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Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

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Abstract

In this paper we study non-nilpotent non-Lie Leibniz F-algebras with one-dimensional derived subalgebra, where F is a field with char(F)≠2. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by L_n, where n=dimL_n. This generalizes the result found in [11], which is only valid when F=C . Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of L_n. Eventually, we solve the coquecigrue problem for L_n by integrating it into a Lie rack.
arXiv:2307.09102v2 [math.RA] 8 Jun 2024
Non-nilpotent Leibniz algebras with
one-dimensional derived subalgebra
Alfonso Di Bartolo, Gianmarco La Rosa, Manuel Mancini
Dipartimento di Matematica e Informatica
Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
alfonso.dibartolo@unipa.it, ORCID: 0000-0001-5619-2644
gianmarco.larosa@unipa.it, ORCID: 0000-0003-1047-5993
manuel.mancini@unipa.it, ORCID: 0000-0003-2142-6193
Abstract
In this paper we study non-nilpotent non-Lie Leibniz F-algebras with
one-dimensional derived subalgebra, where Fis a field with char(F)6= 2.
We prove that such an algebra is isomorphic to the direct sum of the two-
dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra.
We denote it by Ln, where n= dimFLn. This generalizes the result found
in [11], which is only valid when F=C. Moreover, we find the Lie algebra
of derivations, its Lie group of automorphisms and the Leibniz algebra of
biderivations of Ln. Eventually, we solve the coquecigrue problem for Ln
by integrating it into a Lie rack.
Introduction
Leibniz algebras were introduced by J.-L. Loday in [19] as a non-skew symmetric
version of Lie algebras. Earlier such algebraic structures were also considered by
Keywords: Leibniz algebra, Lie algebra, Derivation, Biderivation, Coquecigrue problem.
2020 Mathematics Subject Classification: 16W25, 17A32, 17B30, 17B40, 20M99, 22A30.
The authors are supported by University of Palermo and by the “National Group for Alge-
braic and Geometric Structures, and their Applications” (GNSAGA INdAM). The first au-
thor is also supported by UNIPA FFR 2024 VQR 2024. The second and third authors are
also supported by the National Recovery and Resilience Plan (NRRP), Mission 4, Component
2, Investment 1.1, Call for tender No. 1409 published on 14/09/2022 by the Italian Ministry
of University and Research (MUR), funded by the European Union NextGenerationEU
Project Title Quantum Models for Logic, Computation and Natural Pro cesses
(QM4NP) CUP B53D23030160001 Grant Assignment Decree No. 1371 adopted on
01/09/2023 by the Italian Ministry of Ministry of University and Research (MUR), and by the
Sustainability Decision Framework (SDF) Research Project CUP B79J23000540005
Grant Assignment Decree No. 5486 adopted on 04/08/2023.
1
A. Blokh, who called them D-algebras [5] for their strict connection with deriva-
tions. Leibniz algebras play a significant role in different areas of mathematics
and physics.
Many results of Lie algebras are still valid for Leibniz algebras. One of them
is the Levi decomposition, which states that any Leibniz algebra over a field
Fof characteristic zero is the semidirect sum of its radical and a semisimple
Lie algebra. This makes clear the importance of the problem of classification
of solvable and nilpotent Lie / Leibniz algebras, which has been dealt with
since the early 20th century (see [2], [3], [4], [9], [10], [11], [13] and [14], just for
giving a few examples).
In [16] and [17] nilpotent Leibniz algebras Lwith one-dimensional derived
subalgebra [L, L]were studied and classified. It was proved that, up to iso-
morphism, there are three classes of indecomposable Leibniz algebras with these
properties, namely the Heisenberg algebras lA
2n+1, which are parameterized by
their dimension 2n+ 1 and by a matrix Ain canonical form, the Kronecker
algebra knand the Dieudonné algebra dn, both parameterized by their dimen-
sion only. We want to complete this classification by studying non-nilpotent
Leibniz F-algebras with one-dimensional derived subalgebra, where Fis a field
with char(F)6= 2. Using the theory of non-abelian extensions of Leibniz alge-
bras introduced in [18], we prove that a non-nilpotent non-Lie Leibniz algebra
Lwith dimFL=nand dimF[L, L] = 1 is isomorphic to the direct sum of the
two-dimensional non-nilpotent non-Lie Leibniz algebra S2, i.e. the algebra with
basis {e1, e2}and multiplication table given by [e2, e1] = e1, and an abelian
algebra of dimension n2. We denote it by Ln. This generalizes the result
found in Theorem 2.6 of [11], where the authors proved that a complex non-split
non-nilpotent non-Lie Leibniz algebra with one-dimensional derived subalgebra
is isomorphic to S2.
