Applied Mathematics, 2011, 2, 1027-1030
doi:10.4236/am.2011.28142 Published Online August 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Enveloping Lie Algebras of Low Dimensional
Leibniz Algebras
Massoud Amini1,2, Isamiddin Rakhimov2, Seyed Jalal Langari2
1Department of Mat hem at i cs, Tarbiat Modares University, Tehran, Iran
2Institute for Math ematical Research (INSPEM) & Department of Mathematics, Selang or Darul Eh sa n, Malaysia
E-mail: mamini@modares.ac.ir, massoud@putra.upm.ed u.my, isamiddin@science.upm.edu.rny,
jalal_langari@yahoo.com
Received April 26, 2010; revised November 9, 2010; accepted June 25, 2011
Abstract
We calculate the enveloping Lie algebras of Leibniz algebras of dimensions two and three. We show how
these Lie algebras could be used to distinguish non-isomorphic (nilpotent) Leibniz algebras of low dimen-
sion in some cases. These results could be used to associate geometric objects (loop spaces) to low dimen-
sional Leibniz algebras.
Keywords: Leibniz Algebra, Enveloping Lie Algebra, Nilpotent Algebra
1. Introduction
In this paper, we work with vector spaces (and algebras)
over a field F of characteristic 0, although our results can
be extended in obvious way to the case of vector spaces
over a field of positive characteristic (not equal 2), or
even over a commutative ring with unit. By an alge-
bra , we mean a vector space L over F with a (not

,L
necessarily associative) bilinear operation . :LL L
For ,
x
L

:
x
LL
; denotes the left yxy
multiplication map. Let denotes the Lie

Der L
sub-algebra of

g
lLconsisting of the derivations on
L. Recall that a linear map

g
lL
is a derivation
of if and only if

,L
 
,
x
x



for all
x
L. Here, we work with a class of algebras in
which the left multiplication map has a stronger com-
patibility relation with derivations. These are Leibniz
algebras, introduced by J. L. Loday [1], as non-anti-
symmetric generalizations of Lie algebras.
Definition 1.1. A Leibniz algebra is a vector
space over a field F equipped with a bilinear map
L
:LL L
satisfying the Leibniz identity
 
.
x
yzxyzy xz , for all ,,
x
yz L
.
Obviously, a Lie algebra is a Leibniz algebra. A Leib-
niz algebra is a Lie algebra if and only if
0( )
x
xxL

Also, an algebra
,L
is a Leibniz algebra if and only if
x
Der L
or equivalently,


:, ,[,]LglL

is a homomorphism. Thus we have a homomorphism
:, ,[,]LglL

when
,L is a Leibniz alge-
bra.
Definition 1.2. If
,L
is a Leibniz algebra. We may
define 11
(LLL k,
kk
LL
1)
 . The series
123
LLL
is called the descending central series of . If the L
series terminates for some positive integer s, then
the Leibniz algebra is said to be nilpotent. L
2. Methods
The main tool to classify Low dimensional Leibniz alge-
bras is to find the corresponding enveloping Lie algebra
and fit them in the Beck-Kolman list of low dimensional
Lie algebras. Since these Lie algebras are realized as
certain quotients of the given Leibniz algebras, we first
need the following fact.
1028 M. AMINI ET AL.
Theorem 1.1. Let and be Leibniz alge-
1,L

2,L
bras. If then
1
LL2112 2
JLJ L, where
;
ii
J
xxxL is the ideal generated by squares in
i
L, for i = 1, 2.
Proof. Let 21
:LL
be an isomorphism. We de-
fine 11
:LJ L
2 2
J
that

such
21
x
Jx

 J.
It is easy to show that
is well defined and noto,
and
2
 


  
 
112 2
12 1122
122 122
112 2
xJ xJ
xx JxxJ
x
xJxJ xJ
xJ xJ

 





Also,
 

 

12
121
;0
;
Kerx JxJ
xJx JJ
xJxJ


 
 
 
11
;0
This theorem could be used to prove that some (nilpo-
tent) Leibniz algebras are non-isomorphic. This is im-
portant, as the nilpotent low dimensional Lie algebras are
al ready classified [2].
Example 1.1. Let ,and
111 21 3
:,Lee ee e 
2
e
211 123213
:,,eeeeeeeee .
3
L
Then 11
LJ is a one-dimensional abelian Lie alge-
bra, but 22
LJis a two-dimensional abele algebra, ian Li
therefore 11
LJis not isomorphic to 2
LJ.
2By the
previous theorem, 1
L is not isomorphic to 2
L.
We noted that Leibniz algebras are non-antisymmetric
in general. Hence, it is natural to consider the skew-
symmetrization of a Leibniz algebra

