Monoidal structures arising from $B_{\infty }$-algebras with applications to Hopf algebras

Abstract

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 $({\mathcal{D}}_{\infty }^{\mathrm{r}}(A),{⊠}_A,A)$ 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 $\mathcal{Y}(\mathbb{k},\mathbb{k})$ 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 $\mathcal{Y}(H,\mathbb{k})$ is a monoidal triangulated equivalence. If $H$ is local, this localizing subcategory is all of $\mathcal{K}(\mathrm{Inj}\text{-}\mathrm{H})$. 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.