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 .
The Lie group is also known as a (compact) symplectic group.
definition
For each and every body with characteristics other than two, the symplectic group is a subgroup of the general linear group
With
where the identity matrix and 0 denotes the n x n zero matrix .
For is a Lie group and the Lie algebra of is
- .
Finite groups
If the body is finite with elements, one writes instead of . You get a finite group with
Elements. The center of this group consists of , therefore it has two elements for odd and is trivial for even .
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
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 .
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
receive.
This group is not a symplectic group in the sense of the previous section. but is the compact real form of .
is a -dimensional compact Lie group and is simply connected . Your Lie algebra is
- ,
where denotes the quaternionic conjugate transposed matrix.
It applies .
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
- ^ 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).