Editing Unitary group (section)

== Generalizations ==
From the point of view of [[Lie theory]], the classical unitary group is a real form of the [[Steinberg group (Lie theory)|Steinberg group]] <sup>2</sup>A<sub>n</sub>, which is an [[algebraic group]] that arises from the combination of the ''diagram automorphism'' of the general linear group (reversing the [[Dynkin diagram]] A<sub>''n''</sub>, which corresponds to transpose inverse) and the ''[[field automorphism]]'' of the extension '''C'''/'''R''' (namely [[complex conjugation]]). Both these automorphisms are automorphisms of the algebraic group, have order 2, and commute, and the unitary group is the fixed points of the product automorphism, as an algebraic group. The classical unitary group is a real form of this group, corresponding to the standard [[Hermitian form]] Ψ, which is positive definite.

This can be generalized in a number of ways:
* generalizing to other Hermitian forms yields indefinite unitary groups {{nowrap|U(''p'', ''q'')}};
* the field extension can be replaced by any degree 2 separable algebra, most notably a degree 2 extension of a finite field;
* generalizing to other diagrams yields other [[groups of Lie type]], namely the other [[Steinberg group (Lie theory)|Steinberg groups]] <sup>2</sup>D<sub>''n''</sub>, <sup>2</sup>E<sub>6</sub>, <sup>3</sup>D<sub>4</sub>, (in addition to <sup>2</sup>A<sub>''n''</sub>) and [[Suzuki–Ree group]]s
*: <math>{}^2\!B_2\left(2^{2n+1}\right), {}^2\!F_4\left(2^{2n+1}\right), {}^2\!G_2\left(3^{2n+1}\right);</math>
* considering a generalized unitary group as an algebraic group, one can take its points over various algebras.

=== Indefinite forms ===
Analogous to the [[indefinite orthogonal group]]s, one can define an '''indefinite unitary group''', by considering the transforms that preserve a given Hermitian form, not necessarily positive definite (but generally taken to be non-degenerate). Here one is working with a [[vector space]] over the complex numbers.

Given a Hermitian form Ψ on a complex vector space ''V'', the unitary group U(Ψ) is the group of transforms that preserve the form: the transform ''M'' such that {{nowrap|1=Ψ(''Mv'', ''Mw'') = Ψ(''v'', ''w'')}} for all {{nowrap|''v'', ''w'' ∈ ''V''}}. In terms of matrices, representing the form by a matrix denoted Φ, this says that {{nowrap|1=''M''<sup>∗</sup>Φ''M'' = Φ}}.

Just as for [[Symmetric bilinear form|symmetric forms]] over the reals, Hermitian forms are determined by [[Signature of a quadratic form|signature]], and are all [[Matrix congruence|unitarily congruent]] to a diagonal form with ''p'' entries of 1 on the diagonal and ''q'' entries of −1. The non-degenerate assumption is equivalent to {{nowrap|1=''p'' + ''q'' = ''n''}}. In a standard basis, this is represented as a quadratic form as:
: <math>\lVert z \rVert_\Psi^2 = \lVert z_1 \rVert^2 + \dots + \lVert z_p \rVert^2 - \lVert z_{p+1} \rVert^2 - \dots - \lVert z_n \rVert^2</math>

and as a symmetric form as:
: <math>\Psi(w, z) = \bar w_1 z_1 + \cdots + \bar w_p z_p - \bar w_{p+1}z_{p+1} - \cdots - \bar w_n z_n.</math>

The resulting group is denoted {{nowrap|U(''p'',''q'')}}.

