While preparing for a test I found the next question which i cannot fully answer:
Assume $k$ is an algebraically closed field, and $g_{1},g_{2}$ are $k$-Lie algebras and let $g=g_{1}\times g_{2}$. The first part of the question asks: If $V_{1},\, V_{2}$ are irreducible $g_{1},\, g_{2}$ modules (respectively) then $V=V_{1}\otimes V_{2}$ is an irreducible $g$-module. That I could do.
The second part however asks: Show that any ireducible g-module is isomorphic to some $V_{1}\otimes V_{2}$ as above. This I don't know how to prove.
There is also a question in the end which asks what is the difference if $k$ is not algebraically closed.
Thanks!