Find the dimension of the real Lie algebra su(n)={A∈sln(C)|A+A∗ =0}, A∗ =At. Also have to Show that the map A∗ =At. C⊗R su(n)→sln(C), z⊗A→zA su(n)={A∈sln(C)|A+A∗ =0}, is an isomorphism (of complex vector spaces or complex Lie algebras). The algebra C ⊗R su(n) is called the complexification of the real Lie algebra su(n).
I know that the inverse map is A → 1 ⊗(A−A∗)− i ⊗i(A+A∗). How do I verify this and compare dimensions? Thanks in advance!
By the definition you gave, $\mathfrak{su}(n)$ consists of skew-Hermitian matrices. This vector space has real dimension $n^2-1$, so that we have $$ \dim \mathfrak{su}(n)=n^2-1. $$