# Question about definition related to root system of semisimple Lie algebras

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.

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.