DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 371–378

THE DECOMPOSITION OF TENSOR SPACES

LENKA LAKOMÁ, JOSEF MIKEŠ, LYDIE MIKUŠOVÁ

Abstract. The problem of F -decomposition of tensor spaces is considered. F -decomposition of tensor spaces of the type (1, 1), (1, 2), (1, 3) is computed. Also F -decomposition for e-structures, when special conditions hold, is studied in this paper and explicit formulas for decomposition are given.

1. Introduction

Let E_{q}^{p} be a tensor space of tensors of the type (p, q) on E. (E is a real n-dimensional vector space.) If A ∈ E_{q}^{p} we will write A = (A_{j_{1}j_{2} ··· j_{q}}^{i_{1}i_{2} ··· i_{p}}), where

A_{j_{1}j_{2} ··· j_{q}}^{i_{1}i_{2} ··· i_{p}} are components of the tensor A ∈ E_{q}^{p} with respect to the basis of E_{q}^{p}.

D. Krupka [1], [2], [3] started to study traceless decomposition. Tensor A ∈ E_{q}^{p} is called traceless if all traces are equal zero. D. Krupka proved next theorem, [3].

Theorem 1.1. Let A_{j_{1}j_{2} ··· j_{q}}^{i_{1}i_{2} ··· i_{p}} be a tensor of the type (p, q). If n + 1 ≥ p + q, then there exists a unique decomposition of A of the form

A_{j_{1}j_{2} ··· j_{q}}^{i_{1}i_{2} ··· i_{p}} = B_{j_{1}j_{2} ··· j_{q}}^{i_{1}i_{2} ··· i_{p}} + ∑_{t=1}^{min\{p,q\}} \begin{matrix}∑ \\ ⨁ \end{matrix} δ_{j_{σ_{1}} j_{σ_{2}} ··· j_{σ_{t}}}^{i_{ρ_{1}} i_{ρ_{2}} ··· ı_{ρ_{t}}} B_{\{σ_{1}σ_{2} ··· σ_{t}\}···}^{\{ρ_{1}ρ_{2} ··· ρ_{t}\}···}

where

⨁ = \{ \begin{matrix}∀ρ_{1}, ρ_{2} ⋯ ρ_{t} ∈ \overline{1, p} & (ρ_{1} < ρ_{2} < ⋯ < ρ_{t}) \\ ∀σ_{1}, σ_{2} ⋯ σ_{t} ∈ \overline{1, q} & (σ_{i} are different to each other) \end{matrix}

and

δ_{j_{σ_{1}} j_{σ_{2}} ··· j_{σ_{t}}}^{i_{ρ_{1}} i_{ρ_{2}} ··· ı_{ρ_{t}}} = δ_{j_{σ_{1}}}^{i_{ρ_{1}}} δ_{j_{σ_{2}}}^{i_{ρ_{2}}} ⋯ δ_{j_{σ_{t}}}^{i_{ρ_{t}}}

such that tensors B_{j_{1}j_{2} ··· j_{q}}^{i_{1}i_{2} ··· i_{p}}; B_{{σ_{1}σ_{2} ··· σ_{t}}···}^{{ρ_{1}ρ_{2} ··· ρ_{t}}···} are traceless.

1991 Mathematics Subject Classification. 53A45, 53B35.

Key words and phrases. Tensor spaces, decomposition, e-structure.

Supported by grant No. 201/96/0227 of The Grant Agency of Czech Republic.

This paper is in final form and no version of it will be submitted for publication elsewhere.

This theorem was generalized by J. Mikeš in [4], [5]. J. Mikeš introduced the concept F -traceless tensor in the following way.

Let F_{j}^{i} be an arbitrary affinor (F_{j}^{i} ∈ E_{1}^{1} such that F_{α}^{α} = 0.) Tensor A ∈ E_{q}^{p} is called F -traceless if the following condition holds.

∀k = \overline{1, p}; ∀r = \overline{1, q}; A_{··· j_{r−1}βj_{r+1}···}^{··· i_{k−1}αi_{k+1} ···}F_{α}^{β} = 0; A_{··· j_{r−1}αj_{r+1}···}^{··· i_{k−1}αi_{k+1} ···} = 0.

We also will work with e-structures. For these structures the condition F_{α}^{i}F_{j}^{α} = eδ_{j}^{i} holds, where e = ±δ_{j}^{i}.

