Symplectic group: Difference between revisions
No edit summary |
en>BD2412 |
||
Line 1: | Line 1: | ||
{{Group theory sidebar |Topological}} | |||
{{Lie groups |Classical}} | |||
In [[mathematics]], the '''unitary group''' of degree ''n'', denoted U(''n''), is the [[Group (mathematics)|group]] of ''n'' × ''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'''). '''Hyperorthogonal group''' is an archaic name for the unitary group, especially over finite fields. | |||
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 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 ''n'' × ''n'' [[skew-Hermitian matrix|skew-Hermitian matrices]], with the [[Lie algebra|Lie bracket]] given by the [[commutator]]. | |||
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 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 determinant 1. 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 [[Splitting lemma|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'') can be taken to consists of matrices, which are diagonal, have ''e''<sup>''iθ''</sup> in the upper left corner and ''1'' on the rest of the diagonal. | |||
The unitary group U(''n'') is [[nonabelian group|nonabelian]] for ''n'' > 1. The [[center of a group|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 abelian [[normal subgroup]] of U(''n''), the unitary group is not [[Semisimple algebraic group|semisimple]]. | |||
==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''<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(''n'', '''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 U(''n'') → U(''n''+1) is an isomorphism on π<sub>1</sub>. (It has quotient the [[Stiefel manifold]].) | |||
The determinant map det: U(''n'') → U(1) induces an isomorphism of fundamental groups, with the splitting U(1) → U(''n'') inducing the inverse. | |||
==Related groups== | |||
===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>{{cite book |last=Arnold |first=V.I. |title=Mathematical Methods of Classical Mechanics |publisher=Springer |edition=Second |year=1989 |page=225}}</ref><ref>{{cite web |url=http://www.math.ucr.edu/home/baez/symplectic.html |title=Symplectic, Quaternionic, Fermionic |last=Baez |first=John |accessdate=1 February 2012}}</ref> | |||
At the level of equations, this can be seen as follows: | |||
:{| class="wikitable" | |||
! '''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 ''h'' = ''g'' + ''i''ω, where ''h'' is the Hermitian form, ''g'' is the [[Riemannian metric]], ''i'' is the [[almost complex manifold|almost complex structure]], and ω is the [[almost symplectic manifold|almost symplectic structure]]. | |||
From the point of view of [[Lie group]]s, this can partly be explained as follows: O(2''n'') is the [[maximal compact subgroup]] of GL(2''n'', '''R'''), and U(''n'') is the maximal compact subgroup of both GL(''n'', '''C''') and Sp(2''n''). Thus the intersection O(2''n'') ∩ GL(''n'', '''C''') or O(2''n'') ∩ Sp(2''n'') 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(2''n'') = U(''n''). | |||
===Special unitary and projective unitary groups=== | |||
[[File:PSU-PU.svg|right]] | |||
{{Main|Projective unitary group}} | |||
Just as the orthogonal group 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 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 [[#Finite_fields|unitary groups over finite fields]], one similarly obtains special unitary and projective unitary groups, but in general <math>\operatorname{PSU}(n,q^2) \neq \operatorname{PU}(n,q^2)</math>. | |||
==G-structure: almost Hermitian== | |||
In the language of [[G-structure]]s, a manifold with a U(''n'')-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]] 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 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]] <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 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*''Φ''M'' = Φ. | |||
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 ''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: | |||
:<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 U(''p,q''). | |||
===Finite fields=== | |||
Over the [[finite field]] with ''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 ''c ∈ '''F'''<sub>''q''<sup>2</sup></sub>. 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 ''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 U(''n'', ''q'') or 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 SU(''n'', ''q'') or SU(''n'', ''q''<sup>2</sup>). For convenience, this article will use the U(''n'', ''q''<sup>2</sup>) convention. The center of U(''n'', ''q''<sup>2</sup>) has order ''q''+1 and consists of the scalar matrices which 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 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'', ''q''<sup>2</sup>), and the quotient of the special unitary group by its center is the '''[[projective special unitary group]]''' PSU(''n'', ''q''<sup>2</sup>). In most cases (''n'' > 1 and (''n'', ''q''<sup>2</sup>) ∉ {(2,2^2), (2,3^2), (3,2^2)}), SU(''n'', ''q''<sup>2</sup>) is a [[perfect group]] and 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 ''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 Φ = I the equations are given in matrices as ''A*A'' = ''I'', where <math>A^* = \overline A^\mathrm{T}</math> is the [[conjugate transpose]]. Given a different form, they are ''A*''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>\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> | |||
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 - On modules with quadratic forms, pp. 55-66 in Lecture Notes in Mathematics, Vol. 108, Springer, Berlin Heidelberg New York, 1969</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>\Lambda_{min} := \{r \in R \ : \ r-r^J\varepsilon\},</math> | |||
:<math>\Lambda_{max} := \{r \in R \ : \ r^J\varepsilon = -r\}.</math> | |||
Let Λ ⊆ ''R'' be an additive subgroup of ''R'', then Λ is called ''form parameter'' if <math>\Lambda_{min} \subseteq \Lambda \subseteq \Lambda_{max}</math> and <math>r^J \Lambda r \subseteq \Lambda</math>. 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. <math>f(xr,ys)=r^Jf(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'' (''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'' | |||
:<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 Λ = Λ<sub>max</sub>, with ''J'' any non-trivial involution (i.e. <math>J\neq id_R, J^2 = id_R</math> and ε = −1 gives 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: | |||
:<math>C_1 = (u^2+v^2)+(w^2+x^2)+(y^2+z^2)+... </math> | |||
:<math>C_2 = (uv-vu)+(wx-xw)+(yz-zy)+...</math> | |||
These are easily seen to be the real and imaginary parts of the complex form <math>Z \overline{Z}</math>. The two invariants separately are invariants of O(2''n'') and Sp(2''n'', R). 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== | |||
The [[classifying space]] for U(''n'') is described in the article [[classifying space for U(n)]]. | |||
== See also == | |||
*[[projective unitary group]] | |||
*[[orthogonal group]] | |||
*[[symplectic group]] | |||
== Notes == | |||
<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 | mr=1859189 | year=2002 | volume=39}} | |||
{{DEFAULTSORT:Unitary Group}} | |||
[[Category:Lie groups]] |
Revision as of 00:53, 28 December 2013
Template:Group theory sidebar Template:Lie groups
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields.
