Editing Tensor algebra (section)

==Hopf algebra==
The [[Hopf algebra]] adds an antipode to the bialgebra axioms.  The antipode <math>S</math> on <math>k\in K=T^0V</math> is given by
:<math>S(k)=k</math>
This is sometimes called the "anti-identity". The antipode on <math>v\in V=T^1V</math> is given by
:<math>S(v)=-v</math>
and on <math>v \otimes w\in T^2V</math> by
:<math>S(v \otimes w) = S(w) \otimes S(v) = w\otimes v</math>
This extends homomorphically to
:<math>
\begin{align}
S(v_1 \otimes \cdots \otimes v_m) 
&= S(v_m) \otimes\cdots\otimes S(v_1) \\
&= (-1)^m v_m \otimes\cdots\otimes v_1
\end{align}</math>

=== Compatibility ===
Compatibility of the antipode with multiplication and comultiplication requires that
:<math>\nabla \circ (S \boxtimes \mathrm{id}) \circ \Delta
= \eta \circ \epsilon
= \nabla \circ (\mathrm{id} \boxtimes S) \circ \Delta</math>
This is straightforward to verify componentwise on <math>k\in K</math>:
:<math>
\begin{align}
(\nabla \circ (S \boxtimes \mathrm{id}) \circ \Delta)(k)
&= (\nabla \circ (S \boxtimes \mathrm{id})) (1\boxtimes k) \\
&= \nabla(1 \boxtimes k) \\
&= 1 \otimes k \\
&= k
\end{align}</math>
Similarly, on <math>v\in V</math>:
:<math>
\begin{align}
(\nabla \circ (S \boxtimes \mathrm{id}) \circ \Delta)(v)
&= (\nabla \circ (S \boxtimes \mathrm{id})) (v\boxtimes 1 + 1 \boxtimes v) \\
&= \nabla(-v \boxtimes 1 + 1 \boxtimes v) \\
&= -v \otimes 1 + 1 \otimes v \\
&= -v + v\\
&= 0
\end{align}</math>
Recall that
:<math>(\eta \circ \epsilon)(k)=\eta(k)=k</math>
and that 
:<math>(\eta \circ \epsilon)(x)=\eta(0)=0</math>
for any <math>x\in TV</math> that is ''not'' in <math>K.</math>

One may proceed in a similar manner, by homomorphism, verifying that the antipode inserts the appropriate cancellative signs in the shuffle, starting with the compatibility condition on <math>T^2V</math> and proceeding by induction.