We study in detail the properties of the algebra Lnand we compute the Lie
algebra of derivations Der(Ln), its Lie group of automorphism Aut(Ln) and the
Leibniz algebra of biderivations Bider(Ln).
Finally, we solve the coquecigrue problem for the Leibniz algebra Ln. We
mean the problem, formulated by J.-L. Loday in [19], of finding a generalization
of Lie third theorem to Leibniz algebras. Using M. K. Kinyon’s results for the
class of real split Leibniz algebras (see [15]), we show how to explicitly integrate
Lninto a Lie rack defined over the vector space Rn.
1 Preliminaries
We assume that Fis a field with char(F)6= 2. For the general theory we refer
to [1].
Definition 1.1. Aleft Leibniz algebra over Fis a vector space Lover Fendowed
with a bilinear map (called commutator or bracket)[,] : L×LLwhich
2
satisfies the left Leibniz identity
[x, [y, z]] = [[x, y], z] + [y, [x, z ]] ,x, y, z L.
In the same way we can define a right Leibniz algebra, using the right Leibniz
identity
[[x, y], z] = [[x, z], y] + [x, [y, z]] ,x, y, z L.
Given a left Leibniz algebra L, the multiplication [x, y]op = [y, x]defines a right
Leibniz algebra structure on L.
A Leibniz algebra that is both left and right is called symmetric Leibniz
algebra. From now on we assume that dimFL < .
We have a full inclusion functor i:Lie Leib that embeds Lie algebras
over Finto Leibniz algebras over F. Its left adjoint is the functor π:Leib
Lie, which associates to each Leibniz algebra Lthe quotient L/ Leib(L), where
Leib(L)is the smallest bilateral ideal of Lsuch that the quotient L/ Leib(L)
becomes a Lie algebra. Leib(L)is defined as the subalgebra generated by all
elements of the form [x, x], for any xL, and it is called the Leibniz kernel of
L.
We define the left and the right center of a Leibniz algebra
Zl(L) = {xL|[x, L] = 0},Zr(L) = {xL|[L, x] = 0}.
The intersection of the left and right center is called the center of Land it
is denoted by Z(L). In general for a left Leibniz algebra L, the left center
Zl(L)is a bilateral ideal, meanwhile the right center is not even a subalgebra.
Furthermore, one can check that Leib(L)Zl(L).
The definition of derivation for a Leibniz algebra is the same as in the case
of Lie algebras.
Definition 1.2. A linear map d:LLis a derivation of Lif
d([x, y]) = [d(x), y] + [x, d(y)],x, y L.
An equivalent way to define a left Leibniz algebra Lis to saying that the left
adjoint maps adx= [x, ]are derivations. Meanwhile the right adjoint maps
Adx= [, x]are not derivations in general. The set Der(L)of all derivations
of Lis a Lie algebra with the usual bracket [d, d] = ddddand the set
Inn(L)spanned by the left adjoint maps, which are called inner derivations,
is an ideal of Der(L). Moreover Aut(L)is a Lie group and its Lie algebra is
precisely Der(L).
In [19] J.-L. Loday introduced the notion of anti-derivation and biderivation
for a Leibniz algebra.
Definition 1.3. A linear map D:LLis an anti-derivation of Lif
D([x, y]) = [x, D(y)] [y , D(x)],x, y L.
3
The space ADer(L) of anti-derivations of Lhas a Der(L)-module structure
with the extra multiplication d·D=dDDd, for any derivation dand for
any anti-derivation D, and one can check that the right adjoint maps Adxare
anti-derivations.
Definition 1.4. Abiderivation of Lis a pair (d, D)Der(L)×ADer(L)such
that
[d(x) + D(x), y] = 0,x, y L.
The set Bider(L)of all biderivations of Lhas a Leibniz algebra structure
with the bracket
[(d, D),(d, D)] = ([d, d], d ·D)
and it is defined a Leibniz algebra homomorphism
LBider(L), x 7→ (adx,Adx).
The pair (adx,Adx)is called the inner biderivation associated with xLand
the set of all inner biderivations of Lforms a Leibniz subalgebra of Bider(L).
We recall the definitions of solvable and nilpotent Leibniz algebras.
