Hyper-para-Kähler Lie algebras with abelian complex structures and their classification up to dimension 8
In the paper under review the authors consider the hyper-para-Kähler algebras \(\mathfrak{g}\) whose complex structure \(J\) is abelian, that is, the eigenspaces of \(J\) in the complexification of \(\mathfrak{g}\) are abelian Lie algebras. These algebras are characterized in terms of certain triples consisting of an associative, commutative algebra \(V\) such that \(V^{3} = \{ 0 \}\), a skew-symmetric form on it, and a semilinear map. This allows the authors to give a classification of hyper-para-Kähler algebras of dimension \(8\), and of those of arbitrary dimension for which \(V^{2}\) has dimension one.
scientific article; zbMATH DE number 6894771
scientific article