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\def\GII{7}
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\def\Zfin{14}
\def\Zad{15}




%%%%%%%%%%%%%%%%%




 \centerline{ {\ver Towards a Socle for Lie Algebras}}


\bigskip

\centerline{{\bf Antonio Fern\'andez L\'opez$\ ^1$}, {\bf Esther
Garc\'\i a$\ ^2$},}
 \centerline{{\bf and Miguel G\'omez Lozano$\ ^3$}}

\footnote{}{\rms $\ ^{1,}\  ^3$ Partially supported by the MCYT and the Fondos FEDER, BFM2001-1938-C02-01, and the Junta de Andaluc\'{\i}a FQM264.}

\footnote{}{\rms $\ ^2$ Partially supported by the MCYT and the Fondos FEDER, BFM2001-1938-C02-02.}
\bigskip


\centerline{\it $^1$Departamento de \'Algebra, Geometr\'{\i}a y Topolog\'{\i}a, Universidad de M\'alaga,}

\centerline{\it 29071 M\'alaga, Spain.}

\centerline{\it E-mail: {\tt emalfer@agt.cie.uma.es}}


\centerline{\it $^2$Departamento de \'Algebra, Universidad Complutense de Madrid,}

\centerline{\it  28040 Madrid, Spain.}


\centerline{\it E-mail: {\tt egarciag@mat.ucm.es}}


%\centerline{{\ver and} MIGUEL G\'OMEZ LOZANO $^1$}

\centerline{\it $^3$Departamento de \'Algebra, Geometr\'{\i}a y Topolog\'{\i}a, Universidad de M\'alaga,}

\centerline{\it 29071 M\'alaga, Spain.}

\centerline{\it E-mail: {\tt magomez@agt.cie.uma.es}}



\bigskip
\noindent{\rms ABSTRACT: A notion of socle is introduced for
3-graded Lie algebras (over a ring of scalars $\Phi$ containing
${1\over 6}$) whose associated Jordan pairs are nondegenerate.
This notion of socle does not depend on certain 3-gradings and
this allows us to define a Jordan socle for non-necessarily
3-graded Lie algebras. The Jordan socle turns out to be a 3-graded
ideal and is the sum of minimal 3-graded inner ideals each of
which is a central extension of the TKK-algebra of a division
Jordan pair. Nondegenerate Lie algebras having a large Jordan
socle are essentially determined by TKK-algebras of simple Jordan
pairs with minimal inner ideals and their derivation algebras,
which are also 3-graded.}

\bigskip

\centerline{\bf INTRODUCTION}

\bigskip


The aim of this paper is to summarize the results contained in our
two papers [\FGGo, \FGGII]. In the first one, inspired by the
Jordan notion of socle, we define a socle for nondegenerate
3-graded Lie algebras, and study its main properties. In the
second work, motivated by the dependence of that socle on the
considered 3-grading, we introduce the so-called Jordan socle,
which extends the notion of socle for 3-graded Lie algebras but
makes sense also for non-necessarily 3-graded Lie algebras.

The idea of studying Lie algebras by means of Jordan methods is by
no means a novelty; on the contrary, fundamental contributions to
this topic can be found in papers like [\A, \BeZ, \N3g, \Zfin].
Let us say that in [\N3g], the most related to our approach of
these papers, E. Neher describes Lie algebras graded by a 3-graded
root system: a Lie algebra $L$ is graded by a 3-graded root system
$R$ if and only if it is a central extension of the
Tits-Kantor-Koecher algebra of a Jordan pair $V$ ($\TKK(V)$ for
short) covered by a grid whose associated 3-graded root system is
isomorphic to $R$. He gives the classification of Jordan pairs
covered by a grid and describes their Tits-Kantor-Koecher
algebras. We note that any simple Jordan pair covered by a grid
with division coordinate algebra coincides with its socle, so in
this case the socle theory and the grid theory agree.

Our notion of Jordan socle gives rise to a structure theory for nondegenerate Lie algebras having essential socles. Indeed, these Lie algebras can be
described in terms of Tits-Kantor-Koecher algebras of simple Jordan pairs with minimal inner ideals and their derivation algebras. In particular, any
simple finite-dimensional Lie algebra over an algebraically closed field of characteristic $0$ which is not  of type $E_8, F_4$ or $G_2$ has a
non-trivial 3-grading and  coincides with its socle.