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 n2. The Lie algebra of U(n) consists of n × n 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 A such that A*A 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
The kernel of this homomorphism is the set of unitary matrices with determinant 1. This subgroup is called the special unitary group, denoted SU(n). We then have a short exact sequence of Lie groups:
This short 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) can be taken to consists of matrices, which are diagonal, have eiθ in the upper left corner and 1 on the rest of the diagonal.
The unitary group U(n) is nonabelian for n > 1. The center of U(n) is the set of scalar matrices λI 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 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(n, C), the set of all n × n complex matrices, which is itself homeomorphic to a 2n2-dimensional Euclidean 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 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
A path in U(n) from the identity to A is then given by
The unitary group is not simply connected; the fundamental group of U(n) is infinite cyclic for all n:
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 U(n) → U(n+1) is an isomorphism on π1. (It has quotient the Stiefel manifold.)
The determinant map det: U(n) → U(1) induces an isomorphism of fundamental groups, with the splitting U(1) → 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:
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.[1][2]
At the level of equations, this can be seen as follows:
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).
Special unitary and projective unitary groups
![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/12/PSU-PU.svg/394px-PSU-PU.svg.png)
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
Just as the orthogonal group 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 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 .
G-structure: almost Hermitian
In the language of G-structures, a manifold with a U(n)-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 , which is an algebraic group that arises from the combination of the diagram automorphism of the general linear group (reversing the Dynkin diagram An, 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 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 groups (in addition to ) and Suzuki-Ree groups
- considering a generalized unitary group as an algebraic group, one can take its points over various algebras.
Indefinite forms
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:
and as a symmetric form as:
The resulting group is denoted U(p,q).
Finite fields
Over the finite field with q = pr elements, Fq, there is a unique quadratic extension field, Fq2, with order 2 automorphism (the rth power of the Frobenius automorphism). This allows one to define a Hermitian form on an Fq2 vector space V, as an Fq-bilinear map such that and for c ∈ Fq2. 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
where represent the coordinates of w,v ∈ V in some particular Fq2-basis of the n-dimensional space V Template:Harv.
Thus one can define a (unique) unitary group of dimension n for the extension Fq2/Fq, 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 which are unitary, that is those matrices cIV with . 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) ∉ {(2,2^2), (2,3^2), (3,2^2)}), SU(n, q2) is a perfect group and PSU(n, q2) is a finite simple group, Template:Harv.
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 which is an involution and fixes exactly k ( if and only if a ∈ k).[3] 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 Φ = I the equations are given in matrices as A*A = I, where 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:
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:
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 algebraic groups. The definition goes back to Anthony Bak's thesis.[4]
To define it, one has to define quadratic modules first:
Let R be a ring with anti-automorphism J, such that for all r in R and . Define
Let Λ ⊆ R be an additive subgroup of R, then Λ is called form parameter if and . 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. for any and ). Define and , 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
The special case where Λ = Λmax, with J any non-trivial involution (i.e. and ε = −1 gives 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:
These are easily seen to be the real and imaginary parts of the complex form . The two invariants separately are invariants of O(2n) and Sp(2n, R). 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
The classifying space for U(n) is described in the article classifying space for U(n).
See also
Notes
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Template:Cite web
- ↑ Milne, Algebraic Groups and Arithmetic Groups, p. 103
- ↑ Bak, Anthony - On modules with quadratic forms, pp. 55-66 in Lecture Notes in Mathematics, Vol. 108, Springer, Berlin Heidelberg New York, 1969
References
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010