# property of adjoint

Let $\mathfrak{g}$ be real Lie algebra of matrices over $\mathbb{C}$ that is closed under conjugate transpose $(\cdot)^*$. Why is it the case that for each $X \in \mathfrak{g}$, $\mathrm{ad}(X^*) = \mathrm{ad}(X)^*$? We are using the inner product $<X,Y> := \mathrm{Re} \mathrm{tr}(XY^*)$ on $\mathfrak{g}$.

• What is $\text{ad}(X)^{\ast}$? If it's the adjoint with respect to some inner product on matrices, which inner product? – Qiaochu Yuan Jul 12 '17 at 21:43
• $<X,Y> := \text{Re} \text{tr}(XY*)$. – nigel Jul 12 '17 at 22:04
• @nigel And $\mathrm{Retr}$ is... ? – José Carlos Santos Jul 12 '17 at 22:09

## 1 Answer

Just write out the definitions and simplify. The definition of adjoints with respect to an inner product says that the equation you want is equivalent to saying that, for all $Y$ and $Z$, $$\langle\text{ad}(X^*)(Y),Z\rangle=\langle Y,\text{ad}(X)(Z)\rangle.$$ Now use the definition of ad to rewrite this as $$\langle[X^*,Y],Z\rangle=\langle Y,[X,Z]\rangle.$$ Next insert the definition of your inner product to rewrite this as $$\text{Retr}([X^*,Y]\cdot Z^*)=\text{Retr}(Y,[X,Z]^*).$$ Next use the fact that, in a Lie algebra of matrices, the Lie bracket is the ordinary commutator, and remember that the transpose of a product is the product of the transposes in the reverse order, to put the desired equation into the equivalent form $$\text{Retr}(X^*YZ^*-YX^*Z^*)=\text{Retr}(YZ^*X^*-YX^*Z^*).$$ Next, note that Retr must mean real part of the trace (not real part of the transpose, which can't be an inner product because it isn't a scalar). Also remember that the trace is linear, so that your desired equation amounts to $$\text{Retr}(X^*YZ^*)-\text{Retr}(YX^*Z^*)= \text{Retr}(YZ^*X^*)-\text{Retr}(YX^*Z^*).$$ Finally, remember that the trace of a matrix product $AB$ is the same as the trace of $BA$. Apply that with $A=X^*$ and $B=YZ^*$ to see that the first terms on the two sides of your desired equation match. The second terms are identical, so the equation holds.

• I suspect that what makes this question confusing is that, in the equation $\text{ad}(X^*)=\text{ad}(X)^*$, the star on the left side means the conjugate transpose of the matrix $X$whereas the star on the right side means the adjoint with respect to your inner product. – Andreas Blass Jul 12 '17 at 23:23