=== Finite fields ===
Over the [[finite field]] with {{nowrap|1=''q'' = ''p''<sup>''r''</sup>}} elements, '''F'''<sub>''q''</sub>, there is a unique quadratic extension field, '''F'''<sub>''q''<sup>2</sup></sub>, with order 2 automorphism <math>\alpha\colon x \mapsto x^q</math> (the ''r''th power of the [[Frobenius automorphism]]). This allows one to define a Hermitian form on an '''F'''<sub>''q''<sup>2</sup></sub> vector space ''V'', as an '''F'''<sub>''q''</sub>-bilinear map <math>\Psi\colon V \times V \to K</math> such that <math>\Psi(w, v) = \alpha \left(\Psi(v, w)\right)</math> and <math>\Psi(w, cv) = c\Psi(w, v)</math> for {{nowrap|''c'' ∈ '''F'''<sub>''q''<sup>2</sup></sub>}}.{{clarify|What is K?|date=June 2014}} Further, all non-degenerate Hermitian forms on a vector space over a finite field<!-- clear from context, but I think it fails infinite fields of pos char--> are unitarily congruent to the standard one, represented by the identity matrix; that is, any Hermitian form is unitarily equivalent to
: <math>\Psi(w, v) = w^\alpha \cdot v = \sum_{i=1}^n w_i^q v_i</math>
where <math>w_i,v_i</math> represent the coordinates of {{nowrap|''w'', ''v'' ∈ ''V''}} in some particular '''F'''<sub>''q''<sup>2</sup></sub>-basis of the ''n''-dimensional space ''V'' {{harv|Grove|2002|loc=Thm. 10.3}}.

Thus one can define a (unique) unitary group of dimension ''n'' for the extension '''F'''<sub>''q''<sup>2</sup></sub>/'''F'''<sub>''q''</sub>, denoted either as {{nowrap|U(''n'', ''q'')}} or {{nowrap|U(''n'', ''q''<sup>2</sup>)}} depending on the author. The subgroup of the unitary group consisting of matrices of determinant 1 is called the '''special unitary group''' and denoted {{nowrap|SU(''n'', ''q'')}} or {{nowrap|SU(''n'', ''q''<sup>2</sup>)}}. For convenience, this article will use the {{nowrap|U(''n'', ''q''<sup>2</sup>)}} convention.  The center of {{nowrap|U(''n'', ''q''<sup>2</sup>)}} has order {{nowrap|''q'' + 1}} and consists of the scalar matrices that are unitary, that is those matrices ''cI<sub>V</sub>'' with <math>c^{q+1} = 1</math>.  The center of the special unitary group has order {{nowrap|gcd(''n'', ''q'' + 1)}} and consists of those unitary scalars which also have order dividing ''n''.  The quotient of the unitary group by its center is called the '''[[projective unitary group]]''', {{nowrap|PU(''n'', ''q''<sup>2</sup>)}}, and the quotient of the special unitary group by its center is the '''[[projective special unitary group]]''' {{nowrap|PSU(''n'', ''q''<sup>2</sup>)}}. In most cases ({{nowrap|''n'' > 1}} and {{nowrap|(''n'', ''q''<sup>2</sup>) ∉ {(2, 2<sup>2</sup>), (2, 3<sup>2</sup>), (3, 2<sup>2</sup>)}{{void}}}}), {{nowrap|SU(''n'', ''q''<sup>2</sup>)}} is a [[perfect group]] and {{nowrap|PSU(''n'', ''q''<sup>2</sup>)}} is a finite [[simple group]], {{harv|Grove|2002|loc=Thm. 11.22 and 11.26}}.

=== Degree-2 separable algebras ===
More generally, given a field ''k'' and a degree-2 separable ''k''-algebra ''K'' (which may be a field extension but need not be), one can define unitary groups with respect to this extension.

First, there is a unique ''k''-automorphism of ''K'' <math>a \mapsto \bar a</math> which is an involution and fixes exactly ''k'' (<math>a = \bar{a}</math> if and only if {{nowrap|''a'' ∈ ''k''}}).<ref>Milne, [http://www.jmilne.org/math/CourseNotes/aag.html Algebraic Groups and Arithmetic Groups], p. 103</ref> This generalizes complex conjugation and the conjugation of degree 2 finite field extensions, and allows one to define Hermitian forms and unitary groups as above.

=== 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).