NEW ALGEBRAS USING ADDITIVE ABELIAN GROUPS I

Abstract

The simple non-associative algebra <TEX>$N(e^{A_S},q,n,t)_k$</TEX> and its simple sub-algebras are defined in the papers [1], [3], [4], [5], [6], [12]. We define the non-associative algebra <TEX>$\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$</TEX> and its antisymmetrized algebra <TEX>$\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$</TEX>. We also prove that the algebras are simple in this work. There are various papers on finding all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra (see [3], [5], [6], [9], [12], [14], [15]). We also find all the derivations <TEX>$Der_{anti}(WN(e^{{\pm}x^r},0,2)_B^-)$</TEX> of te antisymmetrized algebra <TEX>$WN(e^{{\pm}x^r}0,2)_B^-$</TEX> and every derivation of the algebra is outer in this paper.