\bigskip

\parrafo{3-GRADED LIE ALGEBRAS AND JORDAN PAIRS}


\re{\atmos}{} Throughout  this paper, we will be dealing with Lie algebras $L$ and Jordan pairs $V=(V^+,V^-)$  over
a ring of scalars $\Phi$ containing ${1\over 6}$. As usual, $[x,y]$ will denote the Lie product  and $\ad x$ the
adjoint mapping determined by $x$. Jordan products will be denoted by $Q_xy$, for any $x\in V^\sigma$, $y\in
V^{-\sigma}$, $\sigma = \pm$, with linearizations $Q_{x,z}y=\{ x, y, z\}=D_{x,y}z$.  The reader is referred to [\Jlie,
\L, \N3g] for basic results, notation and terminology. Nevertheless, we will stress some notions and basic properties
for both Jordan pairs and Lie algebras.


\re{\regLie}{}An element $x\in V^\sigma$ is called an {\it absolute zero divisor} if $Q_x=0$. Then $V$ is said to be {\it nondegenerate} if it has no
nonzero absolute zero divisors, {\it semiprime}  if $Q_{B^\pm}B^\mp =0$ implies $B=0$, and {\it prime} if $Q_{B^\pm}C^\mp =0$ implies $B=0$ or $C=0$,
for $B=(B^+, B^-)$,
 $C=(C^+, C^-)$ ideals of $V$. Similarly, $x\in L$ is an {\it
absolute zero divisor}  of $L$ if $(\ad x)^2=0$, and $L$ is {\it nondegenerate} if it has no nonzero absolute zero divisors, {\it semiprime} if
$[I,I] = 0$ implies $I=0$, and {\it prime} if $[I,J]= 0$ implies $I=0$ or $J=0$, for $I, J$ ideals of $L$. A Jordan pair or Lie algebra is {\it
strongly prime} if it is prime and nondegenerate.

\re{\annihilator}{} Given a subset $S$ of $L$, the {\it
annihilator} $\Ann(S)$  of $S$ in $L$ consists on the elements
$x\in L$ such that $[x,S]=0$, and  it is an ideal as soon as $S$
is. Indeed,  the annihilator of a  nondegenerate ideal $I$ in a
Lie algebra $L$ coincides with the set of elements $x\in L$ such
that $[x,[x, I]]=0$ by [\FGGII, 2.5].

\re{\idealnondegen}{} Nonzero ideals of nondegenerate  Jordan pairs inherit nondegeneracy ([\L, JP3]). The same is true for Lie algebras: every
nonzero ideal of a nondegenerate  Lie algebra is nondegenerate  [\Zad, Lemma 4; \Gprime, 0.4]. Conversely, a Lie algebra is nondegenerate as soon as
it has a nondegenerate ideal (\annihilator).




\re{\tresg}{} A {\it $3$-grading} of a Lie algebra $L$ is a decomposition $L=L_1  \oplus L_0 \oplus L_{-1}$, where each $L_i$ is a submodule of  $L$
satisfying $[L_i,L_j]\subset L_{i+j}$,  and where $L_{i+j}=0$ if $i+j \neq 0, \pm 1$.  A Lie algebra is {\it $3$-graded} if it has a 3-grading. We
will write $(L,\pi)$ to denote the Lie algebra $L$ with the particular 3-grading $\pi=(\pi_1,\pi_0,\pi_{-1})$, where each  $\pi_i$ is the projection
of $L$ onto $L_i$, $i=0,\pm1$.

\re{\jpass}{}Given $(L,\pi)$ we have that $\pi (L):=(L_1, L_{-1})$
is a Jordan pair for the triple products defined by $\{x, y, z\}
:=[[x,\, y], \, z]$ for all $x, z \in L_\sigma$, $y \in
L_{-\sigma}$, $\sigma = \pm 1$, which is called the {\it
associated Jordan pair} of $(L, \pi)$. We note that if $L$ is
nondegenerate, so is $\pi (L)$ [\Zfin, Lemma 1.8]. A standard
example  of a 3-graded Lie algebra is the  following one:

\re{\deftkk}{} For any Jordan pair $V$, there exists a 3-graded Lie algebra $\TKK(V)=L_1 \oplus L_0 \oplus L_{-1}$, the {\it Tits-Kantor-Koecher
algebra of $V$},  uniquely determined by the following conditions (cf. [\N3g, 1.5(6)]):

\itemitem{(TKK1)} The associated Jordan pair $(L_1,L_{-1})$ of $L$ is
isomorphic to $V$.
\itemitem{(TKK2)}  $[L_1,L_{-1}]=L_0$.
\itemitem{(TKK3)} $[x_0, L_1\oplus L_{-1}]=0$ implies $x_0=0$, for
any $x_0\in L_0$.

