For an arbitrary finite-dimensional algebra $A$, we introduce a general approach to determining when its first Hochschild cohomology ${\rm HH}^1(A)$, considered as a Lie algebra, is solvable. If $A$ is moreover of tame or finite representation type, we are able to describe ${\rm HH}^1(A)$ as the direct sum of a solvable Lie algebra and a sum of copies of $\mathfrak{sl}_2$. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of $A$. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar.

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关于有限维代数的第一霍克斯柴尔德上同调的可解性

对于任意有限维代数 $A$，我们引入了一种通用方法来确定其第一个 Hochschild 上同调 ${\rm HH}^1(A)$（被视为李代数）何时可解。此外，如果 $A$ 是驯服或有限表示类型，我们可以将 ${\rm HH}^1(A)$ 描述为可解李代数的直接和和 $\mathfrak{sl }_2$。我们继续确定此类副本的确切数量，并根据 $A$ 箭袋的某些克罗内克子箭袋链给出该数字的明确公式。作为推论，我们获得了对 Chaparro、Schroll 和 Solotar 提出的问题的准确答案。