Problem 4.4 - Lie Algebras - Humphreys

Question

I read the exercise 4.4 in the book Introduction to Lie algebras and representation theory of J. Humphreys, and I do not quite understand the sentence :

We start with $L\leq\mathfrak{gl}(p,F)$ as in Exercise 4.3, and let $M:=L+F^p$, the direct sum. We then make $M$ into a Lie algebra by decreeing that $F^p$ is abelian, while $L$ has its usual product and acts on $F^p$ in the given way.

Could some one tell me what the Lie bracket inside $M$ is? Many thanks in advance.

Answer 1

The answer is given in Jacobson's book on Lie algebras. $M$ is the split extension of $L$ and $F^p$. That is, the Lie bracket is given by $$ [m_1,m_2]=[l_1+f_1,l_2+f_2]=[l_1,l_2]+l_1.f_2-l_2.f_1+[f_1,f_2] $$ for $l_1,l_2\in L$ and $f_1,f_2\in F^p$. We have $[f_1.f_2]=0$ for all $f_1,f_2$, since $F^p$ is an abelian Lie algebra. The action of $L$ on $F^p$ is given in the exercise before. If $e_1,\ldots e_n$ is a basis of $F^p$, then $E.e_i=e_{i+1}$ and $E.e_p=e_1$ and $F.e_i=(i-1)e_i$, where $L=\langle E,F\rangle$.