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A counterexample to Premet's and Joseph's conjectures
Author(s) -
Yakimova O.
Publication year - 2007
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdm060
Subject(s) - mathematics , counterexample , algebraically closed field , conjecture , lie algebra , nilpotent , locally nilpotent , rank (graph theory) , zero (linguistics) , pure mathematics , field (mathematics) , combinatorics , algebra over a field , polynomial , polynomial ring , simple (philosophy) , nilpotent group , mathematical analysis , linguistics , philosophy , epistemology
Let be a finite‐dimensional reductive Lie algebra of rank l over an algebraically closed field of characteristic zero. Given x ∈ , we denote by x the centraliser of x in . It was conjectured by Premet that the algebra ( x ) x of x ‐invariants is a graded polynomial algebra in l variables. In this note, we show that this conjecture does not hold for the minimal nilpotent orbit in the simple Lie algebra of type E 8 . As a consequence, a conjecture of Joseph on the semi‐invariants of (bi)parabolics is not true either.
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