On Derivations of Nilpotent Lie Algebras

P. Barbari and A. Kobotis

Abstract

The aim of the present paper is to prove that the nilpotent Lie algebras of dimension six and seven in general are not the radical of any other Lie algebra.

Mathematics Subject Classification: 17B30 Key Words: nilpotent Lie algebra, derivation, simple algebra, semi-simple Lie algebra, inner derivation, center and radical.

1

Let g be a Lie algebra over a field k of characteristic zero. It is interesting to study the relation between this Lie algebra and its Lie algebra of derivations and as well as with other Lie algebras obtained via the given Lie algebra.

The purpose of this paper is to prove that any nilpotent Lie algebra over a field k of characteristic zero of dimension six and seven, except those which are characteristically nilpotent, can not be the radical of a Lie algebra g having the property D(g) = I(g) + C(g). This is an extension of the work( [4]).

The whole paper contains four paragraphs each of them is analyzed as follows. The first paragraph is the introduction. Basic elements of Lie algebras are given in the second paragraph. In the third paragraph are studied the nilpotent Lie algebras of dimension six concerning some properties of this Lie algebras. Finally, the last paragraph faces the same problem for the nilpotent Lie algebras of dimension seven.

2

We shall give some basic notions concerning of this paper.

Let g be a Lie algebra over the field k of characteristic zero of dimension n. It is known that from this algebra we can form the following sequences of ideals of g:

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 1-7 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(2.1)  C^{0}g = g, C^{1}g = [g, g], ..., C^{q}g = [g, C^{q−1}g], ...

which is called descending central sequence.

(2.2)  C^{0}g = 0, C^{1}g = centre(g), ..., C^{q}g = centre(g/C^{q−1}g)

which is called increasing central sequence and

(2.3)  D^{0}g = g, D^{1}g = [g, g], ..., D^{q}g = [D^{q−1}g, D^{q−1}g], ...

which is called derived sequence.

If there exists an integer q ≥ 2 such that C^{q}g = {0}, then the Lie algebra g is called nilpotent of nilpotency q.

A linear mapping f on g is called derivation, if it satisfies the relation:

f[x, y] = [fx, y] + [x, fy], ∀x, y ∈ g.

The set of all derivations f on g is denoted by D(g), that is

D(g) = {f / f : g → g, f linear and f[x, y] = [fx, y] + [x, fy]}

The following mapping:

ad_{x} : g → g, ad_{x} : y → ad_{x}y = [x, y]

is a derivation which is called inner derivation. The set of all inner derivations is denoted by I(g) which is an ideal of the Lie algebra of derivations D(g). The other derivations on g, which are not inner, are called outer, which are denoted by D_{i}(g). It is known that

D(g) = I(g) ⊕ D_{i}(g)

We must notice that D_{i}(g) is an ideal of D(g).

The Lie algebra g is called characteristically nilpotent if the Lie algebra D(g) is nilpotent.

A Lie algebra g is said to be semi-simple if does not contain any non-zero abelian ideal.

If g is semi-simple, then

D(g) = I(g).

The classification of Lie algebras of finite dimension over a field k of characteristic zero is reduced to Levi’ s theorem

g = S ⊕ R,

where R is radical of g, which is the maximal solvable ideal of g and S a semisimple subalgebra of g.

Let S be a semi-simple Lie algebra, since S = [S, S] the trace of S is zero. The center Z(g) of a Lie algebra g is an ideal of g, such that

Z(g) = {z ∈ g, [x, z] = 0 for all x ∈ g}.

A Lie algebra g is called abelian if and only if Z(g) = g.

A nilpotent Lie algebra g is called quasi-cyclic provided g has a subspace U such that

g = U ⊕ [g, g]

and such that g is the direct sum of the subspaces U^{i} where U^{1} = U and U^{i} = [U, U^{i−1}] for i ≥ 2.

We denote by C(g) the set of all endomorphisms of g, which map g into the center Z of g and [g, g] into {0}. It can be proved that C(g) is a subalgebra of D(g). Each element of C(g) is called central derivation.

The Lie algebra of derivations D(g) of g can be written as

D(g) = I(g) + C(g).

It is known that I(g) ∩ C(g) is not always empty.

