Editing Tensor algebra (section)

=== Unit ===
The unit for the algebra
:<math>\eta: K\to TV</math>
is just the embedding, so that
:<math>\eta: k\mapsto k</math>
That the unit is compatible with the tensor product <math>\otimes</math> is "trivial": it is just part of the standard definition of the tensor product of vector spaces.  That is, <math>k\otimes x = kx</math> for field element ''k'' and any <math>x\in TV.</math>  More verbosely, the axioms for an [[associative algebra]] require the two homomorphisms (or commuting diagrams):
:<math>\nabla\circ(\eta \boxtimes\mathrm{id}_{TV}) = \eta\otimes \mathrm{id}_{TV} = \eta\cdot \mathrm{id}_{TV}</math>
on <math>K\boxtimes TV</math>, and that symmetrically, on <math>TV\boxtimes K</math>, that
:<math>\nabla\circ(\mathrm{id}_{TV}\boxtimes\eta) = \mathrm{id}_{TV}\otimes\eta = \mathrm{id}_{TV}\cdot\eta</math>
where the right-hand side of these equations should be understood as the scalar product.