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Let $L$ be a semisimple Lie algebra of finite dimension over a field of charcteristic 0 and algebraically closed, and $H$ a maximal toral subalgebra. Let $R$ be the set of roots of $L$ with respect to $H$.

In the statement of the theorem I am reading says:

$\mathbb{Z}[R] := Span_{\mathbb{Z}}R \subseteq H^*$ is a complete lattice.

I am confused with what this notation $\mathbb{Z}[R]$. Is this implying that the $\mathbb{Z}$-linear combination of elements of $R$ is actually an algebra contained in $H^*$? (and so every element is actually a polynomial in the elements of $R$?) Any clarifications would be appreciated. Thank you.

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The definition is $\mathbb{Z}[R]:=Span_{\mathbb{Z}}R$ (as you have written in your post). Elements of $\mathbb{Z}[R]$ are $\mathbb{Z}$-linear combinations of elements of $R$.

There is no multiplication and this is not to be confused with polynomial functions on $R^*$. The notation is just unfortunate.

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  • $\begingroup$ Thank you! I was starting to suspect maybe it was just a rather confusing notation.. $\endgroup$ – Johnny T. Mar 7 at 19:38

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