带有应用于Hopf代数的$B_\u221e$-代数产生的幺半结构
Monoidal structures arising from $B_\infty$-algebras with applications to Hopf algebras
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中文总结 AI 辅助
本文构造了带$B_\u221e$-结构的$A_\u221e$-代数上右$A_\u221e$-模导出范畴的幺半三角结构，将其应用于有限维Hopf代数，证明了Koszul对偶函子的松弛幺半性及等价性，给出Krause猜想的纯代数证明。
中文摘要 AI 辅助
我们给出了配备$B_\u221e$-结构的$A_\u221e$-代数$A$上的右$A_\u221e$-模的导出范畴上的幺半结构的显式构造。给定这样的$B_\u221e$-代数$A$，我们构造了一个归纳函子$\u03b9\colon \mathcal{D}^{\rm{r}}_\infty(A) \longrightarrow \mathcal{D}^{\rm{bi}}_\infty(A)$，从右$A_\u221e$-模到$A_\u221e$-双模，并定义$M\boxtimes_A N=M\overset{\infty}{\otimes}_A\iota(N)$。我们证明$(\mathcal{D}^{\rm{r}}_\infty(A),\boxtimes_A,A)$是一个幺半三角范畴：单位和结合约束由$A_\u221e$-双模的显式拟同构诱导，包括$\u03b9(A)\simeq A$和$\u03b9(M)\overset{\infty}{\otimes}_A\iota(N)\simeq \iota(M\boxtimes_A N)$。我们将该构造应用于有限维Hopf代数$H$，平凡$H$-模的Yoneda dg代数$\mathcal{Y}(\Bbbk,\Bbbk)$带有自然的brace $B_\u221e$-结构，因此其导出范畴带有上述构造的幺半结构。我们证明Koszul对偶函子$\rm{Hom}_H(\mathcal{Y}(H,\Bbbk),-)\colon \mathcal{K}(\rm{Inj}\text{-}H)\longrightarrow \mathcal{D}(\mathcal{Y}(\Bbbk,\Bbbk))$是三角松弛幺半的，且其对由内射预解式$\mathcal{Y}(H,\Bbbk)$生成的局部化子范畴的限制是幺半三角等价。若$H$是局部的，则该局部化子范畴就是整个$\mathcal{K}(\rm{Inj}\text{-}H)$。特别地，这给出了Krause猜想的幺半等价的另一种纯代数证明，该猜想已由Benson--Krause通过分类空间$BG$证明。最后，我们的例子恢复了分次交换代数的通常张量积，并表明所得的brace $B_\u221e$和幺半结构可本质上依赖于所选的Hopf结构。
英文摘要
We give an explicit construction of monoidal structures on derived categories of right $A_\infty$-modules over an $A_\infty$-algebra $A$ equipped with a $B_\infty$-structure. Given such a $B_\infty$-algebra $A$, we construct an induction functor \[ι\colon \mathcal{D}^{\rm{r}}_\infty(A) \longrightarrow \mathcal{D}^{\rm{bi}}_\infty(A)\] from right $A_\infty$-modules to $A_\infty$-bimodules and define \[M\boxtimes_A N=M\overset{\infty}{\otimes}_Aι(N).\] We prove that $(\mathcal{D}^{\rm{r}}_\infty(A),\boxtimes_A,A)$ is a monoidal triangulated category: the unit and associativity constraints are induced by explicit quasi-isomorphisms of $A_\infty$-bimodules, including \[ι(A)\simeq A \qquad\text{and}\qquad ι(M)\overset{\infty}{\otimes}_Aι(N)\simeq ι(M\boxtimes_A N).\] We apply this construction to finite-dimensional Hopf algebras $H$, the Yoneda dg algebra $\mathcal{Y}(\Bbbk,\Bbbk)$ of the trivial $H$-module carries a natural brace $B_\infty$-structure, and hence its derived category carries the monoidal structure constructed above. We show that the Koszul duality functor \[\rm{Hom}_H(\mathcal{Y}(H,\Bbbk),-)\colon \mathcal{K}(\rm{Inj}\text{-}H)\longrightarrow \mathcal{D}(\mathcal{Y}(\Bbbk,\Bbbk))\] is triangulated lax monoidal, and that its restriction to the localizing subcategory generated by the injective resolution $\mathcal{Y}(H,\Bbbk)$ is a monoidal triangulated equivalence. If $H$ is local, this localizing subcategory is all of $\mathcal{K}(\rm{Inj}\text{-}H)$. In particular, this gives an alternative, purely algebraic proof of the monoidal equivalence conjectured by Krause and established by Benson--Krause through the classifying space $BG$. Finally, our examples recover the usual tensor product for graded commutative algebras and show that the resulting brace $B_\infty$ and monoidal structures can depend essentially on the chosen Hopf structure.
