We study and classify the 3-dimensional Hom-Lie algebras over $\mathbb{C}$. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space $\mathfrak{g}$. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative, we provide a complete set of canonical forms for the linear maps $\mathfrak{g} \to \mathfrak{g}$ that turn $g$ into a Hom-Lie algebra, thus characterizing the corresponding isomorphism classes. As by-products, Hom-Lie algebras for which the linear maps $\mathfrak{g} \to \mathfrak{g}$ are not homomorphisms for their products, are exhibited. Examples also arise of non-isomorphic families of HomLie algebras which share, however, a fixed Lie-algebra product on $\mathfrak{g}$. In particular, this is the case for the complex simple Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on $\mathfrak{g}$.

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关于 3 维复数 Hom-Lie 代数

我们在 $\mathbb{C}$ 上研究和分类 3 维 Hom-Lie 代数。我们首先为在 3 维复向量空间 $\mathfrak{g}$ 上定义的斜对称双线性乘积的同构类提供了一组完整的代表。3 维李代数的众所周知的李括号包含在此类乘积代表的适当同构类中。对于每个产品代表，我们为线性映射 $\mathfrak{g} \to \mathfrak{g}$ 提供了一套完整的规范形式，将 $g$ 转换为 Hom-Lie 代数，从而表征相应的同构类。作为副产品，展示了线性映射 $\mathfrak{g} \to \mathfrak{g}$ 不是它们产品的同态的 Hom-Lie 代数。也出现了 HomLie 代数的非同构族的例子，它们共享，然而，$\mathfrak{g}$ 上的一个固定李代数积。尤其是复杂的简单李代数 $\mathfrak{sl}_2(\mathbb{C})$ 的情况。类似地，有一些同构类的偏对称双线性乘积永远不可能是 $\mathfrak{g}$ 上的李代数括号。