Definition 1.5. Let Lbe a left Leibniz algebra over Fand let
L0=L, Lk+1 = [Lk, Lk],k0
be the derived series of L.Lis nstep solvable if Ln16= 0 and Ln= 0.
Definition 1.6. Let Lbe a left Leibniz algebra over Fand let
L(0) =L, L(k+1) = [L, L(k)],k0
be the lower central series of L.Lis nstep nilpotent if L(n1) 6= 0 and
L(n)= 0.
When Lis two-step nilpotent, it lies in different varieties of non-associative
algebras, such as associative, alternative and Zinbiel algebras. In this case we
refer at Las a two-step nilpotent algebra and we have the following.
Proposition 1.7.
(i) If Lis a two-step nilpotent algebra, then L(1) = [L, L]Z(L)and Lis a
symmetric Leibniz algebra.
(ii) If Lis a left nilpotent Leibniz algebra with dimF[L, L] = 1, then Lis
two-step nilpotent.
In [16] the classification of nilpotent Leibniz algebras with one-dimensional
derived subalgebra was established. The classification revealed that, up to iso-
morphism, there exist only three classes of indecomposable nilpotent Leibniz
algebras of this type.
4
Definition 1.8. [16] Let f(x)F[x]be a monic irreducible polynomial.
Let kNand let A= (aij )i,j be the companion matrix of f(x)k. The
Heisenberg algebra lA
2n+1 is the (2n+ 1)-dimensional Leibniz algebra with basis
{e1,...,en, f1, . . . , fn, z}and the brackets are given by
[ei, fj] = (δij +aij )z, [fj, ei] = (δij +aij )z, i, j = 1,...,n.
When Ais the zero matrix, then we obtain the (2n+1)dimensional Heisen-
berg Lie algebra h2n+1 .
Definition 1.9. [16] Let nN. The Kronecker algebra knis the (2n+ 1)-
dimensional Leibniz algebra with basis {e1,...,en, f1,...,fn, z}and the brack-
ets are given by
[ei, fi] = [fi, ei] = z, i= 1,...,n
[ei, fi1] = z, [fi1, ei] = z , i= 2,...,n.
Definition 1.10. [16] Let nN. The Dieudonné algebra dnis the (2n+ 2)-
dimensional Leibniz algebra with basis {e1,...,e2n+1 , z }and the brackets are
given by
[e1, en+2] = z,
[ei, en+i] = [ei, en+i+1] = z , i= 2,...,n,
[en+1, e2n+1] = z,
[ei, ein] = z, [ei, ein1] = z , i=n+ 2,...,2n+ 1.
We want to extend this classification by studying non-nilpotent Leibniz al-
gebras with one-dimensional derived subalgebra.
2 Non-nilpotent Leibniz algebras with one-
dimensional derived subalgebra
Let Lbe a non-nilpotent left Leibniz algebra over Fwith dimFL=nand
dimF[L, L] = 1. We observe that such an algebra is two-step solvable since the
derived subalgebra [L, L]is abelian.
It is well known that a non-nilpotent Lie algebra with one-dimensional de-
rived subalgebra is isomorphic to the direct sum of the two-dimensional non-
abelian Lie algebra and an abelian algebra (see [12, Section 3]). Thus we are
interested in the classification of non-Lie Leibniz algebras with these properties.
In [11, Theorem 2.6] the authors prove that a complex non-split non-
nilpotent non-Lie Leibniz algebra with one-dimensional derived subalgebra is
isomorphic to the two-dimensional algebra with basis {e1, e2}and multiplica-
tion table [e2, e1] = [e2, e2] = e1. Here we generalize this result when Fis a
general field with char(F)6= 2.
5
Proposition 2.1. Let Lbe a non-nilpotent left Leibniz algebra over Fwith
dimF[L, L] = 1. Then Lhas a two-dimensional bilateral ideal Swhich is iso-
morphic to one of the following Leibniz algebras:
(i) S1=he1, e2iwith [e2, e1] = [e1, e2] = e1;
(ii) S2=he1, e2iwith [e2, e1] = [e2, e2] = e1.
Proof. Let [L, L] = Fz.Lis not nilpotent, then
[L, [L, L]] 6= 0,
i.e. z /Zr(L). Since [L, L]is an abelian algebra, there exists a vector xL,
which is linearly independent than z, such that [x, z ]6= 0. Thus
[x, z] = γ z,
for some γF. The subspace S=hx, z iis an ideal of Land it is not nilpotent:
in fact
06=γz = [x, z ][S, [S, S]] .
