Editing Exterior algebra (section)

== Alternating tensor algebra ==
For a field of characteristic not 2,<ref>See {{harvtxt|Bourbaki|1989|loc=§III.7.5}} for generalizations.</ref> the exterior algebra of a vector space <math>V</math> over <math>K</math> can be canonically identified with the vector subspace of <math>\mathrm{T}(V)</math> that consists of [[antisymmetric tensor]]s. For characteristic 0 (or higher than {{tmath|\dim V}}), the vector space of <math>k</math>-linear antisymmetric tensors is transversal to the ideal  {{tmath|I}}, hence, a good choice to represent the quotient. But for nonzero characteristic, the vector space of {{tmath|K}}-linear antisymmetric tensors could be not transversal to the ideal (actually, for {{tmath|k \geq \operatorname{char} K}}, the vector space of <math>K</math>-linear antisymmetric tensors is contained in <math>I</math>); nevertheless, transversal or not, a product can be defined on this space such that the resulting algebra is isomorphic to the exterior algebra: in the first case the natural choice for the product is just the quotient product (using the available projection), in the second case, this product must be slightly modified as given below (along Arnold setting), but such that the algebra stays isomorphic with the exterior algebra, i.e. the quotient of <math>\mathrm{T}(V)</math> by the ideal <math>I</math> generated by elements of the form {{tmath|x \otimes x}}. Of course, for characteristic {{tmath|0}} (or higher than the dimension of the vector space), one or the other definition of the product could be used, as the two algebras are isomorphic (see V. I. Arnold or Kobayashi-Nomizu).

Let <math>\mathrm{T}^r(V)</math> be the space of homogeneous tensors of degree <math>r</math>.  This is spanned by decomposable tensors
: <math> v_1 \otimes \cdots \otimes v_r,\quad v_i \in V. </math>

The '''antisymmetrization''' (or sometimes the '''skew-symmetrization''') of a decomposable tensor is defined by
: <math>
\operatorname{\mathcal{A}^{(r)}}(v_1 \otimes \cdots \otimes v_r)
=\sum_{\sigma \in \mathfrak{S}_r} \operatorname{sgn}(\sigma) v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(r)}
</math>
and, when <math>r! \neq 0</math> (for nonzero characteristic field <math>r!</math> might be 0):
: <math>
\operatorname{Alt}^{(r)}(v_1 \otimes \cdots \otimes v_r)
= \frac{1}{r!}\operatorname{\mathcal{A}^{(r)}}(v_1 \otimes \cdots \otimes v_r)
</math>
where the sum is taken over the [[symmetric group]] of permutations on the symbols {{tmath|1=\{ 1, \dots, r \} }}. This extends by linearity and homogeneity to an operation, also denoted by <math>\mathcal{A}</math> and <math>\rm{Alt}</math>, on the full tensor algebra {{tmath|\mathrm{T}(V)}}.

Note that
: <math>
\operatorname{\mathcal{A}^{(r)}}\operatorname{\mathcal{A}^{(r)}}=r!\operatorname{\mathcal{A}^{(r)}}.
</math>

Such that, when defined, <math>\operatorname{Alt}^{(r)} </math> is the projection for the exterior (quotient) algebra onto the r-homogeneous alternating tensor subspace.
On the other hand, the image <math>\mathcal{A}(\mathrm{T}(V))</math> is always the '''alternating tensor graded subspace''' (not yet an algebra, as product is not yet defined), denoted {{tmath|A(V)}}. This is a vector subspace of {{tmath|\mathrm{T}(V)}}, and it inherits the structure of a graded vector space from that on {{tmath|\mathrm{T}(V)}}. Moreover, the kernel of <math>\mathcal{A}^{(r)}</math> is precisely {{tmath|I^{(r)} }}, the homogeneous subset of the ideal {{tmath|I}}, or the kernel of  <math>\mathcal{A}</math> is {{tmath|I}}.  When <math>\operatorname{Alt} </math> is defined, <math>A(V)</math> carries an associative graded product <math> \widehat{\otimes} </math> defined by (the same as the wedge product)
: <math>t\wedge s=t~\widehat{\otimes}~s = \operatorname{Alt}(t \otimes s). </math>