J. Mikeš proved the following Theorem which generalizes Theorem 1.1 see [4], [5].

Theorem 1.2. Let A_{j_{1}j_{2} ··· j_{q}}^{i_{1}i_{2} ··· i_{p}} be a tensor of the type (p, q). If n > 2(p + q), then there exists a unique decomposition of A of the form

A_{j_{1}j_{2} ··· j_{q}}^{i_{1}i_{2} ··· i_{p}} = B_{j_{1}j_{2} ··· j_{q}}^{i_{1}i_{2} ··· i_{p}} + ∑_{t=1}^{min\{p,q\}} \begin{matrix}∑ \\ ⨁ \end{matrix} Q^{∗} _{j_{σ_{1}} j_{σ_{2}} ··· j_{σ_{t}}}^{i_{ρ_{1}} i_{ρ_{2}} ··· i_{ρ_{t}}} B^{⋄} _{···}^{···}

where

⨁ = \{ \begin{matrix}∀ρ_{1}, ρ_{2} ⋯ ρ_{t} ∈ \overline{1, p} & (ρ_{1} < ρ_{2} < ⋯ < ρ_{t}) \\ ∀σ_{1}, σ_{2} ⋯ σ_{t} ∈ \overline{1, q} & (σ_{i} are different to each other) \\ τ_{1}, τ_{2} ⋯ τ_{t} ∈ \{0, 1\} \end{matrix}

∗ = \{τ_{1}, τ_{2} ⋯ τ_{t}\}; ⋄ = \{ \begin{matrix}ρ_{1}, & ρ_{2} & ⋯ & ρ_{t} \\ σ_{1}, & σ_{2} & ⋯ & σ_{t} \\ τ_{1}, & τ_{2} & ⋯ & τ_{t} \end{matrix} \} .

Tensors B_{···}^{···} and B^{⋄} _{···}^{···} are F -traceless and

Q^{∗} _{j_{σ_{1}} j_{σ_{2}} ··· j_{σ_{t}}}^{i_{ρ_{1}} i_{ρ_{2}} ··· ı_{ρ_{t}}} = F^{{τ_{1}}}_{j_{σ_{1}}}^{i_{ρ_{1}}} F^{{τ_{2}}}_{j_{σ_{2}}}^{i_{ρ_{2}}} ⋯ F^{{τ_{t}}}_{j_{σ_{t}}}^{i_{ρ_{t}}};  F^{{0}}_{j}^{i} = δ_{j}^{i}; F^{{1}}_{j}^{i} = F_{j}^{i}.

2. F -decomposition of tensors of the type (1, 1) and (1, 2)

The F -decomposition of the tensors of the type (1, 1) for e-structures is obvious and it is given by the next theorem.

Theorem 2.1. Let A be a tensor of the type (1, 1) and let F be a traceless estructure. If n > 0 then there exists unique F -decomposition of the tensor A in the form

A_{j}^{i} = B_{j}^{i} + δ_{j}^{i}C + DF_{j}^{i} (1)

where tensors B_{j}^{i}, C, D are given in the form

C = \frac{1}{n}A_{α}^{α}; D = \frac{1}{en}A_{β}^{α}F_{α}^{β} (2)

Proof. Suppose that tensor B is traceless. Contract the decomposition in the formula (1) by the Kronecker’s delta and by the affinor F. We immediately get the formulas (2).

Now we find F -decomposition for tensors of the type (1, 2). We also extend an area of validity of the general theorem 1.2.

Theorem 2.2. Let A be a tensor of the type (1, 2) and let F be a traceless estructure. If n > 2 then there exists unique F -decomposition of the tensor A in the form

A_{ij}^{h} = B_{ij}^{h} + δ_{i}^{h}C_{j} + δ_{j}^{h}D_{i} + E_{i}F_{j}^{h} + G_{j}F_{i}^{h}, (3)

where tensors B, C_{j}, D_{j}, E_{j}, G_{j} have following form

C_{j} = \frac{1}{n(3e+1−n^{2})} ((2e − n^{2})A_{αj}^{α} + nA_{jα}^{α} + nA_{γβ}^{α}F_{j}^{γ}F_{α}^{β} − 2A_{βγ}^{α}F_{j}^{γ}F_{α}^{β}) ,
D_{j} = \frac{1}{n(3e+1−n^{2})} (nA_{αj}^{α} + (2e − n^{2})A_{jα}^{α} − 2A_{γβ}^{α}F_{j}^{γ}F_{α}^{β} + nA_{βγ}^{α}F_{j}^{γ}F_{α}^{β}) ,