Thus Sis a non-nilpotent Leibniz algebra. Using the classification of two-
dimensional Leibniz algebras given by C. Cuvier in [8], Sis isomorphic either
to S1or to S2.
Remark 2.1. The algebras S1and S2are respectively the Leibniz algebras L2
and L4of Section 3.1 in [1]. We observe that S1is a Lie algebra, meanwhile S2
is a non-right left Leibniz algebra.
One can see Las an extension of the abelian algebra L0=L/S
=Fn2by
S[18]
0//Si//Lπ//L0
s
oo//0.(1)
It turns out that there exists an equivalence of Leibniz algebra extensions
0S L0ωS L00
0S L L00
i2
idS
π1
θ
i1
idB
iπ
σ
where L0ωSis the Leibniz algebra defined on the direct sum of vector spaces
L0Swith the bilinear operation given by
[(x, a),(y, b)](l,r,ω)= (0,[a, b] + lx(b) + ry(a) + ω(x, y)),
where
ω(x, y) = [σ(x), σ(y)]Lσ([x, y ]L0) = [σ(x), σ(y)]L
is the Leibniz algebra 2-cocycle associated with (1) and
lx(b) = [σ(x), i(b)]L, ry(a) = [i(a), σ(y)]L
6
define the action of L0on S;i1, i2, π1are the canonical injections and pro jection.
The Leibniz algebra isomorphism θis defined by θ(x, a) = σ(x) + i(a), for every
(x, a)L0S.
By [18, Proposition 4.2], the 2-cocycle ω:L0×L0Sand the linear maps
l, r :L0gl(S)must satisfy the following set of equations
(L1) lx([a, b]) = [lx(a), b] + [x, lx(b)];
(L2) rx([a, b]) = [a, rx(b)] [b, rx(a)];
(L3) [lx(a) + rx(a), b] = 0;
(L4) [lx, ly]gl(S)l[x,y]L0= adω(x,y);
(L5) [lx, ry]gl(S)r[x,y]L0= Adω(x,y);
(L6) ry(rx(a) + lx(a)) = 0;
(L7) lx(ω(y, z)) ly(ω(x, z)) rz(ω(x, y)) =
=ω([x, y]L0, z)ω(x, [y, z]L0) + ω(y, [x, z ]L0)
for any x, y L0and for any a, b S. Notice that these equations where also
studied in [6] in the case of Leibniz algebra split extensions.
Remark 2.2. The first three equations state that the pair (lx, rx)is a bideriva-
tion of the Leibniz algebra S, for any xL0. Biderivations of low-dimensional
Leibniz algebras were classified in [20] and it turns out that
Bider(S1) = {(d, d)|dDer(S1)}and
Der(S1) = (α β
0 0
α, β F);
Bider(S2) = (α α
0 0,0β
0 0
α, β F).
We study now in detail the non-abelian extension (1) in both cases that S
is isomorphic either to S1or to S2.
2.1 Sis a Lie algebra
When S
=S1, we have that ry=ly, for any yL0and the bilinear operation
of L0ωS1becomes
[(x, a),(y, b)](l,ω)= (0,[a, b] + lx(b)ly(a) + ω(x, y )).
The linear map lxis represented by a 2×2matrix
αxβx
0 0
7
with αx,βxF. From equations (L4)-(L5) it turns out that
ω(x, y) = (αxβyαyβx)e1,x, y L0
and the 2-cocycle ωis skew-symmetric. Moreover, equations (L6)-(L7) are au-
tomatically satisfied and the resulting algebra L0ωS1
=Lis a Lie algebra.
We conclude that Lis isomorphic to the direct sum of S1and L0
=Fn2.
2.2 Sis not a Lie algebra
With the change of basis e27→ e2e1,S2becomes the Leibniz algebra with
basis {e1, e2}and the only non-trivial bracket given by [e2, e1] = e1. Now a
biderivation of S1is represented by a pair of matrices
α0
0 0,0β
0 0
with α, β Fand the pair (lx, rx)Bider(S2)is defined by lx(e1) = αxe1and
rx(e2) = βxe1, for any xL0.
Equation (L4) states that [lx, ly]gl(S2)= [ω(x, y),], with
[lx, ly]gl(S2)=lxlylylx=αx0
0 0αy0
0 0αy0
0 0αx0
0 0=
=αxαy0
0 0αxαy0
0 0=0 0
0 0,
for any x, y L0. Thus ω(x, y)Zl(S2) = Fe1.
