The embedding of initial algebras into dialgebras
A. Dauletiyarova, A. Makhlouf, B. Sartayev
TL;DR
This work addresses how to recover a $\mathrm{Var}$-algebra from its dialgebraic extension by introducing initial $\mathrm{Var}$-dialgebras as a universal subvariety of $di$-$\mathrm{Var}$ from which a $\mathrm{Var}$-algebra can be retrieved via the polarization $a \star b = \tfrac{1}{2}(a \vdash b + a \dashv b)$. It develops a general construction: for a single-identity variety $\mathrm{Var}$, the derived $\mathrm{di}$-$\mathrm{Var}^{(\star)}\langle X\rangle$ lies in $\mathrm{Var}$ precisely when a translated identity in the dialgebra operations holds, and demonstrates this via concrete cases including $\mathrm{Com}$ and $\mathrm{Lie}$. The paper provides explicit bases for the free initial Lie and associative dialgebras, analyzes the dimensions of these free objects, and discusses the interplay with white/black Manin products and Koszul dual operads. It also computes the Koszul duals of several initial dialgebra operads, revealing Lie-admissible and left-commutative structures, and outlines several open problems and directions for further study. Overall, the work advances a universal framework for embedding and recovering classical varieties within dialgebraic contexts and clarifies operadic dualities in this setting.
Abstract
In this paper, for a given variety $\Var$, we present a universal algorithm that constructs a subvariety of $\Var$-dialgebras from which one can recover an algebra belonging to $\Var$. We call such a variety an initial $\Var$-dialgebra. In addition, we construct the basis of the free initial Lie and associative dialgebras.