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

We study n-ary commutative superalgebras and 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 n-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative m-dimensional (m−3)-ary algebras with an invariant form, and a classification of real simple m-dimensional Lie (m−3)-algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for L∞-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric n-ary algebras.