Determination of Invariant Inner Product on Seven Dimensional Nilpotent Lie Algebras

Gr. Tsagas, Ch. Christophoridou and A. Synefaki

Abstract The aim of the present paper is to study the invariant bilinear forms on the Nilpotent Lie algebras of dimension seven.These Nilpotent Lie algebras are over C.

Mathematics Subject Classification: 17B40 Key words: Nilpotent Lie algebra, bilinear symmetric form, invariant bilinear form and nullspace.

1 Introduction

Let g be a Nilpotent Lie algebra over a field K of characteristic zero. We assume that the dimension of g is n. The classification of g is known up to dimension eight [5], [8].

The study of the invariant symmetric bilinear forms on g is an open problem. This problem has been completely solved until to dimension six and partially for

dimension eight [8]. The purpose of this article is to determine the invariant symmetric bilinear forms on the Nilpotent Lie algebras of dimension seven and over the field C. It is also determined the null spaces of these bilinear forms.

The paper contains two paragraphs. The first paragraph is the introduction. The determination of invariant symmetric bilinear forms on the Nilpotent Lie algebras of dimension seven are contained in the second paragraph. It also includes the null spaces of these bilinear forms.

2. Let g be a Lie algebra over a field of characteristic zero, whose dimension is n; that is dimg = n. Let f be an invariant symmetric bilinear form on the Lie algebra g, that means a bilinear form f which satisfies the relation:

f([x, y], z) = f(x, [y, z]), ∀x, y, z ∈ g.

Each bilinear form f on g can be represented by a matrix M = (a_{ij}), a_{ij} ∈ C (with respect of the base {e_{1}, e_{2}, ...., e_{n}} of the Lie algebra g, where a_{ij} = f(e_{i}, e_{j}) and a_{ij} = a_{ji}. The nullspace N_{f} of f is defined as follows

N_{f} := {x ∈ g/f(g, x) = 0 = f(x, g)}.

Balkan Journal of Geometry and Its Applications, Vol.4, No.2, 1999, pp. 117-122 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

In this paper we consider invariant symmetric bilinear forms f on a Nilpotent Lie algebra g of dimension seven, and we shall determine all the invariant symmetric bilinear forms f on g and their nullspaces.

We use the classification which is given in [5]. We have estimated all the corresponding matrices of the bilinear forms and the nullspaces of the seven dimensional Nilpotent Lie algebras. These matrices and nullspaces are given below:

