\(n\)-ary generalized Lie-type color algebras admitting a quasi-multiplicative basis (Q2288137)

The object of this paper are \(n\)-ary color algebras satisfying a certain multilinear identity ``of Lie type''. The particular cases are many classes of binary (color) algebras (Lie, Leibniz, associative, Novikov, etc.), as well as Filippov \(n\)-ary algebras, and its super- and color- generalization. A basis of an \(n\)-ary algebra is called quasi-multiplicative, if the \(n\)-ary product of any elements from the basis is a scalar multiple on an element of the basis. This notion is generalized to color \(n\)-ary algebras; the generalization involves a distinguished color subspace \(\mathbb V\), and elements of the basis satisfying the quasi-multiplicative property are chosen from a color subspace complement to \(\mathbb V\). The main result of the paper is the following: if an \(n\)-ary color algebra satisfying the identity of Lie type admits a quasi-multiplicative basis, then it decomposes as a direct sum of \(\mathbb V\) and ideals, and each of those ideals also admits a quasi-multiplicative basis. This generalizes the earlier similar results in the binary case.