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\begin{document}
\title{Lie algebras that can be written as the sum of two nilpotent subalgebras}
\author{Pasha Zusmanovich}
\address{}
\email{pasha.zusmanovich@gmail.com}
\date{first written October 28, 2003; last minor revison May 27, 2018}
\maketitle
This is a short survey about the current state of affairs with Lie algebras $L$
that can be written as the sum of two nilpotent subalgebras $A,B$: $L = A + B$.
The sum is understood in the sense of vector spaces and is not (necessarily) direct.
Motivated by similar questions from Group Theory and Ring Theory,
Kegel \cite{kegel} asked in 1963
whether a Lie ring that can be written as the sum of two nilpotent subrings,
is solvable.
If we restrict our attention to algebras over a field, it is easy to see that
without loss of generality one can assume that the ground field
is algebraically closed. If, further, we restrict our attention to the
finite-dimensional algebras, the existing powerful arsenal of structure theory
immediately yields a positive answer in the zero characteristic case
(see \cite{goto} or \cite{kos}).
As it always happens, in characteristic $p > 0$ the situation becomes more complicated,
and the attention to this question was renewed in 1982 by Kostrikin \cite{kos}.
After that, Petravchuk \cite{pet} gave an example providing a negative
answer to the question in characteristic $2$.
In positive direction, a few particular results
(with restrictions on the index of nilpotency
of one of the summands) were obtained (including \cite{pet}), and at the beginning of
1990s Panyukov \cite{pan1} (for characteristic $p > 2$)
and myself \cite{zus} (for characteristic $p > 5$),
using different approaches, provided a positive answer in the general case.
A further question arise of precise description of a class of
such algebras inside the class of all solvable algebras
(see \cite{zus}, \cite{pan2}, and \cite{bah}).
Numerous results about finite groups that can be decomposed
into a product of two groups with different properties, may be found in \cite{cs},
\cite{fisman}, \cite[Chap. VI]{huppert}, and references therein.
\section*{Degree of solvability}
So, $L$ is solvable. What can be said about its degree of solvability $s(L)$
in terms of degrees of nilpotency $n(A)$ and $n(B)$ of $A$ and $B$?
In \cite[Proposition 1.5]{kolman}, following the well-known result of Ito for groups,
it is proved that if $n(A) = n(B) = 1$ (i.e., both $A$ and $B$ are abelian),
then $s(L) = 2$ (i.e., $L$ is metabelian).
In \cite{pet3} it is proved that if $n(A) = 1$ (i.e., $A$ is abelian) and $n(B) = 2$
(i.e., $B$ is $2$-step nilpotent), then $s(L) \le 10$. This bound is, probably, too
rough. I do not know an example of such algebra with $s(L) > 3$.
%Here is an example with $s(L) = 4$. Let $A$ be $n$-dimensional,
%$A = \langle a_i \vertbar i \in \mathbb Z/n\mathbb Z \rangle$,
%and $B$ be the free $2$-step nilpotent of rank $n$,
%$B = \langle x_i, [x_i,x_j] \vertbar i,j \in \mathbb Z/n\mathbb Z \rangle$,
%for arbitrary integer $n > 1$.
%Let $[a_i,x_j] = a_{i+j} + x_{i+j}$ for any $i,j\in \mathbb Z/n\mathbb Z$,
%and the sum $L = A + B$ be direct.
\begin{question}[Dietrich Burde]
Is it true that if $n(A) = 1$ and $n(B) = 2$, then $s(L) \le 3$?
\end{question}
Examples of groups for which $s(L) > n(A) + n(B)$, are given in \cite{cs}.
Form the results of \cite{pet1991} follows, that if $n(A) = 1$, then
$s(L)$ is bounded by a function of $n(B)$.
\begin{question}
Is it possible to bound $s(L)$ by a linear function of $n(A)$ and $n(B)$?
\end{question}
The same question for groups is asked in Kourovka notebook, question 14.43.
\section*{Infinite-dimensional algebras}
What happens in the infinite-dimensional situation?
\begin{question}
Is it true, that an infinite-dimensional Lie algebra over a field of characteristic
$\ne 2$ that can be written as the sum of two nilpotent subalgebras, is solvable?
\end{question}
This was (re)asked, in particular, in \cite{bah} and by Rutwig Campoamor-Stursberg.
In \cite{pet1991} a positive answer is obtained in the case when one of the summands is
abelian.
In \cite{pet2}, a positive answer to this question is provided in the two cases:
when one of the summands is finite-dimensional, and when commutants
of both summands are finite-dimensional, and in \cite{honda} -
for the class of locally-finite Lie algebras
(i.e. Lie algebras all whose finitely-generated algebras are finite-dimensional).
Both results are obtained by a quick and easy reduction to the finite-dimensional case.
Similar results for infinite groups (with, again, that or another finiteness
conditions) were obtained by N.S. Chernikov (see \cite{chernikov} and references in
\cite{pet2}).
A weaker question is also open:
\begin{question}
Is it true, that an (infinite-dimensional) associative algebra that can be written as the sum of two
Lie-nilpotent associative algebras, is Lie-solvable?
\end{question}
An affirmative answer is known in the cases when one of the
summands is commutative (\cite{petravchuk-ass-abelian}), or is an one-sided ideal
(\cite[Corollary 1]{lp}).
