definition of \(n \times n\) symplectic matrices topological group right action
Topics
About: group
About: topological space
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of \(n \times n\) symplectic matrices topological group right action.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( \mathbb{H}^n\): \(= \text{ the quaternion Euclidean topological space }\), regarded as the set of the rows
\( M_n (\mathbb{H})^{\times}\): \(= \text{ the } n \times n \text{ invertible quaternion matrices topological group }\)
\( f\): \(: \mathbb{H}^n \times M_n (\mathbb{H})^{\times} \to \mathbb{H}^n, (t, M) \mapsto t M\), \(= \text{ the } n \times n \text{ invertible quaternion matrices topological group right action }\)
\( Sp (n)\): \(= \text{ the } n \times n \text{ symplectic matrices topological group }\)
\(*f^`\): \(: \mathbb{H}^n \times Sp (n) \to \mathbb{H}^n\), as the domain restriction of \(f\)
//
Conditions:
//
2: Note
\(f^`\) is indeed a topological group right action, because \(Sp (n)\) is a subgroup topological subspace of \(M_n (\mathbb{H})^{\times}\), and the proposition that for any topological group left or right action, any valid restriction of the action with respect to any subgroup topological subspace and any topological subspace is a topological group left or right action applies.
\(f^`\) is not transitive, because \(Sp (n) (0) = {0} \subset \mathbb{H}^n\), for example.
\(f^`\) is not free, because \(Sp (n)_0 = Sp (n) \neq \{1\}\), for example.
