The same condition without requiring that gives the complex symplectic group
Defined using the quaternions
Since the quaternions define a division algebra, then we can look at the -linear maps.
Using the vector space isomorphism .
Thus, the -linear condition is the same as
So must be linear and commute with the “action” of (using left multiplication).
This action of is the same as the following map
However, it is important to note that this map is -linear, not -linear.
That means, that it cannot be represented using matrices.
The condition that commutes with means that is of the form
This can be seen with a direct computation, or instead, we can remember that .
Therefore, right multiplication by can be represented by the map
This is -linear and can be represented with the block matrix
which is invariant under conjugation.
Using this to rephrase both sides of the -linearity condition gives
Therefore, for a unitary matrix (, i.e. ), then if is the matrix representation of , and is the conjugation function on then the condition that commutes with is equivalent to
This corresponds to the matrices which are -linear which preserve the norm coming naturally from the inner product on using conjugation.