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 \({\mathbbm {k}}\), showing that the automorphism group splits them into
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a class of cardinality \(|{\mathbbm {k}}|-1\) of semisimple ones which correspond to right group-like projections which are not two sided;
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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;
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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.
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Acknowledgments
The authors are grateful to the anonymous referee for paying our attention to the paper of M. Koppinen, [8]. A.C. was partially supported by NSF grant DMS-1801011. PK was partially supported by the NCN (National Center of Science) grant 2015/17/B/ST1/00085.
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Presented by: Vyjayanthi Chari
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Chirvasitu, A., Kasprzak, P. & Szulim, P. Integrals in Left Coideal Subalgebras and Group-Like Projections. Algebr Represent Theor 23, 1499–1522 (2020). https://doi.org/10.1007/s10468-019-09889-1
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DOI: https://doi.org/10.1007/s10468-019-09889-1