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.