E_{j} = \frac{1}{n(3e+1−n^{2})} ((2 − n^{2})A_{jβ}^{α}F_{α}^{β} + nA_{βj}^{α}F_{α}^{β} + nA_{αβ}^{α}F_{j}^{β} − 2A_{βα}^{α}F_{j}^{β}) ,

G_{j} = \frac{1}{n(3e+1−n^{2})} (nA_{jβ}^{α}F_{α}^{β} + (2 − n^{2})A_{βj}^{α}F_{α}^{β} − 2A_{αβ}^{α}F_{j}^{β} + nA_{βα}^{α}F_{j}^{β}) ,
B_{ij}^{h} = A_{ij}^{h} − δ_{i}^{h}C_{j} − δ_{j}^{h}D_{i} − E_{i}F_{j}^{h} − G_{j}F_{i}^{h}.

Proof. Suppose that tensor B is F -traceless. Contract the decomposition in the form (3) by the Kronecker’s delta and by the affinor F. We obtain a system of algebraic equations in unknowns

C_{j}, D_{j}, E_{j}, G_{j}, C_{α}F_{j}^{α}, D_{α}F_{j}^{α}, E_{α}F_{j}^{α}, G_{α}F_{j}^{α} :

\begin{aligned}nC_{j} & + & D_{j} & + & E_{α}F_{j}^{α} & = A_{αj}^{α}, \\ C_{j} & + & nD_{j} & + & G_{α}F_{j}^{α} & = A_{jα}^{α}, \\ enE_{j} & + & eG_{j} & + & C_{α}F_{j}^{α} & = A_{jβ}^{α}F_{α}^{β}, \\ eE_{j} & + & enG_{j} & + & D_{α}F_{j}^{α} & = A_{βj}^{α}F_{α}^{β}, \\ eE_{j} & + & nC_{α}F_{j}^{α} & + & D_{α}F_{j}^{α} & = A_{αβ}^{α}F_{j}^{β}, \\ eG_{j} & + & C_{α}F_{j}^{α} & + & nD_{α}F_{j}^{α} & = A_{βα}^{α}F_{j}^{β}, \\ eC_{j} & + & enE_{α}F_{j}^{α} & + & eG_{α}F_{j}^{α} & = A_{γβ}^{α}F_{j}^{γ}F_{α}^{β}, \\ eD_{j} & + & eE_{α}F_{j}^{α} & + & enG_{α}F_{j}^{α} & = A_{βγ}^{α}F_{j}^{γ}F_{α}^{β}. \end{aligned}

For n > 2, this system has unique solution. After some computing we obtain components C_{j}, D_{j}, E_{j}, G_{j} from theorem 2.2. Then we compute B after substitution in

B_{ij}^{h} = A_{ij}^{h} − δ_{i}^{h}C_{j} − δ_{j}^{h}D_{i} − E_{i}F_{j}^{h} − G_{j}F_{i}^{h}.

By contracting tensor B we check that B is really F -traceless.

The decomposition is more simple in the case when tensor A_{ij}^{h} has a special structure. Consider following cases:

1) If equation A_{ij}^{h} = A_{ji}^{h} is true then

\begin{matrix}C_{j} = D_{j} = \frac{1}{n(3e+1−n^{2})} ((2e + n − n^{2})A_{αj}^{α} + (n − 2)A_{γβ}^{α}F_{j}^{γ}F_{α}^{β}) , \\ E_{j} = G_{j} = \frac{1}{n(3e+1−n^{2})} ((2 + n − n^{2})A_{jβ}^{α}F_{α}^{β} + (n − 2)A_{αβ}^{α}F_{j}^{β}) . \end{matrix}

2) If equation A_{ij}^{h} = −A_{ji}^{h} holds then

\begin{matrix}C_{j} = −D_{j} = & \frac{1}{n(3e+1−n^{2})} ((2e − n − n^{2})A_{αj}^{α} + (n + 2)A_{γβ}^{α}F_{j}^{γ}F_{α}^{β}) , \\ E_{j} = −G_{j} = & \frac{1}{n(3e+1−n^{2})} ((2 − n − n^{2})A_{jβ}^{α}F_{α}^{β} + (n + 2)A_{αβ}^{α}F_{j}^{β}) . \end{matrix}

3. F -decomposition of tensors of the type (1, 3)

We want to compute the F -decomposition of tensors of the type (1, 3) in this section. But when we solve this problem we obtain 48 algebraic equations in 48 unknowns. The authors have been unable to solve this system of equations by the software accessible to them. We only were successful in extending the validity of the general theorem 1.2.

