Symplectic group

The symplectic group is a term from mathematics , in the overlapping area of ​​the areas of linear algebra and group theory . It is the set of linear mappings that leave a symplectic form , that is, a non-degenerate alternating bilinear form , invariant, just as the orthogonal group of length-conforming maps leaves a non-degenerate, symmetric bilinear form invariant. The symplectic group in dimensions is a semi-simple group to the root system C _n . It plays an important role in the study of symplectic vector spaces . ${\displaystyle 2n}$

The Lie group is also known as a (compact) symplectic group. ${\displaystyle Sp(n)}$

definition

For each and every body with characteristics other than two, the symplectic group is a subgroup of the general linear group ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle K}$ ${\displaystyle Sp_{2n}(K)}$ ${\displaystyle \mathrm {GL} (2n,K)}$

${\displaystyle Sp_{2n}(K)\colon =\left\{T\in GL_{2n}(K)\mid \,T^{\text{T}}\,I_{n}\,T=I_{n}\right\}}$

With

${\displaystyle I_{n}={\begin{pmatrix}0&E_{n}\\-E_{n}&0\end{pmatrix}}}$

where the identity matrix and 0 denotes the n x n zero matrix . ${\displaystyle E_{n}}$ ${\displaystyle n\times n}$

For is a Lie group and the Lie algebra of is ${\displaystyle K\in \left\{\mathbb {R} ,\mathbb {C} ,\mathbb {H} \right\}}$ ${\displaystyle Sp(n,K)}$ ${\displaystyle \mathrm {Sp} (2n,K)}$

${\displaystyle {\mathfrak {sp}}(2n,K)=\left\{A\in \mathrm {Mat} (2n,K):I_{n}A+A^{T}I_{n}=0\right\}}$.

Finite groups

If the body is finite with elements, one writes instead of . You get a finite group with ${\displaystyle K}$ ${\displaystyle q}$ ${\displaystyle \mathrm {Sp} (2n,q)}$ ${\displaystyle \mathrm {Sp} (2n,K)}$

${\displaystyle \mathrm {ord} (Sp(2n,q))=q^{n^{2}}\prod _{i=1}^{n}(q^{2i}-1)}$

Elements. The center of this group consists of , therefore it has two elements for odd and is trivial for even . ${\displaystyle \pm \mathrm {id} _{K^{n}}}$ ${\displaystyle q}$ ${\displaystyle q}$

Projective symplectic groups

The factor groups of the symplectic groups according to their center are called projective symplectic groups and are denoted by. In the case of a finite field with elements, is ${\displaystyle PSp(2n,K)}$ ${\displaystyle q}$

${\displaystyle \mathrm {ord} (PSp(2n,q))={\frac {q^{n^{2}}}{\mathrm {ggT} (2,q-1)}}\prod _{i=1}^{n}(q^{2i}-1)}$

and the groups are simple with the exception of and . This gives an infinite series of simple groups. These are groups of Lie type C _n and thus one of a total of 16 infinite series of groups of Lie type. Therefore is also referred to with . ${\displaystyle PSp(2,2),PSp(2,3)}$ ${\displaystyle PSp(4,2)}$ ${\displaystyle PSp(2n,q)}$ ${\displaystyle C_{n}(q)}$

Compact symplectic group

The compact symplectic group is the group of (invertible) quaternionic- linear mappings, which are the scalar product defined on the n-dimensional quaternionic vector space ${\displaystyle Sp(n)}$ ${\displaystyle \mathbb {H} ^{n}}$

${\displaystyle \langle x,y\rangle ={\bar {x}}_{1}y_{1}+\cdots +{\bar {x}}_{n}y_{n}}$

receive.

This group is not a symplectic group in the sense of the previous section. but is the compact real form of . ${\displaystyle Sp(n)}$ ${\displaystyle Sp(2n,\mathbb {C} )}$

${\displaystyle Sp(n)}$is a -dimensional compact Lie group and is simply connected . Your Lie algebra is ${\displaystyle n(2n+1)}$

${\displaystyle {\mathfrak {sp}}(n)=\left\{A\in \mathrm {Mat} (n,\mathbb {H} ):A+A^{\dagger }=0\right\}}$,

where denotes the quaternionic conjugate transposed matrix. ${\displaystyle A^{\dagger }}$

It applies . ${\displaystyle Sp(n)=U(2n)\cap Sp(2n,\mathbb {C} )}$

Although finite sets are also compact, compact symplectic groups usually refer to the Lie groups given here.

literature

• Vladimir L. Popov: Symplectic group . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ^ Roger W. Carter: Simple Groups of Lie Type , John Wiley & Sons 1972, ISBN 0-471-13735-9 , Chapter 1.3: The Symplectic Groups

Web links

• Eric W. Weisstein : Symplectic Group . In: MathWorld (English).