The minimal model of Rota-Baxter operad with arbitrary weight

This paper investigates Rota–Baxter algebras of of arbitrary weight, that is, associative algebras endowed with Rota-Baxter operators of arbitrary weight, from an operadic viewpoint. Denote by \({ _\lambda \mathfrak {RBA}}\) $_{\lambda }^{}\mathfrak{RBA}$ the operad of Rota-Baxter associative algebras of weight \(\lambda \) $\lambda$ . A homotopy cooperad is explicitly constructed, which can be seen as the Koszul dual of \({ _\lambda \mathfrak {RBA}}\) $_{\lambda }^{}\mathfrak{RBA}$ as it is proven that the cobar construction of this homotopy cooperad is exactly the minimal model of \({ _\lambda \mathfrak {RBA}}\) $_{\lambda }^{}\mathfrak{RBA}$ . This enables us to introduce the notion of homotopy Rota-Baxter algebras. The deformation complex of a Rota-Baxter algebra and the underlying \(L_\infty \) $L_{\infty }$ -algebra structure over it are exhibited as well.