\re{\derLie}{}We will denote by $\Der L$ the set of derivations of $L$. If $M$ is an ideal of $L$ with $\Ann(M)=0$, then $L$ can be embedded in $\Der
M$ via the adjoint mapping: $ L\cong \ad_M L\le \Der M.$

Graded derivations, which appear naturally when dealing with graded structures, generalize the adjoint mappings and are characterized by the way they
act on the homogeneous parts (see [\MARfrac, p. 805]). Moreover,

\re{\dtwoiszerocor}{Corollary {\rm [\GG, 1.9]}} The derivation algebra of any nondegenerate  3-graded Lie algebra is 3-graded itself.\endproof\endre


\bigskip
\parrafo{THE SOCLE OF A 3-GRADED LIE ALGEBRA}


\re{\defsocle}{} The {\it socle} of a nondegenerate 3-graded Lie algebra $(L,\pi)$ is defined as the ideal of $L$ generated by the socle of the
associated Jordan pair $\pi(L)$. Denoted by $\Soc_\pi(L)$ to show which grading we are considering, we have that $\Soc_\pi(L) = \Soc (\pi_1(L))\oplus
[\Soc(\pi_1(L)), \Soc(\pi_{-1}(L))] \oplus \Soc(\pi_{-1}(L))$, [\FGGo, 4.3]. Moreover, $\Soc_\pi(L)$ can be decomposed as a direct sum of simple
ideals, $$\Soc_\pi(L)=\bigoplus S^{(i)}=\bigoplus \TKK(\pi(S^{(i)})),$$ where the $\pi(S^{(i)})$ are the simple components of $\Soc(\pi(L))$.
Furthermore, for any 3-graded ideal $(I,\pi_I)$ of $(L,\pi)$ whose grading $\pi_I$ is compatible with $\pi$, we have that
$\Soc_{\pi_I}(I)=\Soc_\pi(L)\cap I$.


\re{\ideinn}{} In [\Zfin, \S 5]  Zelmanov introduces a notion of inner ideals for ${\bf Z}$-graded Lie algebras which in the particular case of a
3-graded Lie algebra  reads as follows (cf. [\GII, 1.1]):  We say that a graded $\Phi$-submodule $B=B_1\oplus B_0\oplus B_{-1}$ of a 3-graded Lie
algebra $(L, \pi)$ is a {\it  $3$-graded inner ideal} if
\item{(i)} $B$ is a subalgebra of $L$,
\item{(ii)} $[[L,B_1],B_1]+[[L,B_{-1}],B_{-1}]\subset B$.

Notice that  condition (ii) implies that both $B_1$ and $B_{-1}$
are inner ideals of the associated Jordan pair $\pi(L)$.
Conversely, if $(B_1, B_{-1}) \subset \pi(L)$ is a pair of inner
ideals of $\pi(L)$, then $B_1\oplus [B_1, B_{-1}] \oplus B_{-1}$
is a 3-graded inner ideal of $L$. In particular, a Jordan pair
idempotent $e=(e^+, e^-)$ of $\pi(L)$ determines the 3-graded
inner ideal $L(e):=V_2(e^+)\oplus [V_2(e^+),V_2(e^-)]\oplus
V_2(e^-)$, where $V_2(e^\sigma)=[e^\sigma,[e^\sigma, L]]$ for
$\sigma=\pm1$ (see [\L, 5.5]). Moreover, any nondegenerate minimal
3-graded inner ideal of $L$ has the form $L(e)$ for a Jordan pair
division idempotent $e\in \pi(L)$ [\FGGo, 4.6].


