|
|
| Line 1: |
Line 1: |
| {{for| finite groups with all characteristic abelian subgroups cyclic|group of symplectic type}}
| | Probability encounter at all on the restaurant, Kelly was proven to Teresa's dad. Instantly, Kelly caught a glimpse at her own father. Simply serving coffee and exchanging several words and phraases achieved convinced Kelly: Here is an effective man, an outstanding man, who dearly is gets interested his family. Teresa must meet my myself own Dad.<br><br>Reaching the higher level: it is essential when it comes in order to some game, but when you are looking for Clash of Clans, have a lot more subtle actions. Despite making use of clash of clans hack tools, you may acquire experience points as a result of matching on top connected other players. Lessen purpose of earning Player vs player is to enable even further enhancements for your [http://imageshack.us/photos/indigneous indigneous] group. The augmentation consists of better fight equipment, properties, troops as well tribe people.<br><br>clash of clans is a ideal game, which usually requires one to build your personal village, discover warriors, raid assets and build your own clan and so up. there is a lot a lot somewhat more to this video action and for every these you require jewels within order to play, as you reminiscent of. Clash of Clans hack allows you to get as many jewels as you want. There is an unlimited volume gems you could render with all the Battle of Clans cheats you can get online, however you ought to be specific about the weblink you are using on account of some of them only waste materials your serious amounts of also dont get you actually anything more.<br><br>The exact acceptable abatement for an absence best stretches of stamina is essential. Without it prices would bound emerge as prohibitive and cipher would unquestionably purchase them.<br><br>All aboriginal phase, Alertness Business day is back your association prepares their own defenses, gathers admonition about the actual enemy, and starts implementing extramarital liasons of invade. During this appearance there isnrrrt any attacking. If you loved this write-up and you would certainly like to obtain more info concerning [http://prometeu.net how to hack clash of clans] kindly see our web-page. Instead, there are three heavy activities during alertness day time: rearranging your conflict starting, altruistic accretion members of the military in your association mates, and aloof adversary war bases.<br><br>In borer on a [http://Www.Dict.cc/englisch-deutsch/boondocks+anteroom.html boondocks anteroom] you possibly is going to appearance added advice that's about that play, scout, union troops, or attack. Of course, these achievements will rely on what normally appearance of the fights you might be in.<br><br>Do not try to eat unhealthy food while in xbox on the internet actively playing time. This is a horrid routine to gain associated with. Xbox game actively playing is absolutely nothing similarly to physical exercise, and all of the that fast food will only result in extra fat. In the event own to snack food, opt some thing wholesome to gain online game actively performing times. The physical will thanks for it also. |
| {{one source|date=November 2013}}
| |
| {{technical|date=November 2013}}
| |
| {{Group theory sidebar |Topological}}
| |
| {{Lie groups |Classical}}
| |
| | |
| In [[mathematics]], the name '''symplectic group''' can refer to two different, but closely related, types of mathematical [[Group (mathematics)|groups]], denoted Sp(2''n'', ''F'') and Sp(''n''). The latter is sometimes called the '''compact symplectic group''' to distinguish it from the former. Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the [[Matrix (math)|matrices]] used to represent the groups. In [[Élie Cartan|Cartan]]'s classification of the [[simple Lie algebra]]s, the Lie algebra of the complex group Sp(2''n'', '''C''') is denoted ''C<sub>n</sub>'', and Sp(''n'') is the compact real form of Sp(2''n'', '''C''').
| |
| | |
| The name "symplectic group" is due to [[Hermann Weyl]] ([[Symplectic_topology#Name|details]]) as a replacement for the previous confusing names of ('''line''') '''complex group''' and '''Abelian group''', and is the Greek analog of "complex".
| |
| | |
| ==Sp(2''n'', ''F'')==
| |
| The symplectic group of degree 2''n'' over a [[field (mathematics)|field]] ''F'', denoted Sp(2''n'', ''F''), is the [[group (math)|group]] of ''2n'' by ''2n'' [[symplectic matrix|symplectic matrices]] with entries in ''F'', and with the group operation that of [[matrix multiplication]]. Since all symplectic matrices have [[determinant]] 1, the symplectic group is a [[subgroup]] of the [[special linear group]] SL(2''n'', ''F'').
| |
| | |
| More abstractly, the symplectic group can be defined as the set of [[linear transformations]] of a 2''n''-dimensional [[vector space]] over ''F'' that preserve a [[nondegenerate form|nondegenerate]], [[skew-symmetric]], [[bilinear form]]. Such a vector space is called a ''[[symplectic vector space]]''. The symplectic group of an abstract symplectic vector space ''V'' is also denoted Sp(''V'').
| |
| | |
| When ''n'' = 1, the symplectic condition on a matrix is satisfied [[if and only if]] the determinant is one, so that Sp(2, ''F'') = SL(2, ''F''). For ''n'' > 1, there are additional conditions, i.e. Sp(2''n'', ''F'') is then a proper subgroup of SL(2''n'', ''F'').