A Lie algebra g is called T when it is not the radical of any other Lie algebra L having the property

D(L) = I(L) + C(L).

3

It is known that there is a classification of nilpotent Lie algebras of dimension six over a field C or R. In this paragraph we study some properties of these nilpotent Lie algebras.

Now, we can prove the following theorem. Theorem 3.1. The nilpotent Lie algebras over a field k of characteristic zero of dimension 6 belong to the Lie algebras of type T. Proof. We consider the classification of nilpotent Lie algebras of dimension 6 which is given in ([5]). Also, we consider two cases (I) quasi cyclic (II) no quasicyclic.

(I) Let g_{6,1} be the nilpotent Lie algebra described in terms of a basis x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} by the following multiplication tables.

[x_{1}, x_{2}] = x_{5}, [x_{1}, x_{4}] = x_{6}, [x_{2}, x_{3}] = x_{6}.

By the definition of a quasi-cyclic nilpotent Lie algebra given in paragraph 2 we obtain

U = U^{1} = {x_{1}, x_{2}, x_{3}, x_{4}}
U^{2} = [U, U^{1}] = {x_{5}, x_{6}}
U^{3} = [U, U^{2}] = {0}

So, the nilpotent Lie algebra g_{6,1} can be written as the direct sum

g_{6,1} = U ⊕ U^{2}

and by the definition given in paragraph 2, we conclude that g_{6,1} is a quasi-cyclic nilpotent Lie algebra.

According to the results in ([4]) we obtain that g_{6,1} is a T -algebra.

In the same way we prove that the nilpotent Lie algebras

g_{6,3}, g_{6,4}, g_{6,6}, g_{6,9}, g_{6,10}, g_{6,14}, g_{6,15}, g_{6,16}, g_{6,18}, g_{6,4}

are T -algebras.

(II) Let g_{6,2}, be the following nilpotent Lie algebra

[x_{1}, x_{2}] = x_{5}, [x_{1}, x_{5}] = x_{6}, [x_{3}, x_{4}] = x_{6}.

The Lie algebra of derivations D(g_{6,2}) of g_{6,2} is represented by the set of matrices

D(g_{6,2}) = \{[ \begin{matrix}a_{11} & 0 & 0 & 0 & 0 & 0 \\ a_{12} & a_{22} & 0 & 0 & 0 & 0 \\ a_{13} & 0 & a_{33} & a_{43} & 0 & 0 \\ a_{14} & 0 & a_{34} & 2a_{11} + a_{22} + a_{33} & 0 & 0 \\ a_{15} & a_{25} & a_{14} & −a_{13} & a_{11} + a_{22} & 0 \\ a_{16} & a_{26} & a_{36} & a_{46} & a_{25} & 2a_{11} + a_{22} \end{matrix} ]\}

The endomorphism of D(g_{6,2}) defined by

Dx_{1} = x_{1}, Dx_{2} = x_{2}, Dx_{3} = x_{3}, Dx_{4} = 2x_{4}, Dx_{5} = 2x_{5}, Dx_{6} = 3x_{6}

is a derivation of D(g_{6,2}) whose trace is not zero.

According to the results in ([4]) we obtain that g_{6,2} is not the radical of a Lie algebra L whose D(L) can be written in the form

D(L) = I(L) + C(L).

Hence g_{6,2} is a T -algebra.

With the same method we prove that the nilpotent Lie algebras

g_{6,5}, g_{6,7}, g_{6,8}, g_{6,11}, g_{6,12}, g_{6,13}, g_{6,17}, g_{6,19}, g_{6,10}

are T -algebras.

4

In this section we are concerned with the nilpotent Lie algebras of dimension seven over a field C or R of characteristic zero. Comparing with the previous paragraph in which we studied some properties of the nilpotent Lie algebras of dimension six, in this last paragraph we additionally study some properties of a second category, that of the characteristically nilpotent Lie algebras of dimension seven.

We can prove the following Theorem 4.1. The nilpotent Lie algebras over a field k of characteristic zero of dimension 7 can be classified in two categories. The first consists of the Lie algebras of type T and the second one of the characteristically nilpotent Lie algebras. Proof. We consider the classification of nilpotent Lie algebras of dimension 7 which is given in ([5]). Also, we consider two cases (I) T -algebras (1. quasi-cyclic 2. not quasi-cyclic) (II) characteristically nilpotent Lie algebras.