M_{1} = [ \begin{matrix}a_{11} & a_{12} & 0 & 0 & 0 & 0 & 0 \\ a_{12} & a_{22} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{2} = [ \begin{matrix}a_{11} & a_{12} & 0 & 0 & a_{15} & 0 & 0 \\ a_{12} & a_{22} & 0 & -a_{15} & 0 & 0 & 0 \\ 0 & 0 & a_{15} & 0 & 0 & 0 & 0 \\ 0 & -a_{15} & 0 & 0 & 0 & 0 & 0 \\ a_{15} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{3} = [ \begin{matrix}a_{11} & a_{12} & a_{13} & 0 & 0 & 0 & 0 \\ a_{12} & a_{22} & a_{23} & 0 & 0 & 0 & 0 \\ a_{13} & a_{23} & a_{33} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{4} = [ \begin{matrix}a_{11} & a_{12} & 0 & 0 & 0 & a_{16} & 0 \\ a_{12} & a_{22} & 0 & -a_{16} & 0 & 0 & 0 \\ 0 & 0 & a_{16} & 0 & 0 & 0 & 0 \\ 0 & -a_{16} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{16} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{5} = [ \begin{matrix}a_{11} & a_{12} & 0 & 0 & 0 & a_{16} & 0 \\ a_{12} & a_{22} & 0 & 0 & 0 & a_{26} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{16} & a_{26} & 0 & 0 & 0 & a_{66} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{6} = [ \begin{matrix}a_{11} & a_{12} & a_{13} & 0 & a_{15} & 0 & 0 \\ a_{12} & a_{22} & a_{23} & 0 & a_{25} & 0 & 0 \\ a_{13} & a_{23} & a_{33} & 0 & a_{35} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{15} & a_{25} & a_{35} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{7} = [ \begin{matrix}a_{11} & a_{12} & 0 & a_{14} & 0 & 0 & 0 \\ a_{12} & a_{22} & 0 & a_{24} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{14} & a_{24} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{8} = [ \begin{matrix}a_{11} & a_{12} & 0 & 0 & a_{15} & 0 & 0 \\ a_{12} & a_{22} & 0 & 0 & a_{25} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{15} & a_{25} & 0 & 0 & a_{55} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{9} = [ \begin{matrix}a_{11} & a_{12} & a_{13} & a_{14} & 0 & 0 & 0 \\ a_{12} & a_{22} & a_{23} & a_{24} & 0 & 0 & 0 \\ a_{13} & a_{23} & a_{33} & a_{34} & 0 & 0 & 0 \\ a_{14} & a_{24} & a_{34} & a_{44} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{10} = [ \begin{matrix}a_{11} & a_{12} & a_{13} & a_{14} & 0 & 0 & 0 \\ a_{12} & a_{22} & a_{23} & 0 & 0 & a_{14} & 0 \\ a_{13} & a_{23} & a_{33} & 0 & -a_{14} & 0 & 0 \\ a_{14} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -a_{14} & 0 & 0 & 0 & 0 \\ 0 & a_{14} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{11} = [ \begin{matrix}a_{11} & a_{12} & 0 & 0 & 0 & 0 & a_{17} \\ a_{12} & a_{22} & 0 & 0 & 0 & a_{17} & 0 \\ 0 & 0 & 0 & 0 & -a_{17} & 0 & 0 \\ 0 & 0 & 0 & a_{17} & 0 & 0 & 0 \\ 0 & 0 & -a_{17} & 0 & 0 & 0 & 0 \\ 0 & a_{17} & 0 & 0 & 0 & 0 & 0 \\ a_{17} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{12} = [ \begin{matrix}a_{11} & a_{12} & a_{13} & 0 & 0 & a_{16} & 0 \\ a_{12} & a_{22} & a_{23} & 0 & -a_{16} & 0 & 0 \\ a_{13} & a_{23} & a_{33} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & a_{16} & 0 & 0 & 0 \\ 0 & -a_{16} & 0 & 0 & 0 & 0 & 0 \\ a_{16} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{13} = [ \begin{matrix}a_{11} & a_{12} & 0 & a_{14} & a_{15} & 0 & 0 \\ a_{12} & a_{22} & 0 & a_{24} & a_{25} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{14} & a_{24} & 0 & a_{44} & a_{45} & 0 & 0 \\ a_{15} & a_{25} & 0 & a_{45} & a_{55} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{14} = [ \begin{matrix}a_{11} & a_{12} & 0 & a_{14} & 0 & 0 & 0 \\ a_{12} & a_{22} & 0 & a_{24} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{14} & a_{24} & 0 & a_{44} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{15} = [ \begin{matrix}a_{11} & a_{12} & a_{13} & 0 & 0 & a_{16} & 0 \\ a_{12} & a_{22} & a_{23} & 0 & -a_{16} & 0 & 0 \\ a_{13} & a_{23} & a_{33} & a_{16} & 0 & 0 & 0 \\ 0 & 0 & a_{16} & 0 & 0 & 0 & 0 \\ 0 & -a_{16} & 0 & 0 & 0 & 0 & 0 \\ a_{16} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{16} = [ \begin{matrix}a_{11} & a_{12} & a_{13} & 0 & 0 & a_{16} & 0 \\ a_{12} & a_{22} & a_{23} & 0 & 0 & a_{26} & 0 \\ a_{13} & a_{23} & a_{33} & 0 & 0 & a_{36} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{16} & a_{26} & a_{36} & 0 & 0 & a_{66} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{17} = [ \begin{matrix}a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & 0 & 0 \\ a_{12} & a_{22} & a_{23} & a_{24} & a_{25} & 0 & 0 \\ a_{13} & a_{23} & a_{33} & a_{34} & a_{35} & 0 & 0 \\ a_{14} & a_{24} & a_{34} & a_{44} & a_{45} & 0 & 0 \\ a_{15} & a_{25} & a_{35} & a_{45} & a_{55} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{18} = [ \begin{matrix}a_{11} & a_{12} & 0 & a_{14} & 0 & a_{16} & 0 \\ a_{12} & a_{22} & 0 & a_{24} & -a_{16} & 0 & 0 \\ 0 & 0 & a_{16} & 0 & 0 & 0 & 0 \\ a_{14} & a_{24} & 0 & a_{44} & 0 & 0 & 0 \\ 0 & -a_{16} & 0 & 0 & 0 & 0 & 0 \\ a_{16} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

M_{19} = [ \begin{matrix}a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} & 0 \\ a_{12} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} & 0 \\ a_{13} & a_{23} & a_{33} & a_{34} & a_{35} & a_{36} & 0 \\ a_{14} & a_{24} & a_{34} & a_{44} & a_{45} & a_{46} & 0 \\ a_{15} & a_{25} & a_{35} & a_{45} & a_{55} & a_{56} & 0 \\ a_{16} & a_{26} & a_{36} & a_{46} & a_{56} & a_{66} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

Table 1

Matrix  Nullspace  Algebras
M_{1}  {e_{3}, e_{4}, e_{5}, e_{6}, e_{7}} 0.1,0.2,0.3,1.1(i_{λ}),1.1(ii),1.1(iii),1.4,1.5,1.6,2.3
M_{2}  \{e_{6}, e_{7}\}  \begin{matrix}0.4 & λ & ,0.5,0.6,0.7,1.02,1.1(iv),1.10,1.13,1.14,1.17, \\ 2.5,2.6,2.7,2.8,2.9 \end{matrix}

