Unitary group

The unitary group is the 3-fold intersection of the orthogonal, complex, and symplectic groups:

${\displaystyle \operatorname {U} (n)=\operatorname {O} (2n)\cap \operatorname {GL} (n,\mathbf {C} )\cap \operatorname {Sp} (2n,\mathbf {R} ).}$

Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be compatible (meaning that one uses the same J in the complex structure and the symplectic form, and that this J is orthogonal; writing all the groups as matrix groups fixes a J (which is orthogonal) and ensures compatibility).

In fact, it is the intersection of any two of these three; thus a compatible orthogonal and complex structure induce a symplectic structure, and so forth.[3][4]

At the level of equations, this can be seen as follows:

${\displaystyle {\begin{array}{r|r}{\text{Symplectic}}&A^{\mathsf {T}}JA=J\\\hline {\text{Complex}}&A^{-1}JA=J\\\hline {\text{Orthogonal}}&A^{\mathsf {T}}=A^{-1}\end{array}}}$

Any two of these equations implies the third.

At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). On an almost Kähler manifold, one can write this decomposition as h = g + iω, where h is the Hermitian form, g is the Riemannian metric, i is the almost complex structure, and ω is the almost symplectic structure.

From the point of view of Lie groups, this can partly be explained as follows: O(2n) is the maximal compact subgroup of GL(2n, R), and U(n) is the maximal compact subgroup of both GL(n, C) and Sp(2n). Thus the intersection O(2n) ∩ GL(n, C) or O(2n) ∩ Sp(2n) is the maximal compact subgroup of both of these, so U(n). From this perspective, what is unexpected is the intersection GL(n, C) ∩ Sp(2n) = U(n).

Just as the orthogonal group O(n) has the special orthogonal group SO(n) as subgroup and the projective orthogonal group PO(n) as quotient, and the projective special orthogonal group PSO(n) as subquotient, the unitary group U(n) has associated to it the special unitary group SU(n), the projective unitary group PU(n), and the projective special unitary group PSU(n). These are related as by the commutative diagram at right; notably, both projective groups are equal: PSU(n) = PU(n).

The above is for the classical unitary group (over the complex numbers) – for unitary groups over finite fields, one similarly obtains special unitary and projective unitary groups, but in general ${\displaystyle \operatorname {PSU} \left(n,q^{2}\right)\neq \operatorname {PU} \left(n,q^{2}\right)}$.

Analogous to the indefinite orthogonal groups, 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 Ψ(Mv, Mw) = Ψ(v, w) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted Φ, this says that M^∗ΦM = Φ.

Just as for symmetric forms over the reals, Hermitian forms are determined by signature, and are all 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 p + q = n. In a standard basis, this is represented as a quadratic form as:

${\displaystyle \lVert z\rVert _{\Psi }^{2}=\lVert z_{1}\rVert ^{2}+\dots +\lVert z_{p}\rVert ^{2}-\lVert z_{p+1}\rVert ^{2}-\dots -\lVert z_{n}\rVert ^{2}}$

and as a symmetric form as:

${\displaystyle \Psi (w,z)={\bar {w}}_{1}z_{1}+\cdots +{\bar {w}}_{p}z_{p}-{\bar {w}}_{p+1}z_{p+1}-\cdots -{\bar {w}}_{n}z_{n}.}$

The resulting group is denoted U(p,q).

Consider U(1,1): The matrices ${\displaystyle {\begin{pmatrix}u&v\\v^{*}&u^{*}\end{pmatrix}}}$ such that ${\displaystyle uu^{*}-vv^{*}\ \neq \ 0}$ form a group under matrix multiplication. In this case the conjugate transpose does not form the inverse of such a matrix, so the group is a pseudo-unitary group.

These matrices arise in representations of the group of units of two important rings: the composition algebra of split-quaternions and the 2 × 2 matrix ring over real numbers, M(2,R). The symmetries of the pseudo-unitary matrices have been applied in physical science, particularly the special unitary group SU(1, 1) where ${\displaystyle uu^{*}-vv^{*}\ =\ 1.}$[5]

Over the finite field with q = p^r elements, F_q, there is a unique quadratic extension field, F_q², with order 2 automorphism ${\displaystyle \alpha \colon x\mapsto x^{q}}$ (the rth power of the Frobenius automorphism). This allows one to define a Hermitian form on an F_q² vector space V, as an F_q-bilinear map ${\displaystyle \Psi \colon V\times V\to K}$ such that ${\displaystyle \Psi (w,v)=\alpha \left(\Psi (v,w)\right)}$ and ${\displaystyle \Psi (w,cv)=c\Psi (w,v)}$ for c ∈ F_q². Further, all non-degenerate Hermitian forms on a vector space over a finite field are unitarily congruent to the standard one, represented by the identity matrix; that is, any Hermitian form is unitarily equivalent to

