends on di,i,1≤i≤k, ha is Dnilpo en i only i di,i =
0,1≤i≤k.
•Fo µ2, he nilpo ency o Ddepends on b1and di,i,2≤i≤k, ha is Dnilpo en i only i
b1=di,i = 0,2≤i≤k.
•Fo µ3, he nilpo ency o Ddepends on a1, a2and di,i,1≤i≤k, ha is Dnilpo en i only i
a1=a2=di,i = 0,1≤i≤k.
Applying he esul in [10, Theo em 3.2], he s a ed inequali ies ollow. 
The ollowing esul s will be used in he desc ip ion o sol able Leibniz algeb as whose nil adicals
a e µi, i = 1,2,3and wi h maximal dimensional complemen ed space o nil adicals.
P oposi ion 4.5. Le Rbe a sol able Leibniz algeb a whose nil adical is a na u ally g aded p- ili o m
non-Lie Leibniz algeb a. Then
{e1, 1,..., k} ∩ Ann (R) = 0 and {e2,...,en−2k, k+1,..., 2k} ⊆ Ann (R),
wi h a26= 0 o he algeb a R(µ3, s).
SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE p-FILIFORM NILRADICALS 7
P oo . Using Theo em 2.7 and he p ope ies o he igh annihila o ( ha is, [x, x],[x, y] + [y, x]∈
Ann (R)) he asse ion easily ollows o R(µ1, s)and R(µ2, s).
Conside he algeb a R(µ3, s).I is easy o see ha
e1, 1,..., k/∈Ann (R)and e3,...,en−2k, k+1,..., 2k∈Ann (R).
Le us suppose a26= 0. Then, om he de i a ion o µ3we ge
[e2, x] =
n−2k
X
i=2
aiei+
k
X
i=1
b2,i k+i,[x, e2] =
n−2k
X
i=1
αiei+
2k
X
i=1
βi i.
The equali y L(x, e2, e1) = 0 implies αi= 0 o 2≤i≤n−2k−1.Since [e2, x] + [x, e2]∈Ann (R)
and a26= 0, we ha e e2∈Ann (R)which comple e he p oo . 
Lemma 4.6. Le Rbe a sol able Leibniz algeb a whose nil adical is a na u ally g aded p- ili o m
Leibniz algeb a. Then he maximal sol able Leibniz subalgeb a wi h nil adical ak=< 1,..., k>o R
is isomo phic o L(γi)wi h γi=−1 o 1≤i≤k.
P oo . Clea ly, ak={ 1, 2,..., k} o ms an abelian subalgeb a o R. By Theo em 3.1 he maximal
sol able Leibniz algeb a wi h nil adical akis isomo phic o L(γi).Since [ i, xi] + [xi, i]∈Ann (R)
wi h 1≤i≤kand i/∈Ann (R)wi h 1≤i≤k, he p oo o lemma is comple e. 
In he ollowing heo em we p esen he desc ip ion o algeb as o he amily R(µ1, k).
Theo em 4.7. An a bi a y algeb a o he amily R(µ1, k)admi s a basis such ha he non- anishing
Leibniz b acke s become:
R(µ1, k)(ai,j, ϕi,j , δi,j) :























[ei, xj] =
n−2k
P
=i+1
a −i+1,je ,1≤i≤n−2k, 1≤j≤k,
[ i, xi] = i,1≤i≤k,
[ k+i, xi] = k+i,1≤i≤k,
[xi, i] = − i,1≤i≤k,
[xi, j] = ϕi,j k+j,1≤i6=j≤k,
[xi, xj] = δi,jen−2k,1≤i, j ≤k.
P oo . Acco ding o P oposi ions 4.1, 4.5, Theo em 4.4 and Lemma 4.6 we ha e he ollowing b acke s
o R(µ1, k):









































































