Symplectic group
The symplectic group is a term from mathematics, in the area of overlap of the territories linear algebra and group theory. It is the set of linear maps which, like the orthogonal group of length-preserving pictures a nondegenerate symmetric bilinear form can be a symplectic form, ie have a nondegenerate alternating bilinear form invariant invariant. The symplectic group in dimension is a semisimple group for the root system Cn. It plays an important role in the study of symplectic vector spaces.
The Lie group is called a ( compact ) symplectic group.
Definition
For each and every field F with characteristic not equal to two, the symplectic group is a subgroup of the general linear group GL (2n, F).
With
The unit matrix and 0 is the nxn zero matrix respectively.
For a Lie group and the Lie algebra of Sp ( 2n, F) is
Compact symplectic group
The compact symplectic group is the group of ( invertible ) quaternionisch - linear maps that the defined on the quaternionic vector space of n- dimensional scalar
Receive.
This group is not a symplectic group in the sense of the previous section. but is the compact real form of.
Is one -dimensional compact Lie group and simply connected. Your Lie algebra is
The quaternionisch - conjugate transpose matrix called.
It is true.