,L. This is done
through the skew-symmetrized binaryation oper


1
[,] 2
x
yyx 
for ,
x
yL. Note that, in general,

,[,]L is not a
Lie algebebniz ra. On the other haLind, by definition of
algebra,

xDerL
,[,]for all
x
L, and


:,[,]LDerL
  is a homo of anti- morphism
commutative algebras. Let
;
J
xxxL 
be the o-sided ideal of

,Ltw
generated bys. all square
Then
J
coic products ntains all symmetr
x
yyx
,
for ,
x
yL, and since
 
,
x
xxx

 

for all
x
L, we have

kerJ
M
Lbe any ideal containing
J
Let , then since
x
y Myxxyyxyx M
for ,
x
yL
,
the Leibniz product in lifts to aL Lie bracket ],[
in
ML . C, if onversely
J
MLis an ideal, then the
quotient MLh
is a Lie algebra. In particular,
J
is
the smallest two-sided ieal of L dsuch that LJis a
Lie algebra.
Let ]),[,(
hbe a Lie, and

,Lbe a Leibniz algebra,
we dry operation one semidirect product efine a bina th
hLby

,, ,[,],
ij ijij
exeyee eyex

 

for ,
ij
e he
, and ,
x
yL
, where ii
eeM [3].
Since
M
contains all squares, it is clear that this opera-
tion is well defined.
Prsition 1.1. If Ls a nilpotent, opo i Leibniz algebra
and
M
kerJMis an ideal such that of L

and hLM
, then
1) tent Lie algebra. h is nilpo
2) NhL
is a nilpotent Lie algebra.
Pro is a nilpote of. 1) nt Leibniz algeb ra, then ther eL
exist nN
such that
23 0
n
LL LL
Therefore
23 0
n
L
MLMLMML
Then 2
hh h

30
n
h Thusis a nilpo- h
tent Lie algebra.
2) Clearly if h an d
L
are nilpotent, then is a N
nilpotent Lie algebra.
The above proposition assoctes two ie algebras h ia L
and N to a Leiiz algebra bn
L
. Here h is a quotient
of
L
, whereas N is its extension. The corresponding
Lie groupsould be employed to associate a geometric c
object to
L
[3]. We would consider the problem of
classification of these geometric objects (loop spaces) in
a forthcoming paper.
3. Results
Next let us remind the classification results for Leibniz
algebras of dimension twand three [4]. We use the o
convention to denote the algebra of dimension i by
th
j
,ij
. L
Theorem 1.2. In dimension two, there are two
non-isomorphicotent Leibniz algebras2
L
nilp, where 1,
is abelian, and 2,2
L is given by the table 11 2
ee e
.
Theorem 1.3. In dimension three, there are five con-
crete and one parametric family of pairwise non iso-
morphic algebras.
3,1
3,21 12
3,3123 213
:,
:,
:,
L Abelian
Leee
Leeeee e

,

Copyright © 2011 SciRes. AM
M. AMINI ET AL.
1029
3,4113223 123
3,5113123213
3,61 12 213
:, ,(
:,,,
:,.
Leeeee eeeeC
Leeeeeeeee
Leeeeee
),
 
 
 
4. Discussion
In this section, we classify the enveloping Lie algebras of
Leibniz algebras of dimension two and three. This is not
a trivial task, as in each case we have to identify the re-
sulting Lie algebra as one of the known low dimensional
Lie algebras [2], by carefully defining the appropriate
change of basis.
In dimension two, we have Then
2,21 12
:Leee.

2,2 2
::
J
eF


and

2,22,22,212,2 12,2
11 2,2
,
,
hLJspan eJeL
span eeL


Therefore is 1-dimensional abelian Lie algebra.
2,2
h
Now, we consider with basis elements
2,22,2 2,2
NhL
 
13 3
0,,0,eE e
11 2
,0Ee,E . The multiplication
table of is given by
2,2
N
[,] 1
E 2
E 3
E
1
E 0 3
E 0
2
E 3
E 0 0
3
E 0 0 0
For example


1 112,212,2
1 12,222,2
,,0,,
,0 ,0
0
EEeJeJ
ee JeJ

 

