Unitary group 173993 224342556 2008-07-08T12:50:13Z Silly rabbit 555135 Sp(2n, R) /* 2 out of 3 property */ {{Groups}} In [[mathematics]], the '''unitary group''' of degree ''n'', denoted U(''n''), is the [[group (mathematics)|group]] of ''n''&times;''n'' [[unitary matrix|unitary matrices]], with the group operation that of [[matrix multiplication]]. The unitary group is a [[subgroup]] of the [[general 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 [[Complex number#Absolute value, conjugation and distance|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''<sup>2</sup>. The [[Lie algebra]] of U(''n'') consists of complex ''n''&times;''n'' [[skew-Hermitian matrix|skew-Hermitian matrices]], with the [[Lie bracket]] given by the [[commutator]]. The '''general unitary group''' (also called the '''group of unitary similitudes''') consists of all matrices <math>A</math> such that <math>A^*A</math> is a nonzero multiple of the [[identity 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 a [[group homomorphism]] :<math>\det\colon \mbox{U}(n) \to \mbox{U}(1)</math> The [[kernel (group theory)|kernel]] of this homomorphism is the set of unitary matrices with unit determinant. This subgroup is called the '''[[special unitary group]]''', denoted SU(''n''). We then have a [[short exact sequence]] of Lie groups: :<math>1\to\mbox{SU}(n)\to\mbox{U}(n)\to\mbox{U}(1)\to 1</math> This short exact sequence [[split exact sequence|splits]] so that U(''n'') may be written as a [[semidirect product]] of SU(''n'') by U(1). Here the U(1) subgroup of U(''n'') consists of matrices of the form <math>\mbox{diag}(e^{i\theta},1,1,\ldots,1)</math>. The unitary group U(''n'') is [[nonabelian group|nonabelian]] for ''n'' &gt; 1. The [[center of a group|center]] of U(''n'') is the set of scalar matrices &lambda;''I'' with &lambda; &isin; 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 abelian [[normal subgroup]] of U(''n''), the unitary group is not [[semisimple]]. ==Topology== The unitary group U(''n'') is endowed with the [[relative topology]] as a subset of ''M''<sub>''n''</sub>('''C'''), the set of all ''n''&times;''n'' complex matrices, which is itself homeomorphic to a 2''n''<sup>2</sup>-dimensional [[Euclidean space]]. As a topological space, U(''n'') is both [[compact space|compact]] and [[connected space|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''<sub>''n''</sub>('''C'''). To show that U(''n'') is connected, recall that any unitary matrix ''A'' can be [[diagonalized]] 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 :<math>A = S\,\mbox{diag}(e^{i\theta_1},\dots,e^{i\theta_n})\,S^{-1}.</math> A [[path (topology)|path]] in U(''n'') from the identity to ''A'' is then given by :<math>t\mapsto S\,\mbox{diag}(e^{it\theta_1},\dots,e^{it\theta_n})\,S^{-1}.</math> The unitary group is not [[simply connected]]; the fundamental group of U(''n'') is infinite cyclic for all ''n'': :<math>\pi_1(U(n)) \cong \mathbf{Z}.</math> The first unitary group U(1) is topologically a [[circle]], which is well known to have a [[fundamental group]] isomorphic to '''Z''', and the inclusion map <math>U(n) \to U(n+1)</math> is an isomorphism on <math>\pi_1</math>. (It has quotient the [[Stiefel manifold]].) The determinant map <math>\mbox{det}\colon \mbox{U}(n) \to \mbox{U}(1)</math> induces an isomorphism of fundamental groups, with the splitting <math>\mbox{U}(1) \to \mbox{U}(n)</math> inducing the inverse. ==2 out of 3 property== The unitary group is the 3-fold intersection of the [[orthogonal group|orthogonal]], [[symplectic group|symplectic]], and complex groups: :<math>U(n) = O(2n) \cap GL(n,\mathbf{C}) \cap Sp(2n, \mathbf{R})</math> 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. <ref>This is discussed in Arnold, "Mathematical Methods of Classical Mechanics".</ref> <ref>[http://www.math.ucr.edu/home/baez/symplectic.html symplectic<!-- Bot generated title -->]</ref> At the level of equations, this can be seen as follows: :'''Symplectic:''' <math>A^TJA = J</math> :'''Complex:''' <math>A^{-1}JA = J</math> :'''Orthogonal:''' <math>A^T=A^{-1}</math> 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 <math>h=g + i\omega</math>, where ''h'' is the Hermitian form, ''g'' is the [[Riemannian metric]], ''i'' is the [[almost complex manifold|almost complex structure]], and <math>\omega</math> is the [[almost symplectic manifold|almost symplectic structure]]. From the point of view of [[Lie group]]s, this can partly be explained as follows: <math>O(2n)</math> is the [[maximal compact subgroup]] of <math>GL(2n,\mathbf{R})</math>, and <math>U(n)</math> is the maximal compact subgroup of both <math>GL(n,\mathbf{C})</math> and <math>Sp(2n)</math>. Thus the intersection of <math>O(2n) \cap GL(n,\mathbf{C})</math> or <math>O(2n) \cap Sp(2n)</math> is the maximal compact subgroup of both of these, so <math>U(n)</math>. From this perspective, what is unexpected is the intersection <math>GL(n,\mathbf{C}) \cap Sp(2n) = U(n)</math>. ==G-structure: almost Hermitian== In the language of [[G-structure]]s, a manifold with a <math>\mbox{U}(n)</math>-structure is an [[almost Hermitian manifold]]. ==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]] <math>{}^2\!A_n</math>, which