Theorem 3.1. Let A be a tensor of the type (1, 3) and let F be traceless estructure. If n > 4 then there exists unique F -decomposition of the tensor A in the form

A_{ijk}^{h} = B_{ijk}^{h} + δ_{i}^{h}C_{jk} + δ_{j}^{h}D_{ik} + δ_{k}^{h}E_{ij} + G_{jk}F_{i}^{h} + H_{ik}F_{j}^{h} + I_{ij}F_{k}^{h}, (4)

where B_{ijk}^{h} is F -traceless tensor, C_{jk}, D_{ik}, E_{ij}, G_{jk}, H_{ik}, I_{ij} are tensors of the type (0, 2).

Proof. Suppose that tensor B is F -traceless. We contract equation (4) by the Kronecker’s delta and by the affinor F. We obtain a system of 48 algebraic equation in 48 unknowns. The determinant of the matrix of this system is expressed by this form

Det(T) = n^{12} (n − 2)^{16} (n + 2)^{16} (n − 4)^{2} (n + 4)^{2}. (5)

This system has unique solution if and only if Det(T) ≠ 0. The dimension of the tensor space with traceless e- structure must be even, it follows from the algebraic theory. Therefore there exists unique decomposition for n > 4.

4. F -decomposition of tensors of the type (1, 3) under special conditions

We compute F -decompositions tensors of the type(1, 3) for special cases in next theorems.

Theorem 4.1. Let A be the tensor of the type (1, 3), n = 6 and let F be traceless e-structure (e = 1). Then there exists unique F -decomposition of the tensor A in the form

A_{ijk}^{h} = B_{ijk}^{h} + δ_{i}^{h}C_{jk} + δ_{j}^{h}D_{ik} + δ_{k}^{h}E_{ij} + G_{jk}F_{i}^{h} + H_{ik}F_{j}^{h} + I_{ij}F_{k}^{h},

where

C_{jk} = \frac{1}{960} [184A_{αjk}^{α} − 36 (A_{jαk}^{α} + A_{kjα}^{α} + A_{γjβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{γβk}^{α}F_{α}^{β}F_{j}^{γ}) +
+9 (A_{jkα}^{α} + A_{kαj}^{α} + A_{γβj}^{α}F_{α}^{β}F_{k}^{γ} + A_{jγβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{kβγ}^{α}F_{α}^{β}F_{j}^{γ}+
+A_{γkβ}^{α}F_{α}^{β}F_{j}^{γ} + A_{βαγ}^{α}F_{j}^{γ}F_{k}^{β} + A_{γβα}^{α}F_{j}^{γ}F_{k}^{β}) − 6 (A_{αkj}^{α} + A_{βγj}^{α}F_{α}^{β}F_{k}^{γ} +
+A_{jβγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βkγ}^{α}F_{α}^{β}F_{j}^{γ} + A_{kγβ}^{α}F_{α}^{β}F_{j}^{γ} + A_{αβγ}^{α}F_{j}^{γ}F_{k}^{β} + A_{βγα}^{α}F_{j}^{γ}F_{k}^{β}+
+A_{γαβ}^{α}F_{j}^{γ}F_{k}^{β}) + 14 (A_{βjγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βγk}^{α}F_{α}^{β}F_{j}^{γ}) + 4A_{αγβ}^{α}F_{k}^{β}F_{j}^{γ}] ,

