$\imath$Hopf algebras associated with self-dual Hopf algebras
Jiayi Chen, Shiquan Ruan
TL;DR
The paper introduces $\imath$Hopf algebras built from symmetrically self-dual Hopf algebras, using Green-type identities to guarantee associativity of a new product defined on the same basis as the original Hopf algebra. A base-change strategy via a symmetric self-duality map $\varphi$ (with $\varphi=\varphi^*$) allows reduction to cases where the structure constants satisfy $F_{ij}^k=G_{ij}^k$, or a scaled version thereof, ensuring a well-defined $\imath$-multiplication $\diamond$ and unit. The construction is formalized for general self-dual Hopf algebras, with a detailed base-change framework and explicit formulas for the new structure constants ${}^{\imath}\widetilde{F}$ in terms of the original constants and a representation matrix $S$ of $\varphi$. As an application, the authors define the $\imath$Taft algebra $H_2^{\imath}(q)$, prove its commutativity, and show it is isomorphic to the group algebra of $\mathbb{Z}/4\mathbb{Z}$ over an algebraically closed field, thereby connecting $\imath$-Hopf theory to concrete finite-dimensional examples and known Hopf algebras. Overall, the work provides a new pathway to realize $\imath$quantum groups via $\imath$Hopf algebras and offers a concrete base-change toolkit for extending the construction to broader classes of self-dual Hopf algebras.
Abstract
Motivated by the construction of $\imath$Hall algebras and $Δ$-Hall algebras, we introduce $\imath$Hopf algebras associated with symmetrically self-dual Hopf algebras. We prove that the $\imath$Hopf algebra is an associative algebra with a unit, where the associativity relies on an analogue of Green's formula in the framework of Hopf algebras. As an application, we construct the $\imath$Taft algebra of dimension 4, which is proved to be isomorphic to the group algebra of $\mathbb{Z}/4\mathbb{Z}$.