Heisenberg Lie superalgebras and their invariant superorthogonal and supersymplectic forms

Abstract

Finite-dimensional complex Lie superalgebras of Heisenberg type obtained from a given Z 2 -homogeneous supersymplectic form defined on a vector superspace, are classified up to isomorphism. Those arising from even supersymplectic forms, have an ordinary Heisenberg Lie algebra as its underlying even subspace, whereas those arising from odd supersymplectic forms get based on abelian Lie algebras. The question of whether this sort of Heisenberg Lie superalgebras do or do not support a given invariant supergeometric structure is addressed, and it is found that none of them do. It is proved, however, that 1-dimensional extensions by appropriate Z 2 -homogeneous derivations do. Such ‘appropriate’ derivations are characterized, and the invariant supergeometric structures carried by the extensions they define are fully described. Furthermore, necessary and sufficient conditions are obtained in order that any two 1-dimensional extensions by Z 2 -homogeneous derivations be isomorphic; also, necessary and sufficient conditions are obtained in order that any two extensions carrying invariant supergeometric structures be isometric.