is an [[algebraic group]] that arises from the combination of the ''diagram automorphism'' of the general linear group (reversing the [[Dynkin diagram]] <math>A_n</math>, which corresponds to transpose inverse) and the ''[[field automorphism]]'' of the extension <math>\mathbf{C}/\mathbf{R}</math> (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]] <math>\Psi</math>, which is positive definite. This can be generalized in a number of ways: * generalizing to other Hermitian forms yields indefinite unitary groups <math>\operatorname{U}(p,q)</math>; * 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]] <math>{}^2\!D_n, {}^2\!E_6, {}^3\!D_4, </math> (in addition to <math>{}^2\!A_n</math>) and [[Suzuki-Ree groups]] <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 <math>\Psi</math> on a complex vector space <math>V</math>, the unitary group <math>U(\Psi)</math> is the group of transforms that preserve the form: the transform <math>M</math> such that <math>\Psi(Mv,Mw)=\Psi(v,w)</math> for all <math>v,w\in V</math>. In terms of matrices, representing the form by a matrix denoted <math>\Phi</math>, this says that <math>M^*\Phi M = \Phi</math>. Just as for [[symmetric form]]s 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 <math>p</math> entries of 1 on the diagonal and <math>q</math> entries of <math>-1</math>. The non-degenerate assumption is equivalent to <math>p+q=n</math>. 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 <math>U(p,q)</math>. ===Finite fields=== Over the [[finite field]] with <math>q=p^r</math> elements, <math>\mathbf{F}_q</math>, there is a unique degree 2 extension field, <math>\mathbf{F}_{q^2}</math>, with order 2 automorphism <math>\alpha\colon x \mapsto x^q</math> (the <math>r</math>th power of the [[Frobenius automorphism]]). This allows one to define a Hermitian form on an <math>\mathbf{F}_{q^2}</math> vector space <math>V</math>, as an <math>\mathbf{F}_q</math>-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 <math>c \in \mathbf{F}_{q^2}</math>. 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 Hermetian 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 <math>w,v \in V</math> in some particular <math>\mathbf{F}_{q^2}</math>-basis of the <math>n</math>-dimensional space <math>V</math> {{harv|Grove|2002|loc=Thm. 10.3}}. Thus one can define a (unique) unitary group of dimension <math>n</math> for the extension <math>\mathbf{F}_{q^2}/\mathbf{F}_q</math>, denoted either as <math>U(n,q)</math> or <math>U\left(n,q^2\right)</math> depending on the author. The subgroup of the unitary group consisting of matrices of determinant 1 is called the '''special unitary group''' and denoted <math>SU(n,q)</math> or <math>SU(n,q^2)</math>. For convenience, this article with use the <math>U(n,q^2)</math> convention. The center of <math>U(n,q^2)</math> has order <math>q+1</math> and consists of the scalar matrices which are unitary, that is those matrices <math>cI_V</math> with <math>c^{q+1}=1</math>. The center of the special unitary group has order <math>\gcd(n,q+1)</math> and consists of those unitary scalars which also have order dividing <math>n</math>. The quotient of the unitary group by its center is called the '''projective unitary group''', <math>PU(n,q^2)</math>, and the quotient of the special unitary group by its center is the '''projective special unitary group''' <math>PSU(n,q^2)</math>. In most cases (<math> n \geq 2</math> and <math>(n,q^2) \notin \{ (2,2^2), (2,3^2), (3,2^2) \}</math>), <math>SU(n,q^2)</math> is a [[perfect group]] and <math>PSU(n,q^2)</math> is a finite [[simple group]], {{harv|Grove|2002|loc=Thm. 11.22 and 11.26}}. ===Degree 2 separable algebras=== More generally, given a field <math>k</math> and a degree 2 separable <math>k</math>-algebra <math>K</math> (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 <math>k</math>-automorphism of <math>K</math> <math>a \mapsto \bar a</math> which is an involution and fixes exactly <math>k</math> (<math>a=\bar a</math> if and only if <math>a \in k</math>)<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 <math>k</math> (but not over <math>K</math>): for the standard form <math>\Phi=I</math> the equations are given in matrices as <math>A^*A=I</math>, where <math>A^*=\overline A^t</math> is the [[conjugate transpose]]. Given a different form, they are <math>A^*\Phi A=\Phi</math>. The unitary group is thus an [[algebraic group]], whose points over a <math>k</math>-algebra <math>R</math> 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 <math>\mathbf{C}/\mathbf{R}</math> and the standard (positive definite) Hermitian form, these yield an algebraic group with real and complex points given by: :<math>\operatorname{U}(n,\mathbf{C}/\mathbf{R})(\mathbf{R}) = \operatorname{U}(n)</math> :<math>\operatorname{U}(n,\mathbf{C}/\mathbf{R})(\mathbf{C}) = \operatorname{GL}(n,\mathbf{C})</math> ==Classifying space== The [[classifying space]] for ''U''(''n'') is described in the article [[classifying space for U(n)]]. == References == <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]] [[Category:Lie groups]] [[de:Unitäre Gruppe]] [[es:Grupo unitario]] [[fr:Groupe unitaire]] [[pt:Grupo unitário]]