Embedding of $$\mathfrak {sl}_2({\mathbb {C}})$$ sl 2 ( C ) -Modules into Four-Dimensional Power-Associative Zero-Algebra Modules

Using the knowledge about the finite-dimensional irreducible modules over $$\mathfrak {sl}_2({\mathbb {C}})$$ , it is possible to associate for any of them an irreducible module over the four-dimensional zero-algebra on the class of commutative power-associative algebras. This association allows to construct an embedding from the category of $$\mathfrak {sl}_2({\mathbb {C}})$$ -modules into the category of the four-dimensional zero-algebra modules. Furthermore, in this paper it is shown that for any n greater than or equal to two, there exist two non-isomorphic families of irreducible modules of dimension 3n over the commutative power-associative algebra of dimension four and zero multiplication.