D_{jk} = \frac{1}{960} [184A_{jαk}^{α} − 36 (A_{αjk}^{α} + A_{jkα}^{α} + A_{jγβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βγk}^{α}F_{α}^{β}F_{j}^{γ}) +
+9 (A_{kjα}^{α} + A_{αkj}^{α} + A_{βγj}^{α}F_{α}^{β}F_{k}^{γ} + A_{γjβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βk γ}^{α}F_{α}^{β}F_{j}^{γ}+
+A_{kγβ}^{α}F_{α}^{β}F_{j}^{γ} + A_{βγα}^{α}F_{j}^{γ}F_{k}^{β} + A_{αβγ}^{α}F_{j}^{γ}F_{k}^{β}) − 6 (A_{kαj}^{α} + A_{βαγ}^{α}F_{j}^{γ}F_{k}^{β}+
+A_{αγβ}^{α}F_{j}^{γ}F_{k}^{β} + A_{γβα}^{α}F_{j}^{γ}F_{k}^{β} + A_{βjγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{γβj}^{α}F_{α}^{β}F_{k}^{γ} + A_{kβγ}^{α}F_{α}^{β}F_{j}^{γ}+
+A_{γkβ}^{α}F_{α}^{β}F_{j}^{γ}) + 14 (A_{jβγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{γβk}^{α}F_{α}^{β}F_{j}^{γ}) + 4A_{γαβ}^{α}F_{j}^{γ}F_{k}^{β}] ,

E_{jk} = \frac{1}{960} [184A_{jkα}^{α} − 36 (A_{jαk}^{α} + A_{αkj}^{α} + A_{jβγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βkγ}^{α}F_{α}^{β}F_{j}^{γ}) +
+9 (A_{αjk}^{α} + A_{kαj}^{α} + A_{γβj}^{α}F_{α}^{β}F_{k}^{γ} + A_{βjγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βγk}^{α}F_{α}^{β}F_{j}^{γ}+
+A_{kβγ}^{α}F_{α}^{β}F_{j}^{γ} + A_{αγβ}^{α}F_{j}^{γ}F_{k}^{β} + A_{βαγ}^{α}F_{j}^{γ}F_{k}^{β}) − 6 (A_{kjα}^{α} + A_{βγj}^{α}F_{α}^{β}F_{k}^{γ}+
+A_{γjβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{kγβ}^{α}F_{α}^{β}F_{j}^{γ} + A_{γβk}^{α}F_{α}^{β}F_{j}^{γ} + A_{αβγ}^{α}F_{j}^{γ}F_{k}^{β} + A_{βγα}^{α}F_{j}^{γ}F_{k}^{β}+
+A_{γαβ}^{α}F_{j}^{γ}F_{k}^{β}) + 14 (A_{jγβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{γkβ}^{α}F_{α}^{β}F_{j}^{γ}) + 4A_{γβα}^{α}F_{j}^{γ}F_{k}^{β}] ,

G_{jk} = \frac{1}{960} [184A_{βjk}^{α}F_{α}^{β} − 36 (A_{jβk}^{α}F_{α}^{β} + A_{kjβ}^{α}F_{α}^{β} + A_{βjα}^{α}F_{k}^{β} + A_{βαk}^{α}F_{j}^{β}) +
+9 (A_{jkβ}^{α}F_{α}^{β} + A_{kβj}^{α}F_{α}^{β} + A_{βαj}^{α}F_{k}^{β} + A_{jβα}^{α}F_{k}^{β} + A_{kαβ}^{α}F_{j}^{β}+
+ A_{βkα}^{α}F_{j}^{β} + A_{γδβ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{δβγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) − 6 (A_{βkj}^{α}F_{α}^{β} + A_{αβj}^{α}F_{k}^{β}+
+A_{jαβ}^{α}F_{k}^{β} + A_{αkβ}^{α}F_{j}^{β} + A_{kβα}^{α}F_{j}^{β} + A_{γβδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{βδγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}+
+A_{δγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) + 14 (A_{αjβ}^{α}F_{k}^{β} + A_{αβk}^{α}F_{j}^{β}) + 4A_{βγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}] ,

H_{jk} = \frac{1}{960} [184A_{jβk}^{α}F_{α}^{β} − 36 (A_{βjk}^{α}F_{α}^{β} + A_{jβα}^{α}F_{k}^{β} + A_{αβk}^{α}F_{j}^{β} + A_{jkβ}^{α}F_{α}^{β}) +
+9 (A_{kjβ}^{α}F_{α}^{β} + A_{βkj}^{α}F_{α}^{β} + A_{αβj}^{α}F_{k}^{β} + A_{βjα}^{α}F_{k}^{β} + A_{αkβ}^{α}F_{j}^{β}+

