User:Hillgentleman/Myberg's theorem

Myberg's theorem is a theorem about quasi-abelian functions in several complex variables. It is a key step in the Belavin-Drinfeld classification of triangular r-matrices.

Let

$P:\mathbb {C} ^{m}--->\mathbb {C} ^{m}$ be a rational function
$f:\mathbb {C} --->\mathbb {C} ^{m}$ $f:\mathbb {C} --->\mathbb {C} ^{m}$ be a meromorphic function:
- $f(u+v)=P(f(u),f(v))$

Then

There exists
- n>0 - integer
- ${\tilde {f}}:\mathbb {C} ^{n}--->\mathbb {C} ^{m}$ - quasiabelian function
- $\mathbf {a} \in \mathbb {C} ^{n}$ -vector
such that
- $f(u)={\tilde {f}}(u\mathbf {a} )$ ;
and, further,
- f can be chosen so that
  - for every $\mathbf {x} ,\mathbf {y} \in \mathbb {C} ^{n}$ $\mathbf {x} ,\mathbf {y} \in \mathbb {C} ^{n}$ ,
    - ${\tilde {f}}(\mathbf {x} ,\mathbf {y} )=P({\tilde {f}}(\mathbf {x} ),{\tilde {f}}(\mathbf {y} ))$ .

references