Unitary group

"U(2)" redirects here. For other topics, seeU2 (disambiguation).

Inmathematics,theunitary groupof degreen,denoted U(n), is thegroupofn × nunitary matrices,with the group operation ofmatrix multiplication.The unitary group is asubgroupof thegeneral linear groupGL(n,C),and it has as a subgroup thespecial unitary group,consisting of those unitary matrices withdeterminant1.

In the simple casen= 1,the group U(1) corresponds to thecircle group,consisting of allcomplex numberswithabsolute value1, under multiplication. All the unitary groups contain copies of this group.

The unitary group U(n) is arealLie groupof dimensionn².TheLie algebraof U(n) consists ofn × nskew-Hermitian matrices,with theLie bracketgiven by thecommutator.

Thegeneral unitary group(also called thegroup of unitary similitudes) consists of allmatricesAsuch thatA^∗Ais a nonzero multiple of theidentity matrix,and is just the product of the unitary group with the group of all positive multiples of the identity matrix.

Unitary groups may also be defined over fields other than the complex numbers. Thehyperorthogonal groupis an archaic name for the unitary group, especially overfinite fields.

Properties

Since thedeterminantof a unitary matrix is a complex number with norm1,the determinant gives agroup homomorphism

$\det \colon \operatorname {U} (n)\to \operatorname {U} (1).$

Thekernelof this homomorphism is the set of unitary matrices with determinant1.This subgroup is called thespecial unitary group,denotedSU(n).We then have ashort exact sequenceof Lie groups:

$1\to \operatorname {SU} (n)\to \operatorname {U} (n)\to \operatorname {U} (1)\to 1.$

The above mapU(n)toU(1)has a section: we can viewU(1)as the subgroup ofU(n)that are diagonal withe^iθin the upper left corner and1on the rest of the diagonal. ThereforeU(n)is asemidirect productofU(1)withSU(n).

The unitary groupU(n)is notabelianforn> 1.ThecenterofU(n)is the set of scalar matricesλIwithλ∈ U(1);this follows fromSchur's lemma.The center is then isomorphic toU(1).Since the center ofU(n)is a1-dimensional abeliannormal subgroupofU(n),the unitary group is notsemisimple,but it isreductive.

Topology

The unitary group U(n) is endowed with therelative topologyas a subset ofM(n,C),the set of alln×ncomplex matrices, which is itself homeomorphic to a 2n²-dimensionalEuclidean space.

As a topological space, U(n) is bothcompactandconnected.To show that U(n) is connected, recall that any unitary matrixAcan bediagonalizedby another unitary matrixS.Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal. We can therefore write

$A=S\,\operatorname {diag} \left(e^{i\theta _{1}},\dots,e^{i\theta _{n}}\right)\,S^{-1}.$

Apathin U(n) from the identity toAis then given by

$t\mapsto S\,\operatorname {diag} \left(e^{it\theta _{1}},\dots,e^{it\theta _{n}}\right)\,S^{-1}.$

The unitary group is notsimply connected;the fundamental group of U(n) is infinite cyclic for alln:

$\pi _{1}(\operatorname {U} (n))\cong \mathbf {Z}.$

To see this, note that the above splitting of U(n) as asemidirect productof SU(n) and U(1) induces a topological product structure on U(n), so that

$\pi _{1}(\operatorname {U} (n))\cong \pi _{1}(\operatorname {SU} (n))\times \pi _{1}(\operatorname {U} (1)).$

Now the first unitary group U(1) is topologically acircle,which is well known to have afundamental groupisomorphic toZ,whereas$\operatorname {SU} (n)$is simply connected.

The determinant mapdet: U(n) → U(1)induces an isomorphism of fundamental groups, with the splittingU(1) → U(n)inducing the inverse.

TheWeyl groupof U(n) is thesymmetric groupS_n,acting on the diagonal torus by permuting the entries:

$\operatorname {diag} \left(e^{i\theta _{1}},\dots,e^{i\theta _{n}}\right)\mapsto \operatorname {diag} \left(e^{i\theta _{\sigma (1)}},\dots,e^{i\theta _{\sigma (n)}}\right)$

Related groups

2-out-of-3 property

The unitary group is the 3-fold intersection of theorthogonal,complex,andsymplecticgroups:

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

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

In fact, it is the intersection of anytwoof these three; thus a compatible orthogonal and complex structure induce a symplectic structure, and so forth.

