Editing Unitary group (section)

== Polynomial invariants ==
The unitary groups are the automorphisms of two polynomials in real non-commutative variables:
: <math>\begin{align}
  C_1 &= \left(u^2 + v^2\right) + \left(w^2 + x^2\right) + \left(y^2 + z^2\right) + \ldots \\
  C_2 &= \left(uv - vu\right) + \left(wx - xw\right) + \left(yz - zy\right) + \ldots
\end{align}</math>

These are easily seen to be the real and imaginary parts of the complex form <math>Z \overline{Z}</math>. The two invariants separately are invariants of O(2''n'') and Sp(2''n''). Combined they make the invariants of U(''n'') which is a subgroup of both these groups. The variables must be non-commutative in these invariants otherwise the second polynomial is identically zero.