From equation (L5) we have [lx, ry]gl(S2)= [, ω(x, y)]S2, with
[lx, ry]gl(S2)=lxryrylx=0αxβy
0 0 0 0
0 0=0αxβy
0 0 .
Thus, for every a=a1e1+a2e2S2and for every x, y L0, we have
[a, ω(x, y)] = [lx, ry] (a) = αxβya2e1,
i.e. ω(x, y) = αxβye1. Finally, equations (L6) and (L7) are identically satisfied.
Summarizing we have
lx αx0
0 0!
ry 0βy
0 0 !
ω(x, y) = αxβye1
8
for every x, y L0and the bilinear operation [,](l,r,ω)becomes
[(x, a),(y, b)](l,r,ω)= (0,(a2b1+αxb1+βya2+αxβy)e1),
for any x,yL0and for any a=a1e1+a2e2,b=b1e1+b2e2S2.
If we fix a basis {f3,...,fn}of L0and we denote by
αi=αfi, βi=βfi,i= 3,...,n
then Lis isomorphic to the Leibniz algebra with basis {e1, e2, f3,...,fn}and
non-zero brackets
[e2, e1] = e1
[e2, fi] = βie1,i= 3,...,n
[fi, e1] = αie1,i= 3,...,n
[fi, fj] = αiβje1,i, j = 3,...,n.
With the change of basis fi7→ f
i=fi
βi
e1, if βi6= 0, we obtain that
[e2, f
i] = e1[e2, e1] = 0,
[f
i, e1] = γie1,where γi=αi
βi
,
[fi, f
j] = αie1[fi, e1] = 0,
[f
i, f
j] = γie11
βi
[fi, e1] = 0.
If we denote again fif
iand αiγiwhen βi6= 0, then Lhas basis
{e1, e2, f3,...,fn}and non-trivial brackets
[e2, e1] = e1,[fi, e1] = αie1,i= 3,...,n.
Finally, when αi6= 0, we can operate the change of basis
fi7→ fi
αi
e2.
One can check that the only non-trivial bracket now is [e2, e1] = e1and Lis
isomorphic to the direct sum of S2and the abelian algebra L0
=Fn2. This
allows us to conclude with the following.
Theorem 2.2. Let Fbe a field with char(F)6= 2. Let Lbe a non-nilpotent
non-Lie left Leibniz algebra over Fwith dimFL=nand dimF[L, L] = 1. Then
Lis isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie
Leibniz algebra S2and an abelian algebra of dimension n2. We denote this
algebra by Ln.
9
If we suppose that Lis a non-split algebra, i.e. Lcannot be written as the
direct sum of two proper ideals, then we obtain the following result, that is a
generalization of [11, Theorem 2.6] and which is valid over a general field Fwith
char(F)6= 2.
Corollary 2.3. Let Lbe a non-split non-nilpotent non-Lie left Leibniz algebra
over Fwith dimFL=nand dimF[L, L] = 1. Then n= 2 and L
=S2.
Now we study in detail the algebra Ln=S2Fn2by describing the Lie
algebra of derivations, its Lie group of automorphisms and the Leibniz algebra
of biderivations. Moreover, when F=R, we solve the coquegigrue problem (see
[7] and [15]) for Lnby integrating it into a Lie rack.
2.3 Derivations, automorphisms and biderivations of Ln
Let n2and let Ln=S2Fn2. We fix the basis Bn={e1, e2, f3,...,fn}
of Lnand we recall that the only non-trivial commutator is [e2, e1] = e1. A
straightforward application of the algorithm proposed in [20] for finding deriva-
tions and anti-derivations of a Leibniz algebra as pair of matrices with respect
to a fixed basis produces the following.
Theorem 2.4.
(i) A derivation of Lnis represented, with respect to the basis Bn, by a matrix
α0
0 0
0 0 ··· 0
0 0 ··· 0
0a3
0a4
.
.
..
.
.
0an
A
where AMn2(F).
(ii) The group of automorphisms Aut(Ln)is the Lie subgroup of GLn(F)of
matrices of the form
β0
0 1
0 0 ··· 0
0 0 ··· 0
0b3
0b4
.
.
..
.
.
0bn
B
where β6= 0 and BGLn2(F).
10
(iii) The Leibniz algebra of biderivations of Lnconsists of the pairs (d, D)of
linear endomorphisms of Lnwhich are represented by the pair of matrices
α0
0 0
0 0 ··· 0
0 0 ··· 0
0a3
0a4
.
