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# Infinitesimal Invariants in a Function Algebra

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Published:2009-08-01
Printed: Aug 2009
• Rudolf Tange
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## Abstract

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 - 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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