# Simple modules for direct sum of simple Lie algebras

I think that the following statement is true, but I do not know how to prove it.

Let $$\mathfrak{g}_1$$ and $$\mathfrak{g}_2$$ be two real simple Lie algebras. If $$M$$ is a (infinite dimensional) complex simple $$\mathfrak{g}_1\oplus\mathfrak{g}_2$$-module, then $$M\cong M_1\otimes M_2$$ where $$M_1$$ and $$M_2$$ are simple modules of $$\mathfrak{g}_1$$ and $$\mathfrak{g_2}$$ respectively.

This ought to be a well known result if it is true. As is known to all, it is true for finite groups and compact groups.

• I assume you want $M$ to be a module for $\mathfrak{g}_1 \oplus \mathfrak{g}_2$? Mar 15, 2019 at 6:59
• @user44191 Yes, it is what I mean.
– Hebe
Mar 15, 2019 at 8:50

This is true more generally for any two finite-dimensional Lie algebras over a field. Recall that a $$\frak{g}$$-module is the same as a module over $$U (\frak{g})$$, the universal enveloping algebra of $$\frak{g}$$, which is an associative algebra. Moreover, $$U ({\frak{g}_1\oplus\frak{g}_2})\simeq U({\frak{g}_1})\otimes U ({\frak{g}_2})$$, so this question reduces to showing that a simple module $$M$$ over the tensor product $$A_1\otimes A_2$$ decomposes as $$M_1\otimes M_2$$, where $$M_i$$ is a simple module over a universal enveloping algebra $$A_i$$. If my memory serves, there is proof along these lines in Dixmier, "Universal enveloping algebras". Another standard reference is McConnell and Robson, "Noncommutative Noetherian rings".