+ A_{kβα}^{α}F_{j}^{β} + A_{δγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{βδγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) − 6 (A_{kβj}^{α}F_{α}^{β} + A_{βαj}^{α}F_{k}^{β}+

+A_{αjβ}^{α}F_{k}^{β} + A_{βkα}^{α}F_{j}^{β} + A_{kαβ}^{α}F_{j}^{β} + A_{βγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{γδβ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}+
+ A_{δβγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) + 14 (A_{jαβ}^{α}F_{k}^{β} + A_{βαk}^{α}F_{j}^{β}) + 4A_{γβδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}] ,

I_{jk} = \frac{1}{960} [184A_{jkβ}^{α}F_{α}^{β} − 36 (A_{jβk}^{α}F_{α}^{β} + A_{jαβ}^{α}F_{k}^{β} + A_{αkβ}^{α}F_{j}^{β} + A_{βkj}^{α}F_{α}^{β}) +
+9 (A_{βjk}^{α}F_{α}^{β} + A_{kβj}^{α}F_{α}^{β} + A_{αjβ}^{α}F_{k}^{β} + A_{βαj}^{α}F_{k}^{β} + A_{αβk}^{α}F_{j}^{β}+

+ A_{kαβ}^{α}F_{j}^{β} + A_{βγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{δβγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) − 6 (A_{kjβ}^{α}F_{α}^{β} + A_{αβj}^{α}F_{k}^{β}+

+A_{βjα}^{α}F_{k}^{β} + A_{βαk}^{α}F_{j}^{β} + A_{kβα}^{α}F_{j}^{β} + A_{γβδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{βδγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}+
+ A_{δγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) + 14 (A_{jβα}^{α}F_{k}^{β} + A_{kαβ}^{α}F_{j}^{β}) + 4A_{γδβ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}] .

Proof. Suppose that tensor B is F -traceless. We substitute n = 6 a e = 1 in system from theorem 4.1. The solution is given by the above mentioned systems. Now we check F -tracelessness of the tensor B, i.e.

B_{αjk}^{α} = B_{jαk}^{α} = B_{jkα}^{α} = B_{βjk}^{α}F_{α}^{β} = B_{jβk}^{α}F_{α}^{β} = B_{jkβ}^{α}F_{α}^{β} = 0.

This complete the proof. □ Next theorem gives F -decomposition of the tensor A

of the type (1, 3) for n = 8 and e = 1.

Theorem 4.2. Let A be tensor of the type (1, 3), n = 8 and let F be traceless e-structure. Then there exists unique F -decomposition of the tensor A in the form

A_{ijk}^{h} = B_{ijk}^{h} + δ_{i}^{h}C_{jk} + δ_{j}^{h}D_{ik} + δ_{k}^{h}E_{ij} + G_{jk}F_{i}^{h} + H_{ik}F_{j}^{h} + I_{ij}F_{k}^{h},

where

C_{jk} = \frac{1}{1440} [193A_{αjk}^{α} − 26 (A_{jαk}^{α} + A_{kjα}^{α} + A_{γjβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{γβk}^{α}F_{α}^{β}F_{j}^{γ}) +
+4 (A_{jkα}^{α} + A_{kαj}^{α} + A_{γβj}^{α}F_{α}^{β}F_{k}^{γ} + A_{jγβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{kβγ}^{α}F_{α}^{β}F_{j}^{γ}+
+A_{γkβ}^{α}F_{α}^{β}F_{j}^{γ} + A_{βαγ}^{α}F_{j}^{γ}F_{k}^{β} + A_{γβα}^{α}F_{j}^{γ}F_{k}^{β}) − 6 (A_{αkj}^{α} + A_{βγj}^{α}F_{α}^{β}F_{k}^{γ} +
+A_{jβγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βkγ}^{α}F_{α}^{β}F_{j}^{γ} + A_{kγβ}^{α}F_{α}^{β}F_{j}^{γ} + A_{αβγ}^{α}F_{j}^{γ}F_{k}^{β} + A_{βγα}^{α}F_{j}^{γ}F_{k}^{β}+
+A_{γαβ}^{α}F_{j}^{γ}F_{k}^{β}) + 14 (A_{βjγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βγk}^{α}F_{α}^{β}F_{j}^{γ}) + A_{αγβ}^{α}F_{k}^{β}F_{j}^{γ}] ,

