Integrals in left coideal subalgebras and group-like projections

TL;DR

This paper develops a comprehensive theory linking right group-like projections with left coideal subalgebras in Hopf algebras, classifies these structures in Taft algebras, and answers a key open question about their symmetry.

Contribution

It introduces a new correspondence between right group-like projections and left coideal subalgebras, including classifications in Taft Hopf algebras and resolving an open question.

Findings

Established a 1-1 correspondence between right group-like projections and left coideal subalgebras in Hopf algebras.

Classified left coideal subalgebras in Taft Hopf algebras, identifying semisimple and non-semisimple cases.

Proved the existence of right group-like projections that are not left group-like projections, answering an open question.

Abstract

We develop a theory of right group-like projections in Hopf algebras linking them with the theory of left coideal subalgebras with two sided counital integrals. Every right group-like projection is associated with a left coideal subalgebra, maximal among the ones containing the given group-like projection as an integral, and we show that that subalgebra is finite dimensional. We observe that in a semisimple Hopf algebra $H$ every left coideal subalgebra has an integral and we prove a 1-1 correspondence between right group-like projections and left coideal subalgebras of $H$. We provide a number of equivalent conditions for a right group-like projections to be left group-like projection and prove a 1-1 correspondence between semisimple left coideal subalgebras preserved by the squared antipode and two sided group-like projections. We also classify left coideal subalgebras in Taft Hopf algebras $H_{n^2}$ over a field $\mathbb{k}$, showing that the automorphism group splits them into - a class of cardinality $|\mathbb{k}|-1$ of semisimple ones which correspond to right group-like projections which are not two sided; - finitely many semisimple singletons, each corresponding to two sided group-like projection; the number of those singletons for $H_{n^2}$ is equal to the number of divisors of $n$; - finitely many singletons, each non-semisimple and admitting no right group-like projection; the number of those singletons for $H_{n^2}$ is equal to the number of divisors of $n$. In particular we answer the question of Landstad and Van Daele showing that there do exist right group-like projections which are not left group-like projections.