Let $\mathbb{G}\_m$ be the multiplicative group and $T$ a maximal torus of a semisimple group. Let $X^\*(T)=\{ \phi: T \to \mathbb{G}\_m \}$ be the set of characters and $X\_\*(T)=\{ \phi^{\vee}: \mathbb{G}\_m \to T \}$ the set of co-characters. Let
\begin{align}
L(T) = \{ \phi: \mathbb{G}\_m \to T \},
\end{align}
and 
\begin{align}
L\_0(T) = \{ \phi \in L(T): \forall \mu \in X^\*(T), \mu \circ \phi \in L\_0(c) \},
\end{align}
where $L\_0(c)$ is the set of all invertible Taylor series in $c$.


In the paper [Geometric and unipotent crystals](https://arxiv.org/pdf/math/9912105.pdf), Section 2.4, page 7, Lemma 2.6, it is said that the multiplicative map $X\_\*(T) \times L\_0(T) \to L(T)$ is a bijection. 


How to show that an element $\phi$ in $L(T)$ can be decompose as a product $\phi = \phi' \phi''$, where $\phi' \in X\_\*(T)$ and $\phi'' \in L\_0(T)$? Thank you very much.



How to decompose an map $\phi: \mathbb{G}_m \to T$ as the product of a cocharacter $\phi'$ and a map $\phi'':\mathbb{G}_m \to T$?
Although the notation and explanation of the paper is terrible, it's not supposed to be a very deep fact.


Indeed, let's assume for simplicity that $T = \mathbb G\_m^n$ is a *split* torus. Then arbitrary morphisms of schemes $\mathbb G\_m \to T$ correspond to $n$-tuples $(f\_1,\ldots,f\_n) \in (k[x^{\pm 1}]^\times)^n$. Under this correspondence, the characters correspond to the monomials
$$\{(x^{a\_1},\ldots,x^{a\_n})\ |\ a\_1,\ldots,a\_n \in \mathbb Z \}.$$
What the authors mean by *formal loops* $\phi \colon \mathbb G\_m \to T$ is morphisms $\operatorname{Spec} k((x)) \to T$, where $\operatorname{Spec} k((x))$ can be thought of as $\operatorname{Spec}\left(k[x^{\pm 1}] \otimes\_{k[x]} k[[x]]\right)$ (i.e. $\mathbb G\_m \times\_{\mathbb A^1} \hat{\mathbb A}^1$; the 'formal completion of $\mathbb G\_m$ at the origin'). This is an algebraic geometry version of looking at very small loops.


Algebraically, the elements of $L(T)$ are given by $n$-tuples $(f\_1,\ldots,f\_n) \in (k((x))^\times)^n$. The condition on $L\_0(T)$ is that the $f\_i$ are invertible elements of the power series ring $k[[x]]$.


Finally, the claim they make just boils down to the statement that any $(f\_1,\ldots,f\_n) \in (k((x))^\times)^n$ can be written uniquely as a product
$$(f\_1,\ldots,f\_n) = (g\_1,\ldots,g\_n) \times (h\_1,\ldots,h\_n),$$
where the $g\_i$ are monomials (coming from $X\_\*(T)$) and the $h\_i$ are invertible elements of $k[[x]]$ (coming from $L\_0(T)$). Explicitly, if
$$f\_i = \sum\_{j\in\mathbb Z} a\_j x^j,$$
then $g\_i = x^{j\_0}$ for the minimal $j\_0$ such that $a\_{j\_0} \neq 0$, and
$$h\_i = \frac{f\_i}{g\_i} = \sum\_{j = 0}^\infty a\_{j+j\_0} x^j$$
is invertible in $k[[x]]$ since its constant term $a\_{j\_0}$ is nonzero.