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

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

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 analmost Kähler manifold,one can write this decomposition ash=g+iω,wherehis the Hermitian form,gis theRiemannian metric,iis thealmost complex structure,andωis thealmost symplectic structure.

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

Special unitary and projective unitary groups

See also:Special unitary groupandProjective unitary group

Just as the orthogonal group O(n) has thespecial orthogonal groupSO(n) as subgroup and theprojective orthogonal groupPO(n) as quotient, and theprojective special orthogonal groupPSO(n) assubquotient,the unitary group U(n) has associated to it thespecial unitary groupSU(n), theprojective unitary groupPU(n), and theprojective special unitary groupPSU(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) – forunitary groups over finite fields,one similarly obtains special unitary and projective unitary groups, but in general$\operatorname {PSU} \left(n,q^{2}\right)\neq \operatorname {PU} \left(n,q^{2}\right)$.

G-structure: almost Hermitian

In the language ofG-structures,a manifold with a U(n)-structure is analmost Hermitian manifold.

Generalizations

From the point of view ofLie theory,the classical unitary group is a real form of theSteinberg group${}^{2}\!A_{n}$,which is analgebraic groupthat arises from the combination of thediagram automorphismof the general linear group (reversing theDynkin diagramA_n,which corresponds to transpose inverse) and thefield automorphismof the extensionC/R(namelycomplex 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 standardHermitian formΨ, which is positive definite.

This can be generalized in a number of ways:

• generalizing to other Hermitian forms yields indefinite unitary groupsU(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 othergroups of Lie type,namely the otherSteinberg groups${}^{2}\!D_{n},{}^{2}\!E_{6},{}^{3}\!D_{4},$(in addition to${}^{2}\!A_{n}$) andSuzuki-Ree groups

  ${}^{2}\!B_{2}\left(2^{2n+1}\right),{}^{2}\!F_{4}\left(2^{2n+1}\right),{}^{2}\!G_{2}\left(3^{2n+1}\right);$

• considering a generalized unitary group as an algebraic group, one can take its points over various algebras.

Indefinite forms

Analogous to theindefinite orthogonal groups,one can define anindefinite 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 spaceV,the unitary group U(Ψ) is the group of transforms that preserve the form: the transformMsuch thatΨ(Mv,Mw) = Ψ(v,w)for allv,w∈V.In terms of matrices, representing the form by a matrix denoted Φ, this says thatM^∗ΦM= Φ.

Just as forsymmetric formsover the reals, Hermitian forms are determined bysignature,and are allunitarily congruentto a diagonal form withpentries of 1 on the diagonal andqentries of −1. The non-degenerate assumption is equivalent top+q=n.In a standard basis, this is represented as a quadratic form as:

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

and as a symmetric form as:

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

The resulting group is denotedU(p,q).

Finite fields

Over thefinite fieldwithq=p^relements,F_q,there is a unique quadratic extension field,F_q²,with order 2 automorphism$\ Alpha \colon x\mapsto x^{q}$(therth power of theFrobenius automorphism). This allows one to define a Hermitian form on anF_q²vector spaceV,as anF_q-bilinear map$\Psi \colon V\times V\to K$such that$\Psi (w,v)=\ Alpha \left(\Psi (v,w)\right)$and$\Psi (w,cv)=c\Psi (w,v)$forc∈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

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

where$w_{i},v_{i}$represent the coordinates ofw,v∈Vin some particularF_q²-basis of then-dimensional spaceV(Grove 2002,Thm. 10.3).

Thus one can define a (unique) unitary group of dimensionnfor the extensionF_q²/F_q,denoted either asU(n,q)orU(n,q²)depending on the author. The subgroup of the unitary group consisting of matrices of determinant 1 is called thespecial unitary groupand denotedSU(n,q)orSU(n,q²).For convenience, this article will use theU(n,q²)convention. The center ofU(n,q²)has orderq+ 1and consists of the scalar matrices that are unitary, that is those matricescI_Vwith$c^{q+1}=1$.The center of the special unitary group has ordergcd(n,q+ 1)and consists of those unitary scalars which also have order dividingn.The quotient of the unitary group by its center is called theprojective unitary group,PU(n,q²),and the quotient of the special unitary group by its center is theprojective special unitary groupPSU(n,q²).In most cases (n> 1and(n,q²) ∉ {(2, 2²), (2, 3²), (3, 2²)}),SU(n,q²)is aperfect groupandPSU(n,q²)is a finitesimple group,(Grove 2002,Thm. 11.22 and 11.26).

