This paper is a natural continuation of Vinogradov [J. Math. Phys. 58, 071703 (2017)] where we proved that any Lie algebra over an algebraically closed field or over R can be assembled in a number of steps from two elementary constituents, called dyons and triadons. Here we consider the problems of the construction and classification of those Lie algebras which can be assembled in one step from base dyons and triadons, called coaxial Lie algebras. The base dyons and triadons are Lie algebra structures that have only one non-trivial structure constant in a given basis, while coaxial Lie algebras are linear combinations of pairwise compatible base dyons and triadons. We describe the maximal families of pairwise compatible base dyons and triadons called clusters, and, as a consequence, we give a complete description of the coaxial Lie algebras. The remarkable fact is that dyons and triadons in clusters are self-organised in structural groups which are surrounded by casings and linked by connectives. We discuss generalisations and applications to the theory of deformations of Lie algebras.

1.
Cabras
,
A.
and
Vinogradov
,
A. M.
, “
Extension of the Poisson bracket to differential forms and multi-vector fields
,”
J. Geom. Phys.
9
(
1
),
75
100
(
1992
).
2.
Fialowski
,
A.
and
Fuchs
,
D.
, “
Construction of miniversal deformations of Lie algebras
,”
J. Funct. Anal.
161
,
76
110
(
1999
).
3.
Filippov
,
V. T.
, “
N-ary Lie algebras
,”
Sibirskii Math. J.
24
,
126
140
(
1985
).
4.
Hanlon
,
P.
and
Wachs
,
M. L.
, “
On Lie k-algebras
,”
Adv. Math.
113
,
206
236
(
1995
).
5.
Kac
,
V. G.
,
Infinite-Dimensional Lie Algebras
(
Cambridge University Press
,
1994
).
6.
Lévy-Nahas
,
M.
, “
Deformation and contraction of Lie algebras
,”
J. Math. Phys.
8
,
1211
1222
(
1967
).
7.
Marmo
,
G.
,
Vilasi
,
G.
, and
Vinogradov
,
A. M.
, “
The local structure of n-Poisson and n-Jacobi manifolds
,”
J. Geom. Phys.
25
,
141
182
(
1998
).
8.
Moreno
,
G.
, “
The Bianchi variety
,”
Differ. Geom. Appl.
28
(
6
),
705
772
(
2010
).
9.
Nijenhuis
,
A.
and
Richardson
,
R. W.
, “
Deformations of Lie algebra structures
,”
J. Math. Mech.
17
,
89
105
(
1967
).
10.
Onishchik
,
A. L.
and
Vinberg
,
E. B.
,
Lie Groups and Lie Algebras
, Volume 41 of Encyclopaedia of Mathematical Sciences (
Springer
,
1991
), Chap. 7.
11.
Vinogradov
,
A. M.
, “
Particle-like structure of Lie algebras
,”
J. Math. Phys.
58
,
071703
(
2017
).
12.
Vinogradov
,
A. M.
and
Vinogradov
,
M. M.
, “
On multiple generalisations of Lie algebras and Poisson manifolds
,”
Contemp. Math.
219
,
273
287
(
1998
).
13.
Weimar-Woods
,
E.
, “
Contractions, generalized Inönü-Wigner contractions and deformations of finite-dimensional Lie algebras
,”
Rev. Math. Phys.
12
,
1505
1529
(
2000
).
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