Combinatorial Algebra and Its Antisymmetrized Algebra I

Abstract

The simple non-associative algebra N (e(AS), 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 (e(uP), n, t)(k) and its antisymmetrized algebra N(e(uP), n, t)(k)(-) 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 (e(uP),n, t)(k) and the Lie subalgebra N(e(+xy), 0,2)([1])(-) of N (e(uP), n, t)(k)(-).