An Allison-Kantor-Koecher-Tits construction for Lie \(H^*\)-algebras (Q1322052)

The authors establish a correspondence between the class of (complex) structurable \(H^*\)-algebras [for the definition see the authors, J. Algebra 147, 19-62 (1992; Zbl 0749.17001)] with zero annihilator and the class of Lie \(H^*\)-algebras with zero annihilator. The construction providing this correspondence is a topological version of the Allison-Kantor construction for general structurable algebras (without an additional Hilbert space structure). Furthermore, it is proved that the lattices of closed ideals of a structurable \(H^*\)-algebra \((A, {}^-)\) and the associated Lie \(H^*\)-algebra \(K(A,{}^-)\) are isomorphic. It follows that \(K(A,{}^-)\) is topologically simple if and only if \((A,{}^-)\) is topologically simple as an algebra with involution. Also, it is shown that any topologically simple Lie \(H^*\)-algebra can be obtained as \(K(A,{}^-)\) for a suitable \((A,{}^-)\).