Commutative n-ary superalgebras with an invariant skew-symmetric form

English

Vishnyakova, Elizaveta[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit > ; Max Planck Institute for Mathematics, Bonn, Germany]

[en] n-ary and L∞-algebras ; Lie n-algebras ; Invariant form

[en] We study nn-ary commutative superalgebras and L∞L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their nn-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative mm-dimensional (m−3)(m−3)-ary algebras with an invariant form, and a classification of real simple mm-dimensional Lie (m−3)(m−3)-algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for L∞L∞-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric nn-ary algebras.