\re{\sumsoc}{Proposition {\rm [\FGGo, 4.7]}} For any  3-graded Lie algebra $(L,\pi)$ with nondegenerate associated Jordan pair, $\Soc_\pi(L)=\sum_e
L(e)$, where the sum is taken over all Jordan pair division idempotents $e\in \pi(L)$.
\endre


In general, the definition of the socle of a nondegenerate
3-graded Lie algebra depends on  the 3-grading, as can be seen in
the following example:

\re{\soclenotunique}{}{\ver Example:} Let $V$ and $W$ be two
Jordan pairs coinciding with their socles, i.e., $V=\Soc (V)$ and
$W=\Soc(W)$. Let $L$ be the Lie algebra built as the direct sum of
the TKK-algebras of $V$ and $W$. Notice that $L$ admits the
gradings $$\matrix{\pi_1(L)=V^+& \pi_0(L)=[V^+,V^-]\oplus \TKK(W)&
\pi_{-1}(L)=V^-,\cr \pi_{1}'(L)=W^+& \pi_{0}'(L)=[W^+,W^-]\oplus
\TKK(V)& \pi_{-1}'(L)=W^-,\cr \pi_1''(L)=V^+\oplus W^+ &
\pi_0''(L)=[V^+\oplus W^+,V^-\oplus W^-]& \pi_{-1}''(L)=V^-\oplus
W^-,\cr} $$ which give three essentially different socles:
$\Soc_\pi(L)=\TKK(V)$, while $\Soc_{\pi'}(L)=\TKK(W)$ and
$\Soc_{\pi''}(L)=L$.

\re{\nicegrading}{} We will show that the socle is indeed independent of the grading of $L$ when the grading is {\it effective} in the sense that
there is no nonzero ideal contained in the zero part of $L$. Notice that this condition is satisfied when $(L, \pi)$ has (TKK3), and, in particular,
when $L$ is graded as the TKK-algebra of a Jordan pair or when $L$ is strongly prime.


\re{\socleideal}{Theorem {\rm [\FGGII, 3.8]}} Let $(L,\pi)$ be a nondegenerate $3$-graded Lie algebra with an effective $3$-grading $\pi$, and let
$(I,\pi')$ be an ideal of $L$ which is graded with respect to a $3$-grading $\pi'$. Then $\Soc_{\pi'}(I)\subset\Soc_{\pi}(L)$.\endre

Therefore, as soon as a nondegenerate Lie algebra $L$ has an effective 3-grading $(L,\pi)$, its socle contains the socle of any other 3-grading of
$L$.

\re{\socle}{Corollary {\rm [\FGGII, 3.9]}} Let $L$ be a nondegenerate Lie algebra admitting an effective $3$-grading $ (L,\pi)$. Then
$\Soc_\pi(L)\supset\Soc_{\pi'}(L)$ for any other $3$-grading $(L,\pi')$ of $L$. In particular, the socle of $L$ does not depend on the effective
3-grading considered.
\endre

\bigskip

\parrafo{THE JORDAN SOCLE}

Motivated by theorem (\socleideal) we are going to introduce a notion of socle, called the {\it Jordan socle}, for nondegenerate Lie algebras which
are not necessarily 3-graded.

\re{\Jsocle}{} Given a nondegenerate Lie algebra $L$, we define its {\it Jordan socle} as the sum of the socles of $(I,\pi)$, where $I$ is any
3-graded ideal of $L$ and $\pi$ denotes any of its possible 3-gradings: $$\eqalign{ &\JSoc(L)=\sum_{(I,\pi)} \Soc_\pi(I).\cr } $$


\re{\sumsimple}{Theorem {\rm [\FGGII, 4.2]}} The Jordan socle of a nondegenerate Lie algebra $L$ is an ideal of $L$. If $\JSoc(L) \neq 0$ then it is
a direct sum of simple ideals each of which is the TKK-algebra of a simple Jordan pair with minimal inner ideals. Therefore, $\JSoc(L) \cong
\TKK(V)$, where $V$ is a nondegenerate Jordan pair coinciding with its socle.
\endre

Thus, the Jordan socle of a nondegenerate Lie algebra is the biggest 3-graded ideal of $L$ that coincides with its socle. Moreover, if $(L,\pi)$ is a
nondegenerate 3-graded Lie algebra with effective 3-grading, then $\Soc_\pi(L)=\JSoc(L)$.

This notion of Jordan socle allows us to give a structure theory
for nondegenerate Lie algebras with essential Jordan socle,
similar to the ones obtained in the associative and Jordan
settings [\JS, \FT]. Recall that an {\it essential subdirect
product} of a family of algebras $\{ L_\alpha \}$ is any subdirect
product of the $L_\alpha$ containing an essential ideal of the
full direct product $\prod A_\alpha$.