D_{jk} = \frac{1}{1440} [193A_{jαk}^{α} − 26 (A_{αjk}^{α} + A_{jkα}^{α} + A_{jγβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βγk}^{α}F_{α}^{β}F_{j}^{γ}) +

+4 (A_{kjα}^{α} + A_{αkj}^{α} + A_{βγj}^{α}F_{α}^{β}F_{k}^{γ} + A_{γjβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βk γ}^{α}F_{α}^{β}F_{j}^{γ}+

+A_{kγβ}^{α}F_{α}^{β}F_{j}^{γ} + A_{βγα}^{α}F_{j}^{γ}F_{k}^{β} + A_{αβγ}^{α}F_{j}^{γ}F_{k}^{β}) − 6 (A_{kαj}^{α} + A_{βαγ}^{α}F_{j}^{γ}F_{k}^{β}+

+A_{αγβ}^{α}F_{j}^{γ}F_{k}^{β} + A_{γβα}^{α}F_{j}^{γ}F_{k}^{β} + A_{βjγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{γβj}^{α}F_{α}^{β}F_{k}^{γ} + A_{kβγ}^{α}F_{α}^{β}F_{j}^{γ}+

+A_{γkβ}^{α}F_{α}^{β}F_{j}^{γ}) + 14 (A_{jβγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{γβk}^{α}F_{α}^{β}F_{j}^{γ}) + A_{γαβ}^{α}F_{j}^{γ}F_{k}^{β}] ,

E_{jk} = \frac{1}{1440} [193A_{jkα}^{α} − 26 (A_{jαk}^{α} + A_{αkj}^{α} + A_{jβγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βkγ}^{α}F_{α}^{β}F_{j}^{γ}) +

+4 (A_{αjk}^{α} + A_{kαj}^{α} + A_{γβj}^{α}F_{α}^{β}F_{k}^{γ} + A_{βjγ}^{α}F_{α}^{β}F_{k}^{γ} + A_{βγk}^{α}F_{α}^{β}F_{j}^{γ}+

+A_{kβγ}^{α}F_{α}^{β}F_{j}^{γ} + A_{αγβ}^{α}F_{j}^{γ}F_{k}^{β} + A_{βαγ}^{α}F_{j}^{γ}F_{k}^{β}) − 6 (A_{kjα}^{α} + A_{βγj}^{α}F_{α}^{β}F_{k}^{γ}+

+A_{γjβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{kγβ}^{α}F_{α}^{β}F_{j}^{γ} + A_{γβk}^{α}F_{α}^{β}F_{j}^{γ} + A_{αβγ}^{α}F_{j}^{γ}F_{k}^{β} + A_{βγα}^{α}F_{j}^{γ}F_{k}^{β}+

+A_{γαβ}^{α}F_{j}^{γ}F_{k}^{β}) + 14 (A_{jγβ}^{α}F_{α}^{β}F_{k}^{γ} + A_{γkβ}^{α}F_{α}^{β}F_{j}^{γ}) + A_{γβα}^{α}F_{j}^{γ}F_{k}^{β}] ,

G_{jk} = \frac{1}{1440} [193A_{βjk}^{α}F_{α}^{β} − 26 (A_{jβk}^{α}F_{α}^{β} + A_{kjβ}^{α}F_{α}^{β} + A_{βjα}^{α}F_{k}^{β} + A_{βαk}^{α}F_{j}^{β}) +

+4 (A_{jkβ}^{α}F_{α}^{β} + A_{kβj}^{α}F_{α}^{β} + A_{βαj}^{α}F_{k}^{β} + A_{jβα}^{α}F_{k}^{β} + A_{kαβ}^{α}F_{j}^{β}+

+ A_{βkα}^{α}F_{j}^{β} + A_{γδβ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{δβγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) − 6 (A_{βkj}^{α}F_{α}^{β} + A_{αβj}^{α}F_{k}^{β}+

+A_{jαβ}^{α}F_{k}^{β} + A_{αkβ}^{α}F_{j}^{β} + A_{kβα}^{α}F_{j}^{β} + A_{γβδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{βδγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}+

