Lie-admissible algebras and Kac-Moody algebras
\textit{A. A. Albert} [Trans. Am. Math. Soc. 64, 552-593 (1948; Zbl 0033.15402)] proposed the problem of classifying all power-associative flexible Lie-admissible algebras whose commutator algebras are the finite dimensional simple Lie algebras over a field of characteristic zero. This problem was solved and strengthened by several authors [see \textit{H. C. Myung}, Malcev-admissible algebras (Prog. Math. 64, Birkhäuser, Basel) (1986; Zbl 0609.17009) and the references therein]. Recently, \textit{H. C. Myung} [J. Korean Math. Soc. 33, 1123-1128 (1996; Zbl 0883.17026)] classified all third power-associative Lie-admissible algebras whose commutator algebra is the Virasoro algebra and posed the problem of extending his results to cover the Kac-Moody algebras. The paper under review is devoted to solving this last problem. The third power-associative Lie-admissible products \(*\) that appear are all constructed from the Lie product in the Kac-Moody algebra \(L\), a linear functional \(\tau: L\rightarrow {\mathbb C}\) and a symmetric bilinear map \(\sigma: L\times L\rightarrow Z(L)\) as follows: \(x*y={1\over 2}[x,y]+\tau(x)y+\tau(y)x+\sigma(x,y)\). Conditions on \(\tau\) and \(\sigma\) to ensure the flexible law are given, too.
scientific article; zbMATH DE number 1104111
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