Canad. J. Math. 61(2009), 950-960
Printed: Aug 2009
Let $G$ be a reductive connected linear algebraic group
over an algebraically closed field of positive
characteristic and let $\g$ be its Lie algebra.
First we extend a well-known result about the Picard group of a
semi-simple group to reductive groups.
Then we prove that if the derived group is simply connected
and $\g$ satisfies a
mild condition, the algebra $K[G]^\g$ of regular functions
on $G$ that are invariant under the action of $\g$ derived
from the conjugation action is a unique factorisation domain.
20G15 - Linear algebraic groups over arbitrary fields
13F15 - Rings defined by factorization properties (e.g., atomic, factorial, half-factorial) [See also 13A05, 14M05]