- Unitary group
In

mathematics , the**unitary group**of degree "n", denoted U("n"), is the group of "n"×"n" unitary matrices, with the group operation that ofmatrix multiplication . The unitary group is asubgroup of thegeneral linear group GL("n",**C**).In the simple case "n" = 1, the group U(1) corresponds to the

circle group , consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group.The unitary group U("n") is a real

Lie group of dimension "n"^{2}. TheLie algebra of U("n") consists of complex "n"×"n" skew-Hermitian matrices, with theLie bracket given by thecommutator .The

**general unitary group**(also called the**group of unitary similitudes**) consists of all matrices $A$ such that $A^*A$ is 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.**Properties**Since the

determinant of a unitary matrix is a complex number with norm 1, the determinant gives agroup homomorphism :$detcolon\; mbox\{U\}(n)\; o\; mbox\{U\}(1)$The kernel of this homomorphism is the set of unitary matrices with unit determinant. This subgroup is called the, denoted SU("n"). We then have aspecial unitary group short exact sequence of Lie groups::$1\; ombox\{SU\}(n)\; ombox\{U\}(n)\; ombox\{U\}(1)\; o\; 1$This short exact sequence splits so that U("n") may be written as asemidirect product of SU("n") by U(1). Here the U(1) subgroup of U("n") consists of matrices of the form $mbox\{diag\}\; (e^\{i\; heta\}$,1,1,...,1).The unitary group U("n") is nonabelian for "n" > 1. The center of U("n") is the set of scalar matrices λ"I" with λ ∈ U(1). This follows from

Schur's lemma . The center is then isomorphic to U(1). Since the center of U("n") is a 1-dimensional abeliannormal subgroup of U("n"), the unitary group is notsemisimple .**Topology**The unitary group U("n") is endowed with the

relative topology as a subset of "M"_{"n"}(**C**), the set of all "n"×"n" complex matrices, which is itself homeomorphic to a 2"n"^{2}-dimensionalEuclidean space .As a topological space, U("n") is both compact and connected. The compactness of U("n") follows from the

Heine-Borel theorem and the fact that it is a closed and bounded subset of "M"_{"n"}(**C**). To show that U("n") is connected, recall that any unitary matrix "A" can bediagonalized by another unitary matrix "S". Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal. We can therefore write:$A\; =\; S,mbox\{diag\}(e^\{i\; heta\_1\},dots,e^\{i\; heta\_n\}),S^\{-1\}.$

A path in U("n") from the identity to "A" is then given by

:$tmapsto\; S,mbox\{diag\}(e^\{it\; heta\_1\},dots,e^\{it\; heta\_n\}),S^\{-1\}.$

The unitary group is not

simply connected ; the fundamental group of U("n") is infinite cyclic for all "n"::$pi\_1(U(n))\; cong\; mathbf\{Z\}.$The first unitary group U(1) is topologically acircle , which is well known to have afundamental group isomorphic to**Z**, and the inclusion map $U(n)\; o\; U(n+1)$ is an isomorphism on $pi\_1$.(It has quotient theStiefel manifold .)The determinant map $mbox\{det\}colon\; mbox\{U\}(n)\; o\; mbox\{U\}(1)$ induces an isomorphism of fundamental groups, with the splitting $mbox\{U\}(1)\; o\; mbox\{U\}(n)$ inducing the inverse.

**Related groups****2 out of 3 property**The unitary group is the 3-fold intersection of the orthogonal, symplectic, and complex groups::$U(n)\; =\; O(2n)\; cap\; GL(n,mathbf\{C\})\; cap\; Sp(2n,\; mathbf\{R\})$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. [

*This is discussed in Arnold, "Mathematical Methods of Classical Mechanics".*] [*[*]*http://www.math.ucr.edu/home/baez/symplectic.html symplectic*]At the level of equations, this can be seen as follows::

**Symplectic:**$A^TJA\; =\; J$:**Complex:**$A^\{-1\}JA\; =\; J$:**Orthogonal:**$A^T=A^\{-1\}$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 $h=g\; +\; iomega$, where "h" is the Hermitian form, "g" is theRiemannian metric , "i" is the almost complex structure, and $omega$ is the almost symplectic structure.From the point of view of

Lie group s, this can partly be explained as follows:$O(2n)$ is themaximal compact subgroup of $GL(2n,mathbf\{R\})$, and $U(n)$ is the maximal compact subgroup of both $GL(n,mathbf\{C\})$ and $Sp(2n)$. Thus the intersection of $O(2n)\; cap\; GL(n,mathbf\{C\})$ or $O(2n)\; cap\; Sp(2n)$ is the maximal compact subgroup of both of these, so $U(n)$. From this perspective, what is unexpected is the intersection $GL(n,mathbf\{C\})\; cap\; Sp(2n)\; =\; U(n)$.**pecial Unitary and Projective Unitary Groups**Just as the orthogonal group has the

special orthogonal group "SO(n)" as subgroup and theprojective orthogonal group "PO(n)" as quotient, and theprojective special orthogonal group "PSO(n)" assubquotient , the unitary group has associated to it thespecial unitary group "SU(n)," theprojective unitary group "PU(n)," and theprojective special unitary group "PSU(n)." These are related as by the commutative diagram at right; notably, both projective groups are equal: $operatorname\{PSU\}(n)\; =\; operatorname\{PU\}(n)$.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 $operatorname\{PSU\}(n,q^2)\; eq\; operatorname\{PU\}(n,q^2)$.