${\displaystyle \Psi (w,v)=w^{\alpha }\cdot v=\sum _{i=1}^{n}w_{i}^{q}v_{i}}$

where ${\displaystyle w_{i},v_{i}}$ represent the coordinates of w, v ∈ V in some particular F_q²-basis of the n-dimensional space V (Grove 2002, Thm. 10.3).

Thus one can define a (unique) unitary group of dimension n for the extension F_q²/F_q, denoted either as U(n, q) or U(n, q²) depending on the author. The subgroup of the unitary group consisting of matrices of determinant 1 is called the special unitary group and denoted SU(n, q) or SU(n, q²). For convenience, this article will use the U(n, q²) convention. The center of U(n, q²) has order q + 1 and consists of the scalar matrices that are unitary, that is those matrices cI_V with ${\displaystyle c^{q+1}=1}$. The center of the special unitary group has order 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, PU(n, q²), and the quotient of the special unitary group by its center is the projective special unitary group PSU(n, q²). In most cases (n > 1 and (n, q²) ∉ {(2, 2²), (2, 3²), (3, 2²)}), SU(n, q²) is a perfect group and PSU(n, q²) is a finite simple group, (Grove 2002, Thm. 11.22 and 11.26).

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 ${\displaystyle a\mapsto {\bar {a}}}$ which is an involution and fixes exactly k (${\displaystyle a={\bar {a}}}$ if and only if a ∈ k).[6] 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.

The equations defining a unitary group are polynomial equations over k (but not over K): for the standard form Φ = I, the equations are given in matrices as A^∗A = I, where ${\displaystyle A^{*}={\bar {A}}^{\mathsf {T}}}$ is the conjugate transpose. Given a different form, they are A^∗ΦA = Φ. The unitary group is thus an algebraic group, whose points over a k-algebra R are given by:

${\displaystyle \operatorname {U} (n,K/k,\Phi )(R):=\left\{A\in \operatorname {GL} (n,K\otimes _{k}R):A^{*}\Phi A=\Phi \right\}.}$

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:

${\displaystyle {\begin{aligned}\operatorname {U} (n,\mathbf {C} /\mathbf {R} )(\mathbf {R} )&=\operatorname {U} (n)\\\operatorname {U} (n,\mathbf {C} /\mathbf {R} )(\mathbf {C} )&=\operatorname {GL} (n,\mathbf {C} ).\end{aligned}}}$

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

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 algebraic groups. The definition goes back to Anthony Bak's thesis.[7]

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

Let R be a ring with anti-automorphism J, ${\displaystyle \varepsilon \in R^{\times }}$ such that ${\displaystyle r^{J^{2}}=\varepsilon r\varepsilon ^{-1}}$ for all r in R and ${\displaystyle \varepsilon ^{J}=\varepsilon ^{-1}}$. Define

${\displaystyle {\begin{aligned}\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{aligned}}}

Let Λ ⊆ R be an additive subgroup of R, then Λ is called form parameter if ${\displaystyle \Lambda _{\text{min}}\subseteq \Lambda \subseteq \Lambda _{\text{max}}}$ and ${\displaystyle r^{J}\Lambda r\subseteq \Lambda }$. A pair (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., ${\displaystyle f(xr,ys)=r^{J}f(x,y)s}$ for any ${\displaystyle x,y\in M}$ and ${\displaystyle r,s\in R}$). Define ${\displaystyle h(x,y):=f(x,y)+f(y,x)^{J}\varepsilon \in R}$ and ${\displaystyle q(x):=f(x,x)\in R/\Lambda }$, then f is said to define the Λ-quadratic form (h, q) on M. A quadratic module over (R, Λ) is a triple (M, h, q) such that M is an R-module and (h, q) is a Λ-quadratic form.

To any quadratic module (M, h, q) defined by a J-sesquilinear form f on M over a form ring (R, Λ) one can associate the unitary group

${\displaystyle 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)\}.}

The special case where Λ = Λ_max, with J any non-trivial involution (i.e., ${\displaystyle J\neq id_{R},J^{2}=id_{R}}$ and ε = −1 gives back the "classical" unitary group (as an algebraic group).