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.

MSC Classifications:

20G15, 13F15 show english descriptions Linear algebraic groups over arbitrary fields
Rings defined by factorization properties (e.g., atomic, factorial, half-factorial) [See also 13A05, 14M05] 20G15 - Linear algebraic groups over arbitrary fields
13F15 - Rings defined by factorization properties (e.g., atomic, factorial, half-factorial) [See also 13A05, 14M05]

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