+A_{δγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) + 14 (A_{αjβ}^{α}F_{k}^{β} + A_{αβk}^{α}F_{j}^{β}) + A_{βγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}] ,

H_{jk} = \frac{1}{1440} [193A_{jβk}^{α}F_{α}^{β} − 26 (A_{βjk}^{α}F_{α}^{β} + A_{jβα}^{α}F_{k}^{β} + A_{αβk}^{α}F_{j}^{β} + A_{jkβ}^{α}F_{α}^{β}) +

+4 (A_{kjβ}^{α}F_{α}^{β} + A_{βkj}^{α}F_{α}^{β} + A_{αβj}^{α}F_{k}^{β} + A_{βjα}^{α}F_{k}^{β} + A_{αkβ}^{α}F_{j}^{β}+

+ A_{kβα}^{α}F_{j}^{β} + A_{δγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{βδγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) − 6 (A_{kβj}^{α}F_{α}^{β} + A_{βαj}^{α}F_{k}^{β}+

+A_{αjβ}^{α}F_{k}^{β} + A_{βkα}^{α}F_{j}^{β} + A_{kαβ}^{α}F_{j}^{β} + A_{βγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{γδβ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}+

+ A_{δβγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) + 14 (A_{jαβ}^{α}F_{k}^{β} + A_{βαk}^{α}F_{j}^{β}) + A_{γβδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}] ,

I_{jk} = \frac{1}{1440} [193A_{jkβ}^{α}F_{α}^{β} − 26 (A_{jβk}^{α}F_{α}^{β} + A_{jαβ}^{α}F_{k}^{β} + A_{αkβ}^{α}F_{j}^{β} + A_{βkj}^{α}F_{α}^{β}) +

+4 (A_{βjk}^{α}F_{α}^{β} + A_{kβj}^{α}F_{α}^{β} + A_{αjβ}^{α}F_{k}^{β} + A_{βαj}^{α}F_{k}^{β} + A_{αβk}^{α}F_{j}^{β}+

+ A_{kαβ}^{α}F_{j}^{β} + A_{βγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{δβγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) − 6 (A_{kjβ}^{α}F_{α}^{β} + A_{αβj}^{α}F_{k}^{β}+

+A_{βjα}^{α}F_{k}^{β} + A_{βαk}^{α}F_{j}^{β} + A_{kβα}^{α}F_{j}^{β} + A_{γβδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ} + A_{βδγ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}+

+ A_{δγδ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}) + 14 (A_{jβα}^{α}F_{k}^{β} + A_{kαβ}^{α}F_{j}^{β}) + A_{γδβ}^{α}F_{α}^{β}F_{j}^{γ}F_{k}^{δ}] .

Proof. Suppose that tensor B is F -traceless. Substitute n = 8 a e = 1 in system from theorem 4.1. The solution is given by above-mentioned systems. Then we check F -tracelessness of the tensor B and this complete the proof. □

References

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[2] Krupka, D., Topics in the calculus of variations: Variational sequences, Diff. Geom. and Its Appl., Proc. Conf., Silesian Univ. Opava, Czech Republic (1993), 473–495.

[3] Krupka, D., The trace decomposition problem, Beiträge zur Algebra und Geometrie (1995), Vol. 36, N. 2, 303–315.

[4] Mikeš, J., On the general trace decomposition problem, Proc. Conf. Diff. Geom. and Appl., 1995, Brno, Masaryk Univ., Brno, 1966, 45–50.

[5] Mikeš, J., Lakomá, L., On the special trace decomposition problem on quaternionic structure, Proc. of the Third International Workshop on Differential Geometry and its Aplication; The first German-Romanian seminar on geometry; Sibiu - Romania, September 18 - 23, 1997.

[6] Mikeš, J., Moldobaev, Dž., O někotorych algebraičeskich svojstvach těnzorov, Issled. po topol. i obobšč. prostranstvam, Sb. naučn. tr. Frunze (1982), 60–64.

[7] Weyl, H., The Classical Groups, Princeton Univ. Press, Princeton (1946).

[8] Yano, K., Differential geometry on complex and almost complex spaces, Oxford, Pergamon Press (1965), 326p.

Dept. Algebra and Geometry, Palacky University Olomouc Tomkova 40, 779 00 Olomouc, CZECH REPUBLIC E-mail: Mikes@risc.upol.cz, lenka.lakoma@upol.cz