%The case when one of the summands is finite-dimensional,
%probably may be dealt with by trying to mimic
%approach of \cite{zus} in the finite-dimensional modular case.
%Namely, consider a counterexample with a minimal possible dimension of one of the
%summands and observe that such counterexample has a solvable maximal subalgebra of finite
%codimension. Then, following Boris Weisfeiler, one may consider the associated
%filtration and associated graded subalgebra. So we will obtain a graded
%algebra whose sum of non-negative component is solvable. As this is a
%rather strong condition, one may hope, following Weisfeiler's arguments \cite{wei}
%and/or embedding of such an algebra into infinite-dimensional Lie
%algebra of vector fields, to get a description of such algebras similar
%to finite-dimensional modular case, and then go back to the initial filtered
%algebra (via deformation-theoretic and cohomological considerations or
%directly).
%
%However, it probably not worths all these troubles, as the
%truly "infinite" case is when both summands are infinite-dimensional (and without
%any other finiteness restrictions).
\begin{thebibliography}{BTT}
\bibitem[BTT]{bah}
Y. Bahturin, M. Tvalavadze, and T. Tvalavadze,
\emph{Sums of simple and nilpotent Lie algebras},
Comm. Algebra \textbf{30} (2002), 4455--4471.
\bibitem[C]{chernikov}
N.S. Chernikov, \emph{Infinite groups that are products of nilpotent subgroups},
Soviet Math. Dokl. \textbf{21} (1980), N3, 701--703.
\bibitem[CS]{cs} J. Cossey and S. Stonehewer,
\emph{On the derived length of finite dinilpotent groups},
Bull. London Math. Soc. \textbf{30} (1998), 247--250.
\bibitem[F]{fisman}
E. Fisman, \emph{Finite factorizable groups}, PhD Thesis, Bar-Ilan University, 1982.
\bibitem[G]{goto}
M. Goto, \emph{Note on a characterization of solvable Lie algebras},
J. Sci. Hiroshima Univ. Ser. A-I \textbf{26} (1962), 1--2.
\bibitem[HS]{honda}
M. Honda and T. Sakamoto, \emph{Lie algebras represented as a sum of two subalgebras},
Math. J. Okayama Univ. \textbf{42} (2000), 73--81.
\bibitem[H]{huppert} B. Huppert, \emph{Endliche Gruppen. I}, Springer, 1967.
\bibitem[Ke]{kegel}
O.H. Kegel, \emph{Zur Nilpotenz gewisser assoziativer Ringe},
Math. Ann. \textbf{149} (1963), 258--260.
\bibitem[Kol]{kolman} B. Kolman, \emph{Semi-modular Lie algebras},
J. Sci. Hiroshima Univ. Ser. A-I \textbf{29} (1965), 149--163.
\bibitem[Kos]{kos}
A.I. Kostrikin, \emph{A solvability criterion for a finite-dimensional Lie algebra},
Moscow Univ. Math. Bull. \textbf{37} (1982), N2, 21--26.
\bibitem[LP]{lp} V.S. Luchko and A.P. Petravchuk,
\emph{On one-sided Lie nilpotent ideals of associative rings},
Algebra Discrete Math. 2007, N4, 102--107; arXiv:0803.0968.
\bibitem[Pa1]{pan1}
V.V. Panyukov, \emph{On the solvability of Lie algebras of positive characteristic},
Russ. Math. Surv. \textbf{45} (1990), N4, 181--182.
\bibitem[Pa2]{pan2}
\bysame{},
\emph{On solvable Lie algebras decomposable into a sum of two nilpotent subalgebras},
Sbornik Math. \textbf{83} (1995), 221-235.
\bibitem[Pe1]{pet}
A.P. Petravchuk,
\emph{Lie algebras which can be decomposed into the sum of an abelian subalgebra
and a nilpotent subalgebra},
Ukrain. Math. J. \textbf{40} (1988), 331--334.
\bibitem[Pe2]{pet1991} \bysame{},
\emph{The solubility of a Lie algebra which decomposes into a direct sum of an abelian
and a nilpotent subalgebra},
Ukrain. Math. J. \textbf{43} (1991), 920--924.
\bibitem[Pe3]{pet2}
\bysame{},
\emph{On the sum of two Lie algebras with finite dimensional commutants},
Ukrain. Math. J. \textbf{47} (1995), N8, 1244--1252.
\bibitem[Pe4]{petravchuk-ass-abelian} \bysame{},
\emph{On the Lie solvability of sums of some associative rings},
Visn. Ky\"iv. Univ. Ser. Fiz.-Mat. Nauki 1999, N1, 78--81 (in Russian).\footnote[2]{
I have not seen that paper and judging it solely by MR2003a:16043.
}
\bibitem[Pe5]{pet3}
\bysame{}, \emph{On derived length of the sum of two nilpotent Lie algebras},
Visn. Mat. Mekh. Ky\"iv. Univ. 2001, No.6, 53--56 (in Ukrainian).
\bibitem[Z]{zus}
P. Zusmanovich,
\emph{A Lie algebra that can be written as a sum of two nilpotent subalgebras, is solvable}, Math. Notes \textbf{50} (1991), 909--912;
arXiv:0911.5418.
\end{thebibliography}
\end{document}
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