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Sorry to ask what is almost certainly a very trivial question, but in Fulton and Harris's first course in representation theory they write down a property of root lattices which I think must be sort of wrong.

In lecture 12 they are talking about representations of $\mathfrak{sl}_3(\mathbb{C})$, and specifically show that the root lattices (lying in the plane $\mathfrak{h}^\star$) are symmetric under reflection in the line given by $\langle H_{1,2},L\rangle = 0$ where $L$ is some functional in $\mathfrak{h}^\star$. This is really confusing me (since they write a variant on this equation a few times) and it really seems to me that $H_{1,2}$ is not in $\mathfrak{h}^\star$ but in $\mathfrak{h}$, so this inner product is meaningless. If they were talking about the dual element this is the right equation, but it just seems like they've changed notation for a reason, since the dual element to this one has been written down many many times.

Is this just another example of the imprecision of this book, or am I missing something? Thanks for the help.

CameronJWhitehead
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1 Answers1

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Let us have a finite dimensional Hilbert space $\mathfrak{h}$ (with an inner product $\langle a,b \rangle$) and its dual $\mathfrak{h}^*$.

Then $f_a(b)=\langle a,b\rangle \in \mathbb{R}$ is a linear operator.

In finite dimensional spaces those are exactly all the linear operators on $\mathfrak{h}$ wich means it is self-dual.

Thus the inner product can be defined on the dual space and with a slight abuse of notation unless specifically explained applied to elements of the the two spaces.

Mr.P
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  • I guess, maybe I'm just being picky but I think it's a bad abuse since $H_{1,2}$ and $L$ don't lie in the same space. Thanks for clarifying though! – CameronJWhitehead Aug 14 '15 at 21:49