Vol. 27, No. 2, 1968

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A counter-example to a fixed point conjecture

Earl J. Taft

Vol. 27 (1968), No. 2, 405–409

Let A be a finite-dimensional commutative Jordan algebra over a field F of characteristic zero. Then we may write A = S + N, S a semisimple subalgebra (Wedderburn factor), N the radical of A, [5], [6]. If G is a completely reducible group of automorphisms of A, then we may choose S to be invariant under G, [4]. If G is finite, then we showed in [10] that any two such G-invariant S were conjugate via an automorphism σ of A which centralizes G and which is a product of exponentials of nilpotent inner derivations of A of the form [Rai,Rxi], xi in N, ai in A, where Ra is multiplication by a in A. It was conjectured in [10] that the various elements xi and ai which occur in the formulation of σ could be chosen as fixed points of G. This conjecture was based on analogous fixed point results proved for associative and Lie algebras, [7], [8], [9]. However, this conjecture is false, and we present in this note a simple counter-example.

Mathematical Subject Classification
Primary: 17.40
Received: 2 August 1967
Published: 1 November 1968
Earl J. Taft