发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在特征p≥5的代数闭域上的Verlinde范畴中，证明线性约化李代数的Chevalley-Eilenberg上同调与对应外代数同构，重现了相关经典定理。
AI 中文摘要
设k为特征p≥5的代数闭域，Verₚ⁺是Verlinde融合范畴Verₚ的偶部，Verₚ是Repₖ(ℤ/p)的半单化。设𝔤是Verₚ⁺中的线性约化李代数，即其有限维表示均为半单的；一个基本例子是当p超过其Coxeter数时，对k上的单李代数赋予主幂幺元的ℤ/p作用后得到的半单化结果。我们证明𝔤是无不变量的，即单位对象不是𝔤的直和项。对满足3≤m≤p-2的奇数m，令𝔤ₘ:=Hom_{Verₚ⁺}(Lₘ,𝔤)，E_𝔤:=⊕_{3≤m≤p-2, m为奇数}𝔤ₘ⁽¹⁾[m]，其中(1)表示Frobenius扭。我们的主要结果是分次代数同构H^•_{CE}(𝔤)≅∧^•E_𝔤^*；还将该代数与群概型G=exp(𝔤)的 de Rham上同调H^•_{dR}(G)等同，并证明诱导的分次Hopf代数结构与外代数上的标准结构一致。此外，若V是𝔤的单模且𝔤在其上非平凡作用，则H^•_{CE}(𝔤,V)=0，因此对每个有限维𝔤模V，有H^•_{CE}(𝔤,V)≅∧^•E_𝔤^*⊗V^𝔤，这重现了复半单李代数上同调的Borel-Chevalley定理及其在足够大正特征下的类似结论。
英文摘要
Let $k$ be an algebraically closed field of characteristic $p\geq 5$, and let $\mathrm{Ver}_p^+$ be the even part of the Verlinde fusion category $\mathrm{Ver}_p$, the semisimplification of $\mathrm{Rep}_k(\mathbb Z/p)$. Let $\mathfrak g$ be a linearly reductive Lie algebra in $\mathrm{Ver}_p^+$, i.e., one whose finite-dimensional representations are semisimple. A basic class of examples is obtained by semisimplifying a simple Lie algebra over $k$ equipped with the action of $\mathbb Z/p$ by a principal unipotent element, when $p$ exceeds its Coxeter number. We prove that $\mathfrak g$ is invariantless, i.e., that the unit object is not a summand of $\mathfrak g$. For odd $m$ with $3\leq m\leq p-2$, set $\mathfrak{g}_m:=\operatorname{Hom}_{\mathrm{Ver}_p^+}(L_m,\mathfrak g)$ and $E_{\mathfrak g}:=\bigoplus_{3\leq m\leq p-2,\ m\ {\rm odd}}\mathfrak g_m^{(1)}[m]$, where $(1)$ denotes Frobenius twist. Our main result is an isomorphism of graded algebras $H^\bullet_{\mathrm{CE}}(\mathfrak g)\cong\bigwedge^\bullet E_{\mathfrak g}^*$. We also identify this algebra with the de Rham cohomology $H^\bullet_{\mathrm{dR}}(G)$ of the group scheme $G=\exp(\mathfrak g)$ and show that the induced graded Hopf algebra structure agrees with the standard one on the exterior algebra. Moreover, if $V$ is a simple $\mathfrak g$-module on which $\mathfrak g$ acts nontrivially, then $H^\bullet_{\mathrm{CE}}(\mathfrak g,V)=0$. Hence for every finite-dimensional $\mathfrak g$-module $V$ one has $H^\bullet_{\mathrm{CE}}(\mathfrak g,V)\cong\bigwedge^\bullet E_{\mathfrak g}^*\otimes V^{\mathfrak g}$. This recovers the theorem of Borel and Chevalley on the cohomology of complex semisimple Lie algebras and its analogue in sufficiently large positive characteristic. We also prove similar results for relative cohomology.
Comments18 pages, latex; v2 contains a new section 6 on relative cohomology
