Editing Unitary group (section)

=== Algebraic groups ===
The equations defining a unitary group are polynomial equations over ''k'' (but not over ''K''): for the standard form {{nowrap|1=Φ = ''I''}}, the equations are given in matrices as {{nowrap|1=''A''<sup>∗</sup>''A'' = ''I''}}, where <math>A^* = \bar{A}^\mathsf{T}</math> is the [[conjugate transpose]]. Given a different form, they are {{nowrap|1=''A''<sup>∗</sup>Φ''A'' = Φ}}. The unitary group is thus an [[algebraic group]], whose points over a ''k''-algebra ''R'' are given by:
: <math>\operatorname{U}(n, K/k, \Phi)(R) := \left\{ A \in \operatorname{GL}(n, K \otimes_k R) : A^*\Phi A = \Phi\right\}.</math>

For the field extension '''C'''/'''R''' and the standard (positive definite) Hermitian form, these yield an algebraic group with real and complex points given by:
: <math>\begin{align}
  \operatorname{U}(n, \mathbf{C}/\mathbf{R})(\mathbf{R}) &= \operatorname{U}(n) \\
  \operatorname{U}(n, \mathbf{C}/\mathbf{R})(\mathbf{C}) &= \operatorname{GL}(n, \mathbf{C}).
\end{align}</math>

In fact, the unitary group is a [[linear algebraic group]].

==== Unitary group of a quadratic module ====
The unitary group of a quadratic module is a generalisation of the linear algebraic group U just defined, which incorporates as special cases many different [[Classical group|classical algebraic groups]]. The definition goes back to Anthony Bak's thesis.<ref>Bak, Anthony (1969), "On modules with quadratic forms", ''Algebraic K-Theory and its Geometric Applications'' (editors&mdash;Moss R. M. F., Thomas C. B.) Lecture Notes in Mathematics, Vol. 108,  pp. 55-66, Springer. {{doi|10.1007/BFb0059990}}</ref>

To define it, one has to define quadratic modules first:

Let ''R'' be a ring with anti-automorphism ''J'', <math>\varepsilon \in R^\times</math> such that <math>r^{J^2} = \varepsilon r \varepsilon^{-1}</math> for all ''r'' in ''R'' and <math>\varepsilon^J = \varepsilon^{-1}</math>. Define
: <math>\begin{align}
  \Lambda_\text{min} &:= \left\{r \in R \ : \ r - r^J\varepsilon\right\}, \\
  \Lambda_\text{max} &:= \left\{r \in R \ : \ r^J\varepsilon = -r\right\}.
\end{align}</math>

Let {{nowrap|Λ ⊆ ''R''}} be an additive subgroup of ''R'', then Λ is called ''form parameter'' if <math>\Lambda_\text{min} \subseteq \Lambda \subseteq \Lambda_\text{max}</math> and <math>r^J \Lambda r \subseteq \Lambda</math>. A pair {{nowrap|(''R'', Λ)}} such that ''R'' is a ring and Λ a form parameter is called ''form ring''.

Let ''M'' be an ''R''-module and ''f'' a ''J''-sesquilinear form on ''M'' (i.e., <math>f(xr, ys) = r^J f(x, y)s</math> for any <math>x, y \in M</math> and <math>r, s \in R</math>). Define <math>h(x, y) := f(x, y) + f(y, x)^J \varepsilon \in R</math> and <math>q(x) := f(x, x) \in R/\Lambda</math>, then ''f'' is said to ''define'' the ''Λ-quadratic form'' {{nowrap|(''h'', ''q'')}} on ''M''. A ''quadratic module'' over {{nowrap|(''R'', Λ)}} is a triple {{nowrap|(''M'', ''h'', ''q'')}} such that ''M'' is an ''R''-module and {{nowrap|(''h'', ''q'')}} is a Λ-quadratic form.

To any quadratic module {{nowrap|(''M'', ''h'', ''q'')}} defined by a ''J''-sesquilinear form ''f'' on ''M'' over a form ring {{nowrap|(''R'', Λ)}} one can associate the ''unitary group''
: <math>U(M) := \{\sigma \in GL(M) \ : \ \forall x, y \in M, h(\sigma x, \sigma y) = h(x, y) \text{ and } q(\sigma x) = q(x) \}.</math>

The special case where {{nowrap|1=Λ = Λ<sub>max</sub>}}, with ''J'' any non-trivial involution (i.e., <math>J \neq id_R, J^2 = id_R</math> and {{nowrap|1=''ε'' = −1}} gives back the "classical" unitary group (as an algebraic group).