[e1, xi] =
n−2k
P
=2
a ,ie +
2k
P
=1
b ,i ,1≤i≤k,
[e2, xi] =
n−2k
P
i=3
a −1,ie +
k
P
=1
b ,i k+ ,1≤i≤k,
[ej, xi] =
n−2k
P
=j+1
a −j+1,ie ,3≤j≤n−2k,
[ i, xi] = ci,ien−2k+ i+
2k
P
=k+1
d
i,i ,1≤i≤k,
[ i, xj] = ci,jen−2k+
2k
P
=k+1
d
i,j ,1≤i6=j≤k,
[ k+i, xi] = k+i,1≤i≤k,
[xi, e1] =
n−2k
P
=2
β ,ie −
k
P
=1
b ,i +
k
P
=1
βn−k+ ,i k+ ,1≤i≤k,
[xi, i] =
n−2k
P
=2
γ
i,ie − i+
2k
P
=k+1
ϕ
i,i ,1≤i≤k,
[xi, j] =
n−2k
P
=2
γ
i,je +
2k
P
=k+1
ϕ
i,j ,1≤i6=j≤k,
[xi, xj] =
n−2k
P
=1
δ
i,je +
2k
P
=k+1
θ
i,j ,1≤i, j ≤k.
8 SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE P-FILIFORM NILRADICALS
By aking he change o basis
e′
1=e1−
k
P
=1
b , , e′
2=e2−
k
P
=1
b , k+ ,
′
i= i−γi,ien−2k−
2k
P
=k+1
ϕ
i,i , x′
i=xi−
k
P
=1
θk+
i, k+ ,1≤i≤k,
we can assume bi,i =γn−2k
i,i =ϕk+
i,i =θk+
i, = 0,1≤i, ≤k.
Applying he Leibniz iden i y, he ollowing ela ions a e ob ained
(L(xi, j, e1) = 0,⇒γ
i,j = 0,1≤i, j ≤k, 2≤ ≤n−2k−1,
L(e1, xi, xj) = 0,⇒bi,j =bk+i,j =δ1
i,i =δ1
i,j = 0,1≤i6=j≤k.
Pu ing e′
1=e1−
k
P
=1
bk+ , k+ , x′
i=xi−
n−2k
P
=2
β ,ie −1,we conclude ha β ,i =bk+i,i = 0 wi h
1≤i≤k, 2≤ ≤n−2k.
Conside ing he Leibniz iden i y, we ob ain he ollowing es ic ions on s uc u e cons an s:















L(xi, i, xi) = 0,⇒ci,i =d
i,i = 0,1≤i≤k, k + 1 ≤ ≤2k,
L(xi, i, xj) = 0,⇒ci,j =d
i,j = 0,1≤i6=j≤k, k + 1 ≤ ≤2k,
L(xi, j, xj) = 0,⇒γn−2k
i,j =ϕk+
i,j = 0,1≤i6=j6= ≤k,
L(xi, e1, xj) = 0,⇒δ
i,j =βn−k+i,j = 0,1≤i, j ≤k, 2≤ ≤n−2k−1,
L(xi, xj, xs) = 0,⇒θk+s
i,j = 0,1≤i, j 6=s≤k.

Below he necessa y and su icien condi ions o he exis ence o an isomo phism be ween wo alge-
b as o he amily R(µ1, k)(ai,j, ϕi,j, δi,j)a e es ablished.
P oposi ion 4.8. Two algeb as R(µ1, k)′(a′
i,j, ϕ′
i,j, δ′
i,j)and R(µ1, k)(ai,j, ϕi,j, δi,j)a e isomo phic i
and only i he e exis s A∈C∗such ha
a′
i,j =ai,j
Ai−1,2≤i≤n−2k+ 1,1≤j≤k,
ϕ′
i,j =ϕi,j
A,1≤i6=j≤k, δ′
i,j =δi,j
An−2k,1≤i, j ≤k.
P oo . Taking in o accoun Lemma 3.2 we conside he gene al change o gene a o basis elemen s o
an algeb a om R(µ1, k)(ai,j , ϕi,j, δi,j):
e′
1=
n−2k
P
i=1
Aiei+
2k
P
i=1
Bi i, ′
i=
n−2k
P
j=1
Ci,jej+Di,i i+
2k
P
j=k+1
Di,j j,
x′
i=
n−2k
P
j=1
Ei,jej+Fi,i i+
2k
P
j=k+1
Fi,j j+xi,1≤i≤k.
F om he ollowing b acke s in R(µ1, k)′(a′
i,j, ϕ′
i,j , δ′
i,j) :
[e′
i, e′
1] = e′
i+1,1≤i≤n−2k−1,[ ′
i, e′
1] = 0,[e′
1, ′
i] = ′
k+i,1≤i≤k,
we de i e
e′
2=A1
n−2k
P
i=2
Ai−1ei+A1
k
P
i=1
Bi k+i, e′
i=Ai−1
1
n−2k
P
j=i
Aj−i+1ej,3≤i≤n−2k,
′
k+i=A1Di,i k+i, Ci,j = 0,1≤i≤k, 1≤j≤n−2k−1.
The ollowing anishing pa ame e s Bi=Ei,j =Ci,n−2k=Di,k+ = 0 wi h 1≤i, ≤kand
1≤j≤n−2k−1ha e been ob ained om he p oduc s:
[x′
i, e′
1] = 0,[x′
i, ′
i] = ′
i,1≤i≤k.
The e o e, we ob ain
e′
1=
n−2k
P
i=1
Aiei+
2k
P
i=k+1
Bi i, e′
i=Ai−1
1
n−2k
P
j=i
Aj−i+1ej,2≤i≤n−2k,
′
i=Di,i i, ′
k+i=A1Di,i k+i, x′
i=Ei,n−2ken−2k+Fi,i i+
2k
P
j=k+1
Fi,j j+xi,1≤i≤k.
SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE p-FILIFORM NILRADICALS 9
Le us conside he p oduc s
[e′
1, x′
j] =
n−2k
X
i=2
a′
i,je′
i,[x′
i, ′
j] = ϕ′
i,j ′
k+j,[x′
i, x′
j] = δ′
i,je′
n−2k,1≤i, j ≤k.
F om which we ge he ollowing es ic ions:









