Abstract

In this paper we give the classification of 5-dimensional complex nilpotent associative algebras. We use $\chi (\mathcal{A})=(\mathit{dim}\mathcal{A},\mathit{dim}{\mathcal{A}}^2,\mathit{dim}{\mathcal{A}}^3,\dots ,\mathit{dim}{\mathcal{A}}^n)$ - the isomorphism invariant. We prove that there are only two possible isomorphism invariants $\chi (\mathcal{A})=(5,2,1,0,0)$ or $\chi (\mathcal{A})=(5,3,1,0,0)$ for any 5-dimensional complex nilpotent associative algebra $\mathcal{A}$ with the condition ${\mathcal{A}}^4=0\wedge {\mathcal{A}}^3\ne 0$ and we give the classification (up to isomorphism) of such type algebras. 5-dimensional complex nilpotent associative algebras with the conditions ${\mathcal{A}}^6=0\wedge {\mathcal{A}}^5\ne 0,{\mathcal{A}}^5=0\wedge {\mathcal{A}}^4\ne 0$ and ${\mathcal{A}}^3=0\wedge {\mathcal{A}}^2\ne 0$ have already been classified. Thus, we complete the classification of 5-dimensional complex nilpotent associative algebras.

抽象的

在本文中，我们给出了五维复幂等缔合代数的分类。我们用$\chi （\mathcal{一种}）=（\mathit{暗淡}\mathcal{一种}，\mathit{暗淡}{\mathcal{一种}}^2，\mathit{暗淡}{\mathcal{一种}}^3，\dots ，\mathit{暗淡}{\mathcal{一种}}^{ñ}）$-同构不变。我们证明只有两个可能的同构不变量$\chi （\mathcal{一种}）=（5，2，\mathrm{1个}，0，0）$ 或者 $\chi （\mathcal{一种}）=（5，3，\mathrm{1个}，0，0）$ 适用于任何5维复杂的幂等缔合代数 $\mathcal{一种}$ 有条件 ${\mathcal{一种}}^4=0\wedge {\mathcal{一种}}^3\ne 0$我们给出了这类代数的分类（直至同构）。具有条件的5维复幂等联想代数${\mathcal{一种}}^6=0\wedge {\mathcal{一种}}^5\ne 0，{\mathcal{一种}}^5=0\wedge {\mathcal{一种}}^4\ne 0$ 和 ${\mathcal{一种}}^3=0\wedge {\mathcal{一种}}^2\ne 0$已经被分类。因此，我们完成了5维复杂幂等缔合代数的分类。