An even Artin group is a group which has a presentation with relations of the form $(st)^n=(ts)^n$ with $n\ge 1$. With a group $G$ we associate a Lie $\mathbb Z$-algebra $\mathcal{TG}r(G)$. This is the usual Lie algebra defined from the lower central series, truncated at the third rank. For each even Artin group $G$ we determine a presentation for $\mathcal{TG}r(G)$. Then we prove a criterion to determine whether two Coxeter matrices are isomorphic. Let $c,d\in\mathbb N$ such that $c\ge1$, $d\ge2$ and $\gcd(c,d)=1$. We show that, if two even Artin groups $G$ and $G'$ having presentations with relations of the form $(st)^n=(ts)^n$ with $n\in\{c\}\cup\{d^k\mid k\ge1\}$ are such that $\mathcal{TG}r(G)\simeq\mathcal{TG}r(G')$, then $G$ and $G'$ have the same presentation up to permutation of the generators. On the other hand, we show an example of two non-isomorphic even Artin groups $G$ and $G'$ such that $\mathcal{TG}r(G)\simeq\mathcal{TG}r(G')$.

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关于偶数Artin群的同构问题

偶数 Artin 群是一个表示形式为 $(st)^n=(ts)^n$ 和 $n\ge 1$ 的群。我们将群 $G$ 与 Lie $\mathbb Z$-代数 $\mathcal{TG}r(G)$ 联系起来。这是从较低的中央级数定义的通常的李代数，在第三级截断。对于每个偶数 Artin 组 $G$，我们为 $\mathcal{TG}r(G)$ 确定一个表示。然后我们证明了一个判断两个 Coxeter 矩阵是否同构的标准。令 $c,d\in\mathbb N$ 使得 $c\ge1$、$d\ge2$ 和 $\gcd(c,d)=1$。我们证明，如果两个偶数 Artin 组 $G$ 和 $G'$ 具有 $(st)^n=(ts)^n$ 与 $n\in\{c\}\cup\ {d^k\mid k\ge1\}$ 使得 $\mathcal{TG}r(G)\simeq\mathcal{TG}r(G')$，那么 $G$ 和 $G'$ 有相同的演示文稿直到生成器的排列。另一方面，