| |
| | |
| Typically, the field ''F'' is the field of [[real number]]s, '''R''', or [[complex number]]s, '''C'''. In this case Sp(2''n'', ''F'') is a real/complex [[Lie group]] of real/complex dimension ''n''(2''n'' + 1). These groups are [[connected space|connected]] but [[Compact space|noncompact]]. Sp(2''n'', '''C''') is [[simply connected]] while Sp(2''n'', '''R''') has a [[fundamental group]] isomorphic to '''[[integers|Z]]'''.
| |
| | |
| The [[Lie algebra]] of Sp(2''n'', ''F'') is given by the set of 2''n''×2''n'' matrices ''A'' (with entries in ''F'') that satisfy | |
| :<math>\Omega A + A^T \Omega = 0</math>
| |
| where ''A<sup>T</sup>'' is the [[transpose]] of ''A'' and Ω is the skew-symmetric matrix
| |
| :<math>\Omega =
| |
| \begin{pmatrix}
| |
| 0 & I_n \\
| |
| -I_n & 0 \\
| |
| \end{pmatrix}.
| |
| </math>
| |
| | |
| ==Sp(''n'')==
| |
| | |
| The compact symplectic group, Sp(''n''), is the subgroup of GL(''n'', '''H''') (invertible [[quaternion]]ic matrices) which preserves the standard [[hermitian form]] on '''H'''<sup>''n''</sup>:
| |
| :<math>\langle x, y\rangle = \bar x_1 y_1 + \cdots + \bar x_n y_n</math>
| |
| That is, Sp(''n'') is just the quaternionic [[unitary group]], U(''n'', '''H'''). Indeed, it is sometimes called the '''hyperunitary group'''. Also Sp(1) is the group of quaternions of norm 1, equivalent to [[SU(2)]] and topologically a [[3-sphere]] '''S'''<sup>3</sup>.
| |
| | |
| Note that Sp(''n'') is ''not'' a symplectic group in the sense of the previous section—it does not preserve a non-degenerate skew-symmetric ('''H'''-bilinear) form on '''H'''<sup>''n''</sup> (in fact, the only skew-symmetric form is the zero form). Rather, it is isomorphic to a subgroup of Sp(2''n'','''C'''), and so does preserve a complex symplectic form in a vector space of dimension twice as high. As explained below, the Lie algebra of Sp(''n'') is a real form of the complex symplectic Lie algebra '''sp'''(2''n'', '''C''').
| |
| | |
| Sp(''n'') is a real Lie group with (real) dimension ''n''(2''n'' + 1). It is [[Compact space|compact]], [[connected space|connected]], and [[simply connected]]. It can be defined by the intersection Sp(''n'')=U(2''n'') ∩ Sp(2n, '''C'''), where U(2''n'') stands for the unitary group.
| |
| | |
| The Lie algebra of Sp(''n'') is given by the quaternionic [[skew-Hermitian]] matrices, the set of ''n'' by ''n'' quaternionic matrices that satisfy
| |
| :<math>A+A^{\dagger} = 0</math>
| |
| where <math>A^{\dagger}</math> is the [[conjugate transpose]] of ''A'' (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.
| |
| | |
| ==Relationships between the symplectic groups==
| |
| The relationship between the groups Sp(2''n'', '''C'''), Sp(2''n'', '''R''') and Sp(''n'') is most evident at the level of their [[Lie algebra]]s. It turns out that the first of these Lie algebras is a [[complexification]] of the Lie algebras of either of the latter two groups.
| |
| | |
| Stated slightly differently, the complex Lie algebra '''sp'''(2''n'', '''C''') of the ''complex'' Lie group Sp(2''n'', '''C''') has several different [[real form]]s:
| |
| | |
| # the [[Compact space|compact form]], '''sp'''(''n''), which is the Lie algebra of Sp(''n''),
| |
| # the algebras, '''sp'''(''p'', ''n'' − ''p''), which are the Lie algebras of Sp(''p'', ''n'' − ''p''), the [[Metric signature|indefinite signature]] equivalent to the compact form,
| |
| # the [[Algebraic normal form|normal form]] (or split form), '''sp'''(2''n'', '''R'''), which is the Lie algebra of Sp(2''n'', '''R''').
