basic tensor

The present entry employs the terminology and notation defined anddescribed in the entry on tensor arrays. To keep things reasonablyself-contained we mention that the symbol ${\mathrm{T}}^{p,q}$ refers to thevector space of type $(p,q)$ tensor arrays, i.e. maps

$$I^p\times I^q\to 𝕂,$$

where $I$ is some finite list ofindex labels, and where $𝕂$ is a field.

We say that a tensor array is a characteristic array, a.k.a. abasic tensor, if all but one of its values are $0$, and theremaining non-zero value is equal to $1$. For tuples $A\in I^p$ and$B\in I^q$, we let

$${\epsilon }_A^B:I^p\times I^q\to 𝕂,$$

denote the characteristicarray defined by

The type $(p,q)$ characteristic arrays form a natural basis for${\mathrm{T}}^{p,q}$.

Furthermore the outer multiplication of two characteristic arraysgives a characteristic array of larger valence. In other words, for

$$A_1\in I^{p_1},B_1\in I^{q_1},A_2\in I^{p_2},B_2\in I^{q_2},$$

we have that

$${\epsilon }_{A_1}^{B_1}{\epsilon }_{A_2}^{B_2}={\epsilon }_{A_1{A}_2}^{B_1{B}_2},$$

wherethe product on the left-hand side is performed by outermultiplication, and where $A_1{A}_2$ on the right-hand side refers tothe element of $I^{p_1+p_2}$ obtained by concatenating the tuples$A_1$ and $A_2$, and similarly for $B_1{B}_2$.

In this way we see that the type $(1,0)$ characteristic arrays${\epsilon }_{(i)},i\in I$ (the natural basis of ${𝕂}^I$), and the type$(0,1)$ characteristic arrays ${\epsilon }^{(i)},i\in I$ (the natural basis of${\left({𝕂}^I\right)}^{*}$) generate the tensor array algebra relative to theouter multiplication operation.

The just-mentioned fact gives us an alternate way of writing andthinking about tensor arrays. We introduce the basic symbols

$${\epsilon }_{(i)},{\epsilon }^{(i)},i\in I$$

subject to the commutation relations

$${\epsilon }_{(i)}{\epsilon }^{(i^{\prime })}={\epsilon }^{(i^{\prime })}{\epsilon }_{(i)},i,i^{\prime }\in I,$$

add and multiply these symbols using coefficients in$𝕂$, and use

$${\epsilon }_{(j_1\mathrm{\dots }{j}_p)}^{(i_1\mathrm{\dots }{i}_q)},i_1,\mathrm{\dots },i_q,j_1,\mathrm{\dots },j_p\in I$$

as a handy abbreviation for

$${\epsilon }^{(i_1)}\mathrm{\dots }{\epsilon }^{(i_q)}{\epsilon }_{(j_1)}\mathrm{\dots }{\epsilon }_{(j_p)}.$$

Wethen interpret the resulting expressions as tensor arrays in theobvious fashion: the values of the tensor array are just thecoefficients of the $\epsilon$ symbol matching the given index. However,note that in the $\epsilon$ symbols, the covariant data is written as asuperscript, and the contravariant data as a subscript. This is doneto facilitate the Einstein summation convention.

By way of illustration, suppose that $I=(1,2)$. We can now write downa type $(1,0)$ tensor, i.e. a column vector

$$u=\left(\begin{array}{c}\hfill u^1\hfill \\ \hfill u^2\hfill \end{array}\right)\in {\mathrm{T}}^{1,0}$$

as

$$u=u^1{\epsilon }_{(1)}+u^2{\epsilon }_{(2)}.$$

Similarly, a row-vector

$$\varphi =({\varphi }_1,{\varphi }_2)\in {\mathrm{T}}^{0,1}$$

can be written down as

$$\varphi ={\varphi }_1{\epsilon }^{(1)}+{\varphi }_2{\epsilon }^{(2)}.$$

In the case of a matrix

we would write

$$M=M_1^1{\epsilon }_{(1)}^{(1)}+M_2^1{\epsilon }_{(1)}^{(2)}+M_1^2{\epsilon }_{(2)}^{(1)}+M_2^2{\epsilon }_{(2)}^{(2)}.$$