\re{\thnongraded}{Theorem {\rm [\FGGII, 4.3]}} For a Lie algebra $L$, the following statements are equivalent:
\item{\rm (i)} $L$ is nondegenerate and has essential Jordan
socle.

\item{\rm (ii)} $L$ is an essential subdirect product of a family of
strongly prime Lie algebras $L_\alpha$ having nonzero Jordan socles.

\item{\rm (iii)} there exists a nondegenerate
Jordan pair $V$ that coincides with its socle such that $\ad(\TKK(V)) \le L \le \Der(\TKK(V))$.

\item{\rm (iv)}  $\bigoplus \ad (\TKK(V_\alpha)) \triangleleft L
\le \prod\Der(\TKK(V_\alpha))$,
       where each $V_\alpha$ is a simple Jordan pair with minimal
inner ideals.



Moreover, in case {\rm (iii)}, $\JSoc(L)=\ad(\TKK(V))$, and $L$ is strongly prime if and only if $\JSoc(L)$ is simple, if and only if $V$ is simple.
We also have that $\Der (\TKK(V))$ is the largest strongly prime Lie algebra having Jordan socle equal to $\ad (\TKK(V))$.
\endre



\bigskip
\centerline{{\bf REFERENCES}}
\bigskip

\item{\A.} {\ver  B. N. Allison}. ``A construction of Lie algebras from
J-ternary algebras". {\it Am. J. Math.} {\bf 98} (1976), 285-294.

\item{\BeZ.} {\ver G. Benkart, E. I. Zelmanov}.  ``Lie algebras
graded by finite root systems and intersection matrix algebras".
  {\it Invent. Math.} {\bf 129} (1996), 1-45.

\item{\FGGo.} {\ver A. Fern\'andez L\'opez, E. Garc\'{\i}a, M.  G\'omez
Lozano}. ``$3$-Graded Lie algebras with Jordan finiteness
conditions".  {\it Comm. Algebra}, to appear



\item{\FGGII.} {\ver A. Fern\'andez L\'opez, E.  Garc\'{\i}a, M. G\'omez
Lozano}. ``The Jordan Socle and Finitary Lie Algebras", {\it
(preprint)}. Posted on the Jordan Theory Preprint Server
http://mathematik.uibk.ac.at/mathematik/jordan/.

\item{\FT.} {\ver  A. Fern\'andez L\'opez, M.  Toc\'on}. ``Strongly prime
Jordan pairs with non-zero socle". {\it Manuscripta Math.}  {\bf
111} (2003), 269-283.


\item{\Gprime.} {\ver   E. Garc\'{\i}a}. ``Inheritance of primeness by
ideals in Lie algebras".  {\it Int. J. Math. Game Theory Algebra},
to appear.

\item{\GII.} {\ver   E. Garc\'{\i}a}. ``Simple, primitive and strongly
prime Jordan 3-graded Lie algebras".  {\it J. Algebra}, to appear.

\item{\GG.} {\ver E. Garc\'{\i}a, M. G\'omez Lozano}. ``Jordan Systems of Martindale-like Quotients".
{\it J. Pure Appl. Algebra}, to appear.


\item{\Jlie.} {\ver  N. Jacobson}.  {\it Lie Algebras}.  Interscience
Publishers; New York, 1962.


\item{\JS.} {\ver  N. Jacobson}.  {\it Structure of Rings}. AMS
Colloquium Publications,  37; Providence, USA, 1968.



\item{\L.} {\ver  O. Loos}. {\it  Jordan Pairs}. Lecture Notes in
Mathematics, 460, Springer-Verlag; New York, 1975.

\item{\MARfrac.} {\ver  C. Mart\'{\i}nez}. ``The ring of fractions of a
Jordan algebra".   {\it J. Algebra} {\bf 237}  (2001), 798-812.

\item{\N3g.} {\ver  E. Neher}. ``Lie algebras graded by 3-graded root
systems and Jordan pairs covered by grids".  {\it Amer. J. Math.}
{\bf 118(2)} (1996),  439-491.


\item{\Zfin.} {\ver  E. I. Zelmanov}. ``Lie algebras with a finite grading".
{\it Math. USSR Sbornik} {\bf 52(2)} (1985), 537-552.

\item{\Zad.} {\ver E. I. Zelmanov}. ``Lie algebras with an algebraic
adjoint representation".  {\it Math. USSR Sbornik} {\bf 49(2)}
(1984), 537-552.






\end