**G-structure: almost Hermitian**In the language of

G-structure s, a manifold with a $mbox\{U\}(n)$-structure is analmost Hermitian manifold .**Generalizations**From the point of view of

Lie theory , the classical unitary group is a real form of the Steinberg group $\{\}^2!A\_n$, which is analgebraic group that arises from the combination of the "diagram automorphism" of the general linear group (reversing theDynkin diagram $A\_n$, which corresponds to transpose inverse) and the "field automorphism " of the extension $mathbf\{C\}/mathbf\{R\}$ (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 form $Psi$, which is positive definite.This can be generalized in a number of ways:

* generalizing to other Hermitian forms yields indefinite unitary groups $operatorname\{U\}(p,q)$;

* 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 othergroups of Lie type , namely the other Steinberg groups $\{\}^2!D\_n,\; \{\}^2!E\_6,\; \{\}^3!D\_4,$ (in addition to $\{\}^2!A\_n$) andSuzuki-Ree groups $\{\}^2!B\_2left(2^\{2n+1\}\; ight),\; \{\}^2!F\_4left(2^\{2n+1\}\; ight),\; \{\}^2!G\_2left(3^\{2n+1\}\; ight)$;

* 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 $Psi$ on a complex vector space $V$, the unitary group $U(Psi)$ is the group of transforms that preserve the form: the transform $M$ such that $Psi(Mv,Mw)=Psi(v,w)$ for all $v,win\; V$. In terms of matrices, representing the form by a matrix denoted $Phi$, this says that $M^*Phi\; M\; =\; Phi$.

Just as for

symmetric form s 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::$lVert\; z\; Vert\_Psi^2\; =\; lVert\; z\_1\; Vert^2\; +\; dots\; +\; lVert\; z\_p\; Vert^2\; -\; lVert\; z\_\{p+1\}\; Vert^2\; -\; dots\; -\; lVert\; z\_n\; Vert^2$and as a symmetric form as::$Psi(w,z)\; =\; ar\; w\_1\; z\_1\; +\; cdots\; +\; ar\; w\_p\; z\_p\; -\; ar\; w\_\{p+1\}z\_\{p+1\}\; -\; cdots\; -\; ar\; w\_n\; z\_n$The resulting group is denoted $U(p,q)$.**Finite fields**Over the

finite field with $q=p^r$ elements, $mathbf\{F\}\_q$, there is a unique degree 2 extension field, $mathbf\{F\}\_\{q^2\}$, with order 2 automorphism $alphacolon\; x\; mapsto\; x^q$ (the $r$th power of theFrobenius automorphism ). This allows one to define a Hermitian form on an $mathbf\{F\}\_\{q^2\}$ vector space $V$, as an $mathbf\{F\}\_q$-bilinear map $Psicolon\; V\; imes\; V\; o\; K$ such that $Psi(w,v)=alphaleft(Psi(v,w)\; ight)$ and $Psi(w,cv)=cPsi(w,v)$ for $c\; in\; mathbf\{F\}\_\{q^2\}$.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 Hermetian 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 of $w,v\; in\; V$ in some particular $mathbf\{F\}\_\{q^2\}$-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 $mathbf\{F\}\_\{q^2\}/mathbf\{F\}\_q$, denoted either as $U(n,q)$ or $Uleft(n,q^2\; ight)$ 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 $SU(n,q^2)$. For convenience, this article with use the $U(n,q^2)$ convention. The center of $U(n,q^2)$ has order $q+1$ and consists of the scalar matrices which are unitary, that is those matrices $cI\_V$ with $c^\{q+1\}=1$. 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, $PU(n,q^2)$, and the quotient of the special unitary group by its center is theprojective unitary group $PSU(n,q^2)$. In most cases ($n\; geq\; 2$ and $(n,q^2)\; otin\; \{\; (2,2^2),\; (2,3^2),\; (3,2^2)\; \}$), $SU(n,q^2)$ is aprojective special unitary group perfect group and $PSU(n,q^2)$ is a finitesimple 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 $K$ $a\; mapsto\; ar\; a$ which is an involution and fixes exactly $k$ ($a=ar\; a$ if and only if $a\; in\; k$) [

*Milne, [*] . 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.*http://www.jmilne.org/math/CourseNotes/aag.html Algebraic Groups and Arithmetic Groups*] , p. 103**Algebraic groups**The equations defining a unitary group are polynomial equations over $k$ (but not over $K$): for the standard form $Phi=I$ the equations are given in matrices as $A^*A=I$, where $A^*=overline\; A^t$ is the

conjugate transpose . Given a different form, they are $A^*Phi\; A=Phi$. The unitary group is thus analgebraic group , whose points over a $k$-algebra $R$ are given by::$operatorname\{U\}(n,K/k,Phi)(R)\; :=\; left\{\; Ain\; operatorname\{GL\}(n,Kotimes\_k\; R)\; :\; A^*Phi\; A=Phi\; ight\}$For the field extension $mathbf\{C\}/mathbf\{R\}$ and the standard (positive definite) Hermitian form, these yield an algebraic group with real and complex points given by::$operatorname\{U\}(n,mathbf\{C\}/mathbf\{R\})(mathbf\{R\})\; =\; operatorname\{U\}(n)$:$operatorname\{U\}(n,mathbf\{C\}/mathbf\{R\})(mathbf\{C\})\; =\; operatorname\{GL\}(n,mathbf\{C\})$

**Classifying space**The

classifying space for "U"("n") is described in the articleclassifying space for U(n) .**References***Citation | last1=Grove | first1=Larry C. | title=Classical groups and geometric algebra | publisher=

American Mathematical Society | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-2019-3 | id=MathSciNet | id = 1859189 | year=2002 | volume=39**See also***

special unitary group

*projective unitary group

*orthogonal group

*symplectic group

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