Induced 3-Lie algebras, superalgebras and induced representations

Abstract

We construct 3-Lie superalgebras on a commutative superalgebra by means of involution and even degree derivation. We construct a representation of induced 3-Lie algebras and superalgebras by means of a representation of initial (binary) Lie algebra, trace and supertrace. We show that the induced representation of 3-Lie algebra, that we constructed, is a representation by traceless matrices, that is, lies in the Lie algebra sl(V ), where V is a representation space. In the case of 2-dimensional representation we find conditions under which the induced representation of induced 3-Lie algebra is irreducible. We give the example of irreducible representation of induced 3-Lie algebra of 2nd order complex matrices.

Keywords

• 3-lie algebra
• 3-lie superalgebra
• filippov–jacobi identity
• commutative algebras with involution and derivation
• representations of 3-lie algebras and superalgebras.