(I_{1}) Let g_{7,1.17} be the following nilpotent Lie algebra of dimension seven,

[x_{1}, x_{2}] = x_{3}, [x_{1}, x_{3}] = x_{4}, [x_{1}, x_{4}] = x_{6}, [x_{1}, x_{6}] = x_{7},
[x_{2}, x_{3}] = x_{5}, [x_{2}, x_{5}] = x_{6}, [x_{2}, x_{6}] = x_{7}, [x_{3}, x_{4}] = −x_{7},
[x_{3}, x_{5}] = x_{7}.

We have

U = U^{1} = {x_{1}, x_{2}}
U^{2} = [U, U^{1}] = {x_{3}}
U^{3} = [U, U^{2}] = {x_{4}, x_{5}}
U^{4} = [U, U^{3}] = {x_{6}}
U^{5} = [U, U^{4}] = {x_{7}}
U^{6} = [U, U^{5}] = {0} .

Hence the nilpotent Lie algebra g_{7,1.17} can be written as the direct sum

g_{7,1.17} = U^{1} ⊕ U^{2} ⊕ U^{3} ⊕ U^{4} ⊕ U^{5}

and by the definition given in paragraph 2, we conclude that g_{6,1.17} is a quasicyclic nilpotent Lie algebra.

According to the results in ([4]) we obtain that g_{6,1} is a T -algebra.

In the same way we prove that the nilpotent Lie algebras g_{7,1.19}, g_{7,2.3}, g_{7,2.4}, g_{7,2.5}, g_{7,2.6}, g_{7,2.7}, g_{7,2.8}, g_{7,2.9}, g_{7,2.12}, g_{7,2.16}, g_{7,2.26}, g_{7,2.34}, g_{7,2.35}, g_{7,2.43}, g_{7,2.44}, g_{7,3.1(i^{λ})}, g_{7,3.1(ii)}, g_{7,3.2}, g_{7,3.3}, g_{7,3.4}, g_{7,3.6}, g_{7,3.10}, g_{7,3.11}, g_{7,3.12}, g_{7,3.16}, g_{7,3.19}, g_{7,3.20}, g_{7,3.21}, g_{7,3.22}, g_{7,3.23}, g_{7,3.24}, g_{7,4.2}, g_{7,4.3}, g_{7,4.4}, g_{7,4.5} are T - algebras. (I_{2}) Let g_{7,1.01(i)} be the nilpotent Lie algebra described in terms of a basis x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} by the following multiplication tables

\begin{matrix}[x_{1}, x_{2}] = x_{4}, & [x_{1}, x_{4}] = x_{5}, & [x_{1}, x_{5}] = x_{6}, & [x_{1}, x_{6}] = x_{7}, \\ [x_{2}, x_{3}] = x_{5} + x_{7}, & [x_{3}, x_{4}] = −x_{6}, & [x_{3}, x_{5}] = −x_{7}. \end{matrix}

The Lie algebra of derivations D(g_{7,1.01(i)}) of g_{7,1.01(i)} is represented by the set of matrices

D(g_{7,1.01(i)}) = \{[ \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{12} & a_{22} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{14} & a_{24} & −a_{12} & a_{22} & 0 & 0 & 0 \\ a_{15} & a_{25} & −a_{14} & a_{24} & a_{22} & 0 & 0 \\ a_{16} & a_{26} & − (a_{12} + a_{15}) & a_{25} & a_{24} & a_{22} & 0 \\ a_{17} & a_{27} & a_{37} & a_{26} & a_{25} & a_{24} & a_{22} \end{matrix} ]\}

The endomorphism of D(g_{7,1.01(i)}) defined by

Dx_{1} = 0, Dx_{2} = x_{2}, Dx_{3} = 0, Dx_{4} = x_{4},
Dx_{5} = x_{5}, Dx_{6} = x_{6}, Dx_{7} = x_{7},

is a derivation of D(g_{7,1.01(i)}) whose trace is not zero.

As a consequence of the results in ([4]) we have that g_{7,1.01(i)} is not the radical of a Lie algebra L, whose the Lie algebra of derivations D(L) can be written as

D(L) = I(L) + C(L).