Assuming <math>K</math> has characteristic 0, as <math>A(V)</math> is a supplement of <math>I</math> in {{tmath|\mathrm{T}(V)}}, with the above given product, there is a canonical isomorphism
: <math> A(V)\cong {\textstyle\bigwedge}(V). </math>
When the characteristic of the field is nonzero, <math>\mathcal{A}</math> will do what <math>\rm{Alt}</math> did before, but the  product cannot be defined as above. In such a case, isomorphism <math> A(V)\cong {\textstyle\bigwedge}(V) </math> still holds, in spite of <math> A(V)</math> not being a supplement of  the ideal {{tmath|I}}, but then, the product should be modified as given below (<math> \dot{\wedge}</math> product, Arnold setting).

Finally, we always get {{tmath|A(V)}} isomorphic with {{tmath|{\textstyle\bigwedge}(V)}}, but the product could (or should) be chosen in two ways (or only one). Actually, the product could be chosen in many ways, rescaling it on homogeneous spaces as <math>c(r+p)/c(r)c(p)</math> for an arbitrary sequence <math>c(r)</math> in the field, as long as the division makes sense (this is such that the redefined product is also associative, i.e. defines an algebra on {{tmath|A(V)}}). Also note, the interior product definition should be changed accordingly, in order to keep its skew derivation property.

=== Index notation ===
Suppose that ''V'' has finite dimension ''n'', and that a basis {{nowrap|'''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>}} of ''V'' is given.  Then any alternating tensor {{nowrap|''t'' ∈ A<sup>''r''</sup>(''V'') ⊂ ''T''<sup>''r''</sup>(''V'')}} can be written in [[index notation]] with the [[Einstein summation convention]] as
: <math>t = t^{i_1i_2\cdots i_r}\, {\mathbf e}_{i_1} \otimes {\mathbf e}_{i_2} \otimes \cdots \otimes {\mathbf e}_{i_r},</math>
where ''t''<sup>''i''<sub>1</sub>⋅⋅⋅''i''<sub>''r''</sub></sup> is [[antisymmetric tensor|completely antisymmetric]] in its indices.

The exterior product of two alternating tensors ''t'' and ''s'' of ranks ''r'' and ''p'' is given by
: <math>
t~\widehat{\otimes}~s
= \frac{1}{(r+p)!}\sum_{\sigma \in {\mathfrak S}_{r+p}}\operatorname{sgn}(\sigma)t^{i_{\sigma(1)} \cdots i_{\sigma(r)}} s^{i_{\sigma(r+1)} \cdots i_{\sigma(r+p)}} {\mathbf e}_{i_1} \otimes {\mathbf e}_{i_2} \otimes \cdots \otimes {\mathbf e}_{i_{r+p}}.
</math>

The components of this tensor are precisely the skew part of the components of the tensor product {{nowrap|''s'' ⊗ ''t''}}, denoted by square brackets on the indices:
: <math>
(t~\widehat{\otimes}~s)^{i_1\cdots i_{r+p}}
= t^{[i_1\cdots i_r}s^{i_{r+1}\cdots i_{r+p}]}.
</math>

<!--For the interior product-->
The [[#Interior product|interior product]] may also be described in index notation as follows.  Let <math> t = t^{i_0i_1\cdots i_{r-1}} </math> be an antisymmetric tensor of rank {{tmath|r}}.  Then, for {{nowrap|''α'' ∈ ''V''<sup>∗</sup>}}, {{tmath|\iota_\alpha t}} is an alternating tensor of rank {{tmath|r - 1}}, given by
: <math>
(\iota_\alpha t)^{i_1\cdots i_{r-1}}
= r\sum_{j=0}^n\alpha_j t^{ji_1\cdots i_{r-1}}.
</math>
where ''n'' is the dimension of ''V''.