Degree-2 separable algebras

More generally, given a fieldkand a degree-2 separablek-algebraK(which may be a field extension but need not be), one can define unitary groups with respect to this extension.

First, there is a uniquek-automorphism ofK$a\mapsto {\bar {a}}$which is an involution and fixes exactlyk($a={\bar {a}}$if and only ifa∈k).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 overk(but not overK): for the standard formΦ =I,the equations are given in matrices asA^∗A=I,where$A^{*}={\bar {A}}^{\mathsf {T}}$is theconjugate transpose.Given a different form, they areA^∗ΦA= Φ.The unitary group is thus analgebraic group,whose points over ak-algebraRare given by:

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

For the field extensionC/Rand the standard (positive definite) Hermitian form, these yield an algebraic group with real and complex points given by:

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

In fact, the unitary group is alinear 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 differentclassical algebraic groups.The definition goes back to Anthony Bak's thesis.

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

LetRbe a ring with anti-automorphismJ,$\varepsilon \in R^{\times }$such that$r^{J^{2}}=\varepsilon r\varepsilon ^{-1}$for allrinRand$\varepsilon ^{J}=\varepsilon ^{-1}$.Define

${\begin{aligned}\Lambda _{\text{min}}&:=\left\{r\in R\:\ r-r^{J}\varepsilon \right\},\\\Lambda _{\text{max}}&:=\left\{r\in R\:\ r^{J}\varepsilon =-r\right\}.\end{aligned}}$

LetΛ ⊆Rbe an additive subgroup ofR,then Λ is calledform parameterif$\Lambda _{\text{min}}\subseteq \Lambda \subseteq \Lambda _{\text{max}}$and$r^{J}\Lambda r\subseteq \Lambda$.A pair(R,Λ)such thatRis a ring and Λ a form parameter is calledform ring.

LetMbe anR-module andfaJ-sesquilinear form onM(i.e.,$f(xr,ys)=r^{J}f(x,y)s$for any$x,y\in M$and$r,s\in R$). Define$h(x,y):=f(x,y)+f(y,x)^{J}\varepsilon \in R$and$q(x):=f(x,x)\in R/\Lambda$,thenfis said todefinetheΛ-quadratic form(h,q)onM.Aquadratic moduleover(R,Λ)is a triple(M,h,q)such thatMis anR-module and(h,q)is a Λ-quadratic form.

To any quadratic module(M,h,q)defined by aJ-sesquilinear formfonMover a form ring(R,Λ)one can associate theunitary group

$U(M):=\{\sigma \in GL(M)\:\ \forall x,y\in M,h(\sigma x,\sigma y)=h(x,y){\text{ and }}q(\sigma x)=q(x)\}.$

The special case whereΛ = Λ_max,withJany non-trivial involution (i.e.,$J\neq id_{R},J^{2}=id_{R}$andε= −1gives back the "classical" unitary group (as an algebraic group).

Polynomial invariants

The unitary groups are the automorphisms of two polynomials in real non-commutative variables:

${\begin{aligned}C_{1}&=\left(u^{2}+v^{2}\right)+\left(w^{2}+x^{2}\right)+\left(y^{2}+z^{2}\right)+\ldots \\C_{2}&=\left(uv-vu\right)+\left(wx-xw\right)+\left(yz-zy\right)+\ldots \end{aligned}}$

These are easily seen to be the real and imaginary parts of the complex form$Z{\overline {Z}}$.The two invariants separately are invariants of O(2n) and Sp(2n). 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.

Classifying space

Theclassifying spacefor U(n) is described in the articleClassifying space for U(n).

See also

• Special unitary group
• Projective unitary group
• Orthogonal group
• Symplectic group

References

• Grove, Larry C. (2002),Classical groups and geometric algebra,Graduate Studies in Mathematics,vol. 39, Providence, R.I.:American Mathematical Society,ISBN978-0-8218-2019-3,MR1859189
• Hall, Brian C. (2015),Lie Groups, Lie Algebras, and Representations: An Elementary Introduction,Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,ISBN978-3319134666