From the above we conclude that g_{7,1.01(i)} is a T -algebra.

In the same way we prove that the following nilpotent Lie algebras are T -algebras.

g_{7,1.01(i)}, g_{7,1.01(ii)}, g_{7,1.02}, g_{7,1.03}, g_{7,1.1(i^{λ})}, g_{7,1.1(ii)}, g_{7,1.1(iii)}, g_{7,1.1(iv)} , g_{7,1.1(v)}, g_{7,1.1(vi)}, g_{7,1.2(i^{λ})}, g_{7,1.2(ii)} , g_{7,1.2(iii)}, g_{7,1.2(iv)}, g_{7,1.3(i^{λ})}, g_{7,1.3(ii)}, g_{7,1.3(iii)}, g_{7,1.3(iv)}, g_{7,1.3(v)}, g_{7,1.4}, g_{7,1.5}, g_{7,1.6}, g_{7,1.7}, g_{7,1.8}, g_{7,1.9}, g_{7,1.10}, g_{7,1.11}, g_{7,1.12}, g_{7,1.13}, g_{7,1.14}, g_{7,1.15}, g_{7,1.16}, g_{7,1.18}, g_{7,1.20}, g_{7,1.21}, g_{7,2.1(i^{λ})}, g_{7,2.1(ii)}, g_{7,2.1(iii)}, g_{7,2.1(iv)} , g_{7,2.1(v)}, g_{7,2.2}, g_{7,2.10}, g_{7,2.11}, g_{7,2.13}, g_{7,2.14}, g_{7,2.15}, g_{7,2.17}, g_{7,2.18}, g_{7,2.19}, g_{7,2.20}, g_{7,2.21}, g_{7,2.22}, g_{7,2.23}, g_{7,2.24}, g_{7,2.25}, g_{7,2.27}, g_{7,2.28}, g_{7,2.29}, g_{7,2.30}, g_{7,2.31}, g_{7,2.32}, g_{7,3.33}, g_{7,2.36}, g_{7,2.37}, g_{7,2.38}, g_{7,2.39}, g_{7,2.40}, g_{7,2.41}, g_{7,2.42}, g_{7,2.45}, g_{7,3.1(iii)}, g_{7,3.14}, g_{7,3.15}, g_{7,3.17}, g_{7,3.18}, g_{7,4.1}.

(II) Finally let

g_{7,0.1}, g_{7,0.2}, g_{7,0.3}, g_{7,0.4}, g_{7,0.5}, g_{7,0.6}, g_{7,0.7}, g_{7,0.8}

be nilpotent Lie algebras over a field k of characteristic zero of dimension seven, given in ([5]).

From the fact that each of the Lie algebras D(g_{7,0.i}), i = 1, 2, ..., 8 of derivations of the Lie algebras g_{7,0.i}, i = 1, 2, ..., 8 is nilpotent, we conclude that the above eight Lie algebras are characteristically nilpotent.

References

[1] J. Dixmier, Sur les representations unitaires des groupes de Lie nilpotent III, Canad. J. Math., 19 (1958), 321-348.

[2] V. Morozov, Classification of nilpotent Lie algebras of sixth dimension, 15V. Vyss Used Zaved Mathematic 190, (1958), 161.

[3] S. Togo, Derivation of Lie algebras, J. Sci. Hiroshima Univ. Ser. A-I, 28 (1964), 133-158.

[4] S. Togo, Lie algebras which have few derivations, J. Sci. Hiroshima Univ. Ser. A-I, 29 (1965), 29-41.

[5] L. Magnin, Adjoint and trivial cohomology tables for indecomposable nilpotent Lie algebras of dimension ≥ 7 / over C., Université de Bourgogne, BP 138, 21004 Dijon Cedex France (1995).

P. Barbari  A. Kobotis
Aristotle University  University of Macedonia
of Thessaloniki  Department of Business Administration
School of Technology  156 Egnatia str.
Mathematics Division  Thessaloniki 54006-GREECE
Thessaloniki 54006-GREECE  tl: +3031-891580
tl: +3031-995946  FAX: +3031-891282
FAX: +3031-416312  E-mail: kompotis@macedonia.uom.gr

E-mail: pbar@vergina.eng.auth.gr