 
0
and



121 2,21
1122
,,0,0
0, 0,
EEe Je
eee E




,
Thus, is a 3-dimensional Lie algebra with
2,2
N
12 3
. Briefly, we have Table 1 for dimension 2 [, ]EE E
(where the last column identifies the Lie algebra in the
Beck-Kolman list [2]).
In dimension three, for each 3, we
, 2,3,4,5,6
k
Lk
want to obtain corresponding Lie algebra 3, For
.
k
N
3,2 one can show that 2,3 is 2-dimensional abelian Lie
Table 1. The enveloping Lie algebras of Leibniz algebras of
dimension two.
Li,j hi,j Ni,j = hi,j¡ÁLi,j Ni,j
L2, 2 1-dimensional abelian
Lie algebra [E1, E2] = E3 g3
123 213
,,,eee eee,

and by
3,3
N
12 315 6246
,,,,,EEEEE EEEE

Finally, for , one can show that is a
3,4
L3, 4
L
2-dimensional abelian Lie algebra and is given by
3, 4
N
1355 24
[,] ,] ,[,]EEEE EE
14
[,EE 5
(E).C
Theref
ore for 3-dimensional Leibniz algebras, we get the Table
2. Note that in rows three and four, the enveloping Lie
algebras are isomorphic, while the original Leibniz alge-
bras are not isomorphic. The isomorphism
2
32,3 CgN 
1123344255
,,,,,eEeEeEeEeE  
is given by
3,3 6,21
Ng
by
1 2213445536
,,,,,e EeEeEeEeEeE
6
.

3,4 5,2
Ng
by
11234334255
,,,,eEeEEeEeEe .E
  
Ng
3,5 5,2
by
11 421324455
,,,eEEe EeEeE,Ee.

Also, for the 2-dimensional abelian Lie algebra
2,12,1
,0J, and 2,1
his an abelian Lie algebra. There- L
fore 2,12,
Nh
1 2,1
L
4-dimensional Lie algebra. Fi- is
nallysional abelian Lie algebra , for the 3-dimen
Table 2. The enveloping Lie algebras of Leibniz algebras of
dimension three.
Li,j hi,j Ni,j = hi,j¡ÁLi,j Ni,j
3, 2 mensional 2-di
abelian Lie
algebra
134
,EE E 2
3
g
C
L
12
[, ]ee 3
21 3
,
[,] ,
e
ee e
12 315 6
24 6
[, ],[, ],
[,]
EEE EEE
EEE

 6,21
g
L3, 3
L3, 4 2-dimensional
abelian Lie
algebra
135 145
24 5
[, ],[, ],
[,]().
EEE EEE
EEE C


 5,2
g
L3, 5 al 2-dimension
abelian Lie
algebra
135 145
23 5
[, ],[, ],
[,]
EEE EEE
EEE

 5,2
g
,Lh
algebra and 3,2 is given by 13 . For 3,3
N4
[, ]EE ELL3, 6 nal
algebra
1-dimensio
abelian Lie12 3
[, ]EE E 2
3
g
C
,
one can prove that therefore is given by

3, 0J3,3,3
h
Copyright © 2011 SciRes. AM
M. AMINI ET AL.
Copyright © 2011 SciRes. AM
1030

0 and
3,1 3,1
,LJ is an abelian Lie algebra.
3,1
h
Therefore
3,1
N3,1 3,1
hL
is a 6-dimensional Lie algebra.
5. Conclusions
We have classified the enveloping Lie algebras of Leib-
niz algebras of dimension two and three. In each case,
we have identified the corresponding Lie algebra as one
of the known low dimensional Lie algebras, by defining
the appropriate change of basis which implements the
canonical isomorphism.
There is one two dimensional Leibnitz algebra (up to
isomorphism) whose corresponding Lie quotient is a
1-dimensional abelian Lie algebra. On the other hand,
th
. References
] J. L. Loday, “Une version non-commutative des algebras
ilpotent
,
ere are exactly five non-isomorphic three dimensional
Leibnitz algebra, which correspond to three 2-dimensional
abelian Lie algebras (two of which are isomorphic), one
1-dimensional abelian Lie algebra, and a 3-dimensional
non-abelian Lie algebra.
6
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[2] R. E. Beck and B. Kolman, “Constructions of N
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Proceedings of 1981 ACM Symposium on Symbolic and
Algebraic Computation, New York, 1981, pp. 169-174.
[3] M. K. Kinyon and A. Weinestein, “Leibniz Algebras
Courant Algebroids, and Multiplications on Homogene-
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No. 3, 2001, pp. 525-550. doi:10.1353/ajm.2001.0017
[4] S. Albeverio, B. A. Omirov and I. S. Rakhimov, “Varie-
ties of Nilpotent Complex Leibniz Algebras of Dimen-
sion Less Than Five,” Communications in Algebra, Vol.
33, No. 5, 2005, pp. 1575-1585.
doi:10.1081/AGB-200061038