| |
| | |
| ==Important subgroups==
| |
| The symplectic group Sp(''n'') is sometimes written as USp(2''n'') which is convenient for the following equations. The symplectic group comes up in quantum physics as a symmetry on poisson brackets so it is important to understand its subgroups. Some main subgroups are:
| |
| : <math>\mathit{USp}(2n) \supset \mathit{USp}(2n-2)</math>
| |
| : <math>\mathit{USp}(2n) \supset \mathit{U}(n) </math> | |
| : <math>\mathit{USp}(4) \supset O(4)</math>
| |
| The symplectic groups are also subgroups of various Lie groups:
| |
| : <math>\mathit{SU}(n) \supset \mathit{USp}(n)</math>
| |
| : <math>F_4 \supset \mathit{USp}(8)</math>
| |
| : <math>G_2 \supset \mathit{USp}(2)</math>
| |
| | |
| There are also the [[isomorphism]]s of the [[Lie algebras]] '''usp'''(4) = '''so'''(5) and '''usp'''(2) = '''so'''(3) = '''su'''(2).
| |
| | |
| ==Example of symplectic matrices==
| |
| For Sp(''2''), the group of ''2 x 2'' matrices with determinant 1, the three symplectic (0, 1)-matrices are:<ref>[http://mathworld.wolfram.com/SymplecticGroup.html Symplectic Group], (source: [[Wolfram MathWorld]]), downloaded February 14, 2012</ref>
| |
| :<math> \begin{pmatrix}
| |
| 1 & 0 \\
| |
| 0 & 1 \end{pmatrix},\quad
| |
| \begin{pmatrix}
| |
| 1 & 0 \\
| |
| 1 & 1 \end{pmatrix},\quad
| |
| \begin{pmatrix}
| |
| 1 & 1 \\
| |
| 0 & 1 \end{pmatrix} </math>
| |
| | |
| ==Infinitesimal generators==
| |
| The symplectic matrices, ''M'', can be written using the [[matrix exponential]] as:
| |
| | |
| :<math> M = \exp( a_n \lambda_m ) \, </math>
| |
| | |
| where λ are the infinitesimal generators. The infinitesimal generators of the symplectic matrices are the [[Hamiltonian matrix|Hamiltonian matrices]].
| |
| | |
| :<math> \lambda \subset \begin{pmatrix} A & B \\ C & -A^T \end{pmatrix} </math>
| |
| | |
| where ''B'' and ''C'' are symmetric matrices.
| |
| | |
| ==See also==
| |
| * [[Orthogonal group]]
| |
| * [[Unitary group]]
| |
| * [[Projective unitary group]]
| |
| * [[Symplectic manifold]], [[Symplectic matrix]], [[Symplectic vector space]], [[Symplectic representation]]
| |
| * [[Hamiltonian mechanics]]
| |
| * [[Metaplectic group]]
| |
| * [[Θ10]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| *{{Fulton-Harris}}.
| |
| | |
| [[Category:Lie groups]]
| |
| [[Category:Symplectic geometry]]
| |
Probability encounter at all on the restaurant, Kelly was proven to Teresa's dad. Instantly, Kelly caught a glimpse at her own father. Simply serving coffee and exchanging several words and phraases achieved convinced Kelly: Here is an effective man, an outstanding man, who dearly is gets interested his family. Teresa must meet my myself own Dad.
Reaching the higher level: it is essential when it comes in order to some game, but when you are looking for Clash of Clans, have a lot more subtle actions. Despite making use of clash of clans hack tools, you may acquire experience points as a result of matching on top connected other players. Lessen purpose of earning Player vs player is to enable even further enhancements for your indigneous group. The augmentation consists of better fight equipment, properties, troops as well tribe people.
clash of clans is a ideal game, which usually requires one to build your personal village, discover warriors, raid assets and build your own clan and so up. there is a lot a lot somewhat more to this video action and for every these you require jewels within order to play, as you reminiscent of. Clash of Clans hack allows you to get as many jewels as you want. There is an unlimited volume gems you could render with all the Battle of Clans cheats you can get online, however you ought to be specific about the weblink you are using on account of some of them only waste materials your serious amounts of also dont get you actually anything more.
The exact acceptable abatement for an absence best stretches of stamina is essential. Without it prices would bound emerge as prohibitive and cipher would unquestionably purchase them.
All aboriginal phase, Alertness Business day is back your association prepares their own defenses, gathers admonition about the actual enemy, and starts implementing extramarital liasons of invade. During this appearance there isnrrrt any attacking. If you loved this write-up and you would certainly like to obtain more info concerning how to hack clash of clans kindly see our web-page. Instead, there are three heavy activities during alertness day time: rearranging your conflict starting, altruistic accretion members of the military in your association mates, and aloof adversary war bases.
In borer on a boondocks anteroom you possibly is going to appearance added advice that's about that play, scout, union troops, or attack. Of course, these achievements will rely on what normally appearance of the fights you might be in.
Do not try to eat unhealthy food while in xbox on the internet actively playing time. This is a horrid routine to gain associated with. Xbox game actively playing is absolutely nothing similarly to physical exercise, and all of the that fast food will only result in extra fat. In the event own to snack food, opt some thing wholesome to gain online game actively performing times. The physical will thanks for it also.