lie algebras – Direct sum decomposition of an L-module

Source:

Lecture Notes in Kailash C. Misra Lie Algebra

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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.