Matrix algebra

Definition

The algebra made of square matrices is typically called the matrix algebra. More precisely, if $F$ is a field (such as the real numbers or the complex numbers) and $n$ is a positive integer, then the matrix algebra over $F$ of size $n \times n$ is denoted by $M (n, F)$ or $M_{n} (F)$ . It consists of all $n \times n$ matrices with entries in $F$ , and the operations of addition and scalar multiplication are defined in the usual way.

Characteristics

It is a central algebra.
...

Action of the projective linear group

Let's focus on $F = C$ . The action of the projective linear group $P G L (n, C)$ on the matrix algebra is given by conjugation. That is, for any $A$ in the matrix algebra and any $g$ in $P G L (n, C)$ , the action of $g$ on $A$ is defined as $g A g^{- 1}$ , where $g^{- 1}$ denotes the inverse of $g$ . This action is well-defined since scalar matrices are in the center of $G L (n, C)$ and are therefore ignored in the quotient.

Automorphisms

The automorphisms of this algebra are all inner automorphism (see here), so they are all elements of $P G L (n, C)$ , because two elements of $G L (n, C)$ give rise to the same inner automorphism if they are multiples of each other by a scalar matrix.

1-parameter group of automorphisms and derivations

Any given 1-parameter group of automorphisms ${φ_{t}}$ is, then, a one-parameter subgroup of $P G L (n, C)$ , and hence is generated by an element of the Lie algebra ${pgl}_{n} (C)$ . So there must exist an $A \in M_{n} (C) ≅ {gl}_{n} (C)$ such that

φ_{t} (B) = \exp (A t) B \exp (- A t),

for an arbitrary matrix $B$ .
The flow $B (t) = φ_{t} (B)$ satisfies $B (0) = B$ and

{\frac{d}{d t} B (t) |}_{t = 0} = A B - B A = [A, B]

So the 1-parameter group of automorphisms ${φ_{t}}$ gives rise to a derivation $D_{φ}$ of the algebra:

B \mapsto D_{φ} (B) := {\frac{d}{d t} B (t) |}_{t = 0}

Conversely, all the derivations on this algebra are inner: for every derivation $D$ it exists a matrix $A$ of size $n \times n$ , such that $D = [-, A]$ .

Then, given a derivation $D$ we can construct a 1-parameter group of automorphisms $φ_{t}^{D}$ in the following way:

φ_{t}^{D} (B) = \exp (A t) B \exp (- A t),

It turns out that

$D_{φ^{D}} = D$ and
$φ^{D_{φ}} = φ$ .

Compare with flow of observables in CM: the algebra is the algebra of smooth functions, the derivations are vector fields.