M_{3}

{e_{4}, e_{5}, e_{6}, e_{7}}

0.8,1.01(i),1.01(ii),1.1(v),1.2(i_{λ})(*),1.2(iii),1.2(iv),
1.3(i_{λ}),1.3(ii),1.3(iii),1.3(iv),1.7,1.8,1.9,1.11,1.12,
1.15,1.18,1.19,1.21,2.1(i_{λ}),2.1(ii),2.1(v),2.10,2.11,

2.12,2.13,2.14,2.15,2.16,2.17,2.19,2.20,2.21,2.22, 2.24,2.26,2.31,2.32,2.33,2.34,2.35,2.37,2.39,2.42, 2.43,3.2,3.3,3.4,3.5,3.20,3.21,3.22

M_{4}  {e_{5}, e_{7}}  1.03
M_{5}  {e_{3}, e_{4}, e_{5}, e_{7}}  1.1(vi)
M_{6}  {e_{4}, e_{6}, e_{7}}  1.3(v),2.1(iii)
M_{7}  {e_{3}, e_{5}, e_{6}, e_{7}}  1.16,2.40
M_{8}  {e_{3}, e_{4}, e_{6}, e_{7}}  1.20
M_{9}  {e_{5}, e_{6}, e_{7}}  2.1(iv),2.23,2.25,2.27,2.28,2.29,2.30,2.36,2.45, 3.7,3.8,3.9,3.10,3.11.,3.12,3.13,3.14,3.15,3.16, 3.17,3.24,4.1,4.2,4.5
M_{10}  {e_{7}}  2.2
M_{11}  {0}  2.4
M_{12}  {e_{7}}  2.18
M_{13}  {e_{3}, e_{6}, e_{7}}  2.38
M_{14}  {e_{3}, e_{5}, e_{6}, e_{7}}  2.41
M_{15}  {e_{7}}  2.44,3.1(i_{λ}), 3.1(ii), 3.6
M_{16}  {e_{4}, e_{5}, e_{6}}  3.1(iii)
M_{17}  {e_{6,}e_{7}}  3.18,3.19,4.3
M_{18}  {e_{7}}  3.23
M_{19}  {e_{7}}  4.4

Consequently, we have proved the following Theorem 2.1. Let g be a Nilpotent Lie algebra of dimension seven. The invariant

symmetric bilinear form on each of these algebras is described by one of the matrices M_{1} − M_{19}.

From the above theorem we conclude Corollary 2.2. Let g be a Nilpotent Lie algebra of dimension seven. The nullspaces

of the invariant symmetric bilinear forms of these Lie algebras are given by the table 1. There are 13 different nullspaces.

References

[1] M.Ancochea Bermudez et M.Goze, Classification des algèbres de Lie nilpotentes complexes de dimension 7, Archiv der Mathematik, Vol 52, No 2 (1989), 175-185.

[2] V.V.Astrakhantsev, Decomposability of metrizable Lie algebra, Funktsional’nyi Analiz i Ego Prilozheniya, Vol.12, No.3 (1978), 64-65.

[3] R.E.Block, H.Zassenhaus, The Lie algebras with a non-degenerate trace form, Illinois Journal of Mathematics 8 (1964), 543-549.

[4] M.Bordemann, Invariante Bilinearformen auf endlich-dimensionalen Algebren, Diplomarbeit, Mathematische Fakultät der Universität Freiburg i. Br. (Februar 1988).

[5] R.Carles, Weight systems for nilpotent Lie algebras of dimension □ 7 over C, Universite de Poitiers, department de Mathematiques, No 47 (1989).

[6] G.Favre, L.J.Santharoubane, Symmetric, invariant, non-degenerate bilinear form on a Lie algebra, Journal of Algebra 105 (1987), 451-464.

[7] J.Patera, R.T.Sharp, P.Winternitz, H.Zassenhaus, Invariants of real low dimension Lie algebras, Journal of Mathematical Physics, Vol 17, No 6 (1976).

[8] Gr.Tsagas, A.Kobotis, Special class of Nilpotent Lie Algebras, Bull. Cal. Math. Soc., 81 (1989), 327-341.

[9] Gr.Tsagas, Cr.Christophoridou, A.Synefaki, Invariant bilinear forms on Nilpotent Lie Algebras , Hadronic Journal, Vol. 22 (1999).

[10] Gr.Tsagas, Cr.Christophoridou, A.Synefaki, Special bilinear forms on eight dimensional Nilpotent Lie, Hadronic Journal, Vol. 22 (1999).

[11] Gr.Tsagas, T.Koukouvinos, Classification of Nilpotent Lie Algebras of dimension eight, whose maximal abelian ideal is of dimension four, Proceedings of the Workshop “Global Analysis, Differential Geometry and Lie Algebras”, Thessaloniki, June 1998.

[12] H.Zassenhaus, On trace bilinear forms on Lie algebras, Proceedings of the Glasgow Math.Assoc. 4 (1959), 62-72.

Department of Mathematics and Physics, Faculty of Technology, Aristotle University of Thessaloniki Thessaloniki 54006, Greece