Integrals in Left Coideal Subalgebras and Group-Like Projections

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 Hn2 over a field k, showing that the automorphism group splits them intoa class of cardinality | 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 Hn2 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 Hn2 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.