# Understanding the proof in Humphreys Lie algebras page 37

I've been reading 'Introduction to Lie algebras and representation theory' of Humphreys. I have two questions.

The first question is about Coroallary 8.2 in page 37. Corollary states that "The restriction of the Killing form to the maximal toral subalgebra $$H$$ is nondegenerate." Then, the book says that for each $$\phi \in H^*$$, there exists the unique $$t_\phi \in H$$ such that $$\phi(h)=\kappa(t_\phi, h)$$ for all $$h\in H$$. I think the Cor. 8.2 only gives a reason for the uniqueness of $$t_\phi$$, not existence. How can we know that every linear functional on $$H$$ is represented by the Killing form?

The second question is about the proof of the Proposition 8.3.-(a). (This question is just related to the linear algebra.) The book says that "If $$\Phi$$ fails to span $$H^*$$, then (by duality) there exits nonzero $$h \in H$$ such that $$\alpha(h)=0$$ for all $$\alpha \in \Phi$$." Here's what I understand.

Reduce $$\Phi$$ to be linealry independenet, say $$\Phi'$$. Consider a subset of $$H$$ which is dual to $$\Phi'$$. Since $$\Phi'$$ cannot span $$H^*$$, this subset is not a basis for $$H$$. Extend it to be a basis for $$H$$. For the basis elements $$h_i$$'s which are added while extending, $$\alpha(h_i)=0$$ for all $$\alpha \in \Phi'$$, and hence for all $$\alpha \in \Phi$$.

Am I correct? Or is there any easier way to understand it?

Any help would be appreciated. Thanks.

You have $$\dim H^*=\dim H$$. So, since the map$$\begin{array}{ccc}H&\longrightarrow&H^*\\t&\mapsto&\kappa(\cdot,h)\end{array}$$is injective, it is also surjective.
And if $$\langle\Phi\rangle\ne H^*$$, take $$\beta\in H^*\setminus\{0\}$$ such that $$(\forall\alpha\in\langle\Phi\rangle):\kappa(\alpha,\beta)=0$$. Now, take $$h\in H$$ such that $$\kappa(\cdot,h)=\beta$$. Then, if $$\alpha\in\langle\Phi\rangle$$, $$\alpha(h)=\kappa(\alpha,\beta)=0$$.
• I'm afraid your notations deviate somewhat from the OP's. Is the first map supposed to be $t \mapsto \kappa(t, \cdot)$? And after, I think your $\phi$ is OP's $\kappa$, whereas they use $\phi$ for elements of $H^\ast$. Apr 17, 2021 at 14:45