# lie algebras – Direct sum decomposition of an L-module

Source:

Lecture Notes in Kailash C. Misra Lie Algebra

a)

Let $$phi: M rightarrow sl (2, mathcal {C})$$ to be defined by

$$E_ {11} -E_ {22} mapsto h$$

$$E_ {12} mapsto e$$

$$E_ {21} mapsto f$$

L is an M-Module under the adjoint action since the adjoint action simply changes the conditions of the L-Module under the conditions of Lie algebra.

b) Here is where I am fighting. Allusions?

• I do not see what the M-sub-modules look like.
• I can not think of a single sub-module decomposition of L.