.
..
.
.
0an
A
,
0α
0 0
0 0 ··· 0
0 0 ··· 0
0a
3
0a
4
.
.
..
.
.
0a
n
A
where A,AMn2(F).
3 The integration of the Leibniz algebra Ln
The coquecigrue problem is the problem formulated by J.-L. Loday in [19] of
finding a generalization of Lie third theorem to Leibniz algebras. Given a real
Leibniz algebra L, one wants to find a manifold endowed with a smooth map,
which plays the role of the adjoint map for Lie groups, such that the tangent
space at a distinguished element, endowed with the differential of this map,
gives a Leibniz algebra isomorphic to L. Moreover, when Lis a Lie algebra, we
want to obatin the simply connected Lie group associated with L. From now
on, we assume that the underlying field of any algebra is F=R.
In [15] M. K. Kinyon shows that it is possible to define an algebraic structure,
called rack, whose operation, differentiated twice, defines on its tangent space
at the unit element a Leibniz algebra structure.
Definition 3.1. Arack is a set Xwith a binary operation :X×XX
which is left autodistributive
x(yz) = (xy)(xz),x, y, z X
and such that the left multiplications xare bijections.
A rack is pointed if there exists an element 1Xsuch that 1x=xand
x1 = 1, for any xX.
A rack is a quandle if the binary operation is idempotent.
The first example of a rack is any group Gendowed with its conjugation
xy=xyx1,x, y G.
We denote this rack by Conj(G)and we observe that it is a quandle.
11
Definition 3.2. A pointed rack (X, ,1) is said to be a Lie rack if Xis a smooth
manifold, is a smooth map and the left multiplications are diffeomorphisms.
M. K. Kinyon proved that the tangent space T1Xat the unit element 1of
a Lie rack X, endowed with the bilinear operation
[x, y] = 2
∂s∂t s,t=0
γ1(s)γ2(t)
where γ1, γ2: [0,1] Xare smooth paths such that γ1(0) = γ2(0) = 1,γ
1(0) =
xand γ
2(0) = y, is a Leibniz algebra.
He also solved the coquecigrue problem for the class of split Leibniz algebras.
Here a Leibniz algebra is said to be split if there exists an ideal
Leib(L)IZl(L)
and a Lie subalgebra Mof Lsuch that L
=(MI, {−,−}), where the bilinear
operation {−,−} is defined by
{(x, a),(y, b)}= ([x, y], ρx(b))
and ρ:M×IIis the action on the M-module I.Lis said to be the
demisemidirect product of Mand I. More precisely, we have the following.
Theorem 3.3. [15] Let Lbe a split Leibniz algebra. Then a Lie rack integrating
Lis X= (HI, ), where His the simply connected Lie group integrating M
and the binary operation is defined by
(g, a)(h, b) = (ghg1, φg(b)),
where φis the exponentiation of the Lie algebra action ρ.
Some years later S. Covez generalized M. K. Kinyon’s results proving that
every real Leibniz algebra admits an integration into a Lie local rack (see
[7]). More recently it was showed in [16] that the integration proposed by S.
Covez is global for any nilpotent Leibniz algebra. Moreover, when a Leibniz
algebra Lis integrated into a Lie quandle X, it turns out that Lis a Lie alge-
bra and X= Conj(G), where Gis the simply connected Lie group integrating L.
Our aim here is to solve the coquecigrue problem for the non-nilpotent Leib-
niz algebra Ln=S2Fn2. One can check that S2is a split Leibniz alge-
bra, in the sense of M. K. Kinyon, with I= Zl(S2)
=Rand M
=R. Thus
L
=(R2,{−,−})with the bilinear operation defined by
{(x1, x2),(y1, y2)}= (0, ρx1(y2))
and ρx1(y2) = x1y2, for any x1, y2R. It turns out that a Lie rack integrating
S2is (R2,), where
(x1, x2)(y1, y2) = (y1, y2+ex1y2).
12
and the unit element is (0,0). Finally, one can check that the binary operation
(x1, x2, x3,...,xn)(y1, y2, y3,...,yn) = (y1, y2+ex1y2, y3,...,yn)
defines on Rna Lie rack structure with unit element 1 = (0,...,0), such that
(T1Rn,)is a Leibniz algebra isomorphic to Ln. This result, combined with the
ones of [16, Section 4], completes the classification of Lie racks whose tangent
space at the unit element gives a Leibniz algebra with one-dimensional derived
subalgebra.
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14
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