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arXiv 1907.04093math.RT
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有限表示型余交换霍普夫代数的第一Hochschild上同调

On the first Hochschild cohomology of cocommutative Hopf algebras of finite representation type

Hao Chang

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摘要：

设$\mathscr{B}_0(\mathcal{G})\subseteq k\mathcal{G}$是特征为${\rm char}(k)=:p\geq 3$的代数闭域$k$上的无穷小群模式$\mathcal{G}$的群代数$k\mathcal{G}$的主块代数.当$\mathscr{B}_0(\mathcal{G})$具有有限表示型时，我们计算了第一Hochschild上同调$\mathcal{L}:={\rm H}^1(\mathscr{B}_0(\mathcal{G}),\mathscr{B}_0(\mathcal{G}))$的限制李代数结构.因此，我们证明了平凡的$\mathcal{G}$ -模$k$的复杂性与$\mathcal{L}$的最大toral秩相一致。

英文摘要：

Let $\mathscr{B}_0(\mathcal{G})\subseteq k\mathcal{G}$ be the principal block algebra of the group algebra $k\mathcal{G}$ of an infinitesimal group scheme $\mathcal{G}$ over an algebraically closed field $k$ of characteristic ${\rm char}(k)=:p\geq 3$. We calculate the restricted Lie algebra structure of the first Hochschild cohomology $\mathcal{L}:={\rm H}^1(\mathscr{B}_0(\mathcal{G}),\mathscr{B}_0(\mathcal{G}))$ whenever $\mathscr{B}_0(\mathcal{G})$ has finite representation type. As a consequence, we prove that the complexity of the trivial $\mathcal{G}$-module $k$ coincides with the maximal toral rank of $\mathcal{L}$.

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