Lie-like Algebras (Superalgeras)

We introduce the notion of a Lie-like algebra⋄ (superalgebra⋄) for ⋄ ∈ { , , 3−rd }. By bundling a family of algebras, we introduce six new generalizations of Lie algebras which are called Lie-like algebras or Lie-like superalgebras in this paper. The examples of the new generalizations of Lie algebras come from an invariant algebra (or an invariant superalgebra), where the notion of an invariant algebra was introduced in Chapter 1 of [5], and an invariant superalgebra is the counterpart of an invariant algebra in the context of superalgebras. Since an invariant algebra (or invariant superalgebra) can be regarded as an associative subalgebra of the associative algebra of linear transformations over a vector space (or super vector space), the passage from an associative algebra to a Lie algebra can be extended to the passage from an invariant algebra (or an invariant superalgebra) to one of the new generalizations of Lie algebras. This paper consists of six sections. In the first three sections, we introduce Lie-like algebras with ⋄ ∈ { , , 3−rd }. The counterparts of Lie-like algebras with ⋄ ∈ { , , 3−rd } in the context of superalgebras are called Lie-like superalgebras and introduced in the last three sections. Throughout this paper, we make the following conventions. • All vector spaces are the vector spaces over a field k. • Let L be a vector space. We assume that a binary operation L × L → L is always bilinear. • A vector space V is called a super vector space if V = V0 ⊕ V1 is the direct sum of its two subspaces V0 and V1, where 0, 1 ∈ Z2 = Z/2Z. The subspaces V0 and V1 are called the even part and the odd part of V respectively.