Group symplectic

In Mathematical, the term group symplectic is used to designate two families different of linear groups. They are noted Sp (2 N , E ) and Sp ( N ), this last being sometimes named compact group symplectic to distinguish it from the first. It should be noted that this notation does not achieve the unanimity and that certain authors use others of them, generally differing from a factor 2. The notation used in this article is in connection with the size of the matrices representing the groups.

Sp (2 N , E )

The group symplectic of degree 2 N on a body E , noted Sp (2 N , E ), is the group symplectic matrices 2n × 2n with coefficients in E , provided with the matric multiplication. As all the symplectic matrices have for determinant 1, the group symplectic is a Sous-groupe linear special Groupe SSL (2 N , E ).

In a more abstract way, the group symplectic can be defined like the whole of the linear transformations of a vector Space of dimension 2 N on E preserving the not-degenerated, antisymmetric and bilinear forms.

If N = 1, the symplectic condition on a matrix is satisfied if and only if its determinant is such as Sp (2, E ) = SSL (2, E ). For N >1, other conditions are added to it.

Typically, the body E is the body of the real numbers $\ mathbb R$ or of the complex numbers $\ mathbb C$ . In this case, Sp (2 N , E ) is a Groupe of Dregs real or complex, of real or complex size N (2 N + 1). These groups are related but not compact. Sp (2 N , $\ mathbb C$ ) is simply related while Sp (2 N , $\ mathbb R$ ) has a fundamental Groupe isomorphous with Z .

The Algèbre of Dregs of Sp (2 N , E ) is given by the whole of matrices 2 N ×2 real N or complex has satisfactory:

JA + A^T J = 0

where

A^T

is transposed of has and J is the antisymmetric matrix

J =

\begin{pmatrix} 0 & I_n \ \ - I_n & 0 \ \ \end{pmatrix}

Sp ( N )

The group symplectic Sp ( N ) is the sub-group of GL (N,

\ mathbb H

) (

\ mathbb H

being the whole of the quaternionic matrices invertible) preserving the standard square form on

\ mathbb H^n

\ langle X, there \ rangle = \ bar x_1 y_1 + \ cdots + \ bar x_n y_n

I.e. Sp ( N ) is simply the unit Groupe quaternionic U ( N ,

\ mathbb H

). It is sometimes called besides group hyperunitaire . Sp ( N ) is not a group symplectic within the meaning of the preceding section: it does not preserve an antisymmetric form on

\ mathbb H^n

(makes some, such a form does not exist).

Sp ( N ) is a group of Dregs of dimension N (2 N + 1). It is compact, related and simply related. The algebra of Dregs of Sp ( N ) is given by the whole of the quaternionic matrices N × satisfactory N

A+A^ {\ dagger} = 0

where

A^ {\ dagger}

is transposed combined has .

Relations between the groups symplectic

The relation between the groups Sp (2 N ,

\ mathbb R

), Sp (2 N ,

\ mathbb C

) and Sp ( N ) is most obvious on the level of their algebra of Dregs. The algebras of Dregs of these three groups, considered as real groups of Dregs, divide same the Complexification. In the classification of the simple algebras of Dregs of Cartan, this algebra is noted C _N.

The algebra of Dregs complexes C _N is right the algebra sp (2 N , $\ mathbb C$ ) of the complex groups of Dregs Sp (2 N , $\ mathbb C$ ). This algebra has two different real forms:

the compact form, sp ( N ), which is the algebra of Dregs of Sp ( N ),
the normal form, sp (2 N , $\ mathbb R$ ), which is the algebra of Dregs of Sp (2 N , $\ mathbb R$ ).

Comparison of the groups symplectic:

See too

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