Combinatorial Algebra and Its Antisymmetrized Algebra I

Abstract

The simple non-associative algebra N(eAS, q, n, t)k and its simple subalgebras are defined in [1, 3, 5–7, 13]. In this work, we define the combinatorial algebra N(e𝔄p, n, t)k and its antisymmetrized algebra [Formula: see text] and their subalgebras. We prove that these algebras are simple. Some authors [2, 5–7, 10, 13, 14, 16, 17] found all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra. We find all the derivations of the subalgebra N(e±x1x2 ⋯xn, 0, n)[1] of N(e𝔄p, n, t)k and the Lie subalgebra [Formula: see text] of [Formula: see text].