Nilpotent symplectic alternating algebras. II. (Q2821829)

Symplectic alternating algebras were introduced by \textit{P. Moravec} and \textit{G. Traustason} [Commun. Algebra 36, No. 11, 4096--4119 (2008; Zbl 1159.20017)] as a tool in the investigation of powerful 2-Engel groups. The authors developed [J. Algebra 423, 615--635 (2015; Zbl 1369.17032)] a rich theory of nilpotent symplectic alternating algebras and found, in particular, the algebras of dimension up to eight. In the present paper and its sequel [Int. J. Algebra Comput. 26, No. 5, 1095--1124 (2016; Zbl 1369.17034)], the authors classify the algebras of dimension 10 over an arbitrary field. The classification proceeds by conditions on the center. If the center has non-isotropic center, then the theory reduces to the known results for algebras of lower dimensions and the algebras are listed. When the center is isotropic, its dimension is between 2 and 5. The present paper deals with the cases when the center has odd dimension. The dimension 5 case is disposed of quickly where it is shown that there is a unique 10 dimensional algebra with isotropic center of dimension 5. In the dimension 3 case, the proof is long and involved with many subcases.