a′
i,j =ai,j
Ai−1
1
,2≤i≤n−2k, 1≤j≤k,
ϕ′
i,j =ϕi,j
A1,1≤i6=j≤k,
δ′
i,j =δi,j
An−2k
1
,1≤i, j ≤k,
whe e A1Fj,j +Bk+j=Fi,k+i=Fi,k+j+ϕi,j Fj,j = 0,1≤i6=j≤k. 
Below we desc ibe sol able Leibniz algeb as R(µ2, k).
Theo em 4.9. An a bi a y algeb a o he amily R(µ2, k)admi s a basis such ha i s mul iplica ion
able has he ollowing o m:
R(µ2, k)(bi, βi, ϕi,j, θi,j) :













































[e1, x1] = 1+b1 k+1,
[e2, x1] = e2+ k+1,
[ej, x1] = (j−1)ej,3≤j≤n−2k,
[x1, e1] = − 1+β1 k+1,
[e1, xi] = bi k+1,2≤i≤k,
[ i, xi] = i,1≤i≤k,
[ k+i, xi] = k+i,2≤i≤k,
[xi, e1] = βi k+1,2≤i≤k,
[xi, i] = − i,1≤i≤k,
[xi, j] = ϕi,j k+j,1≤i≤k, 2≤j≤k, i 6=j,
[xi, xj] = θi,j k+1,1≤i, j ≤k.
P oo . The desc ip ion o R(µ2, k) ollows om P oposi ion 4.2, 4.5, Theo em 4.4 and Lemma
4.6. In ac , i s ly, we conside de i a ions o µ2and since he pa ame e s b1, d2, d3,...,dka e
in he diagonal, we ha e only knil-independen de i a ions which co espond o he alues o
(b1, d2, d3,...dk) : (1,0,0, . . . , 0),(0,1,0,...,0), ..., (0,0,0,...,1). La e , assuming hese de i a-
ions as Rx1,Rx2, ..., Rxk( espec i ely) we comple e he p oo by applying simila a gumen s as
used in he p oo o Theo em 4.7. 
In he nex p oposi ion necessa y and su icien condi ions o he exis ence o an isomo phism be-
ween wo algeb as o he amily R(µ2, k)(bi, βi, ϕi,j, θi,j)a e es ablished.
P oposi ion 4.10. Two algeb as R(µ2, k)′(b′
i, β′
i, ϕ′
i,j, θ′
i,j)and R(µ2, k)(bi, βi, ϕi,j , θi,j)a e isomo -
phic i and only i he e exis s A∈C∗such ha
b′
i=bi
A,1≤i≤k, β′
i=βi
A,1≤i≤k,
ϕ′
1,i =ϕ1,i
A,2≤i≤k, ϕ′
i,j =ϕi,j
A,2≤i6=j≤k,
θ′
i,j =θi,j
A2,1≤i, j ≤k.
P oo . Analogously o he p oo o P oposi ion 4.8. 
To comple e he desc ip ion o sol able Leibniz algeb as wi h he nil adicals µi, i = 1,2,3and
maximal complemen ed space o nil adical, we gi e he ollowing heo em.