Generalized (co)integrals on coideal subalgebras

Given a a Hopf algebra $H$, its left coideal subalgebra $A$ and a non-zero multiplicative functional $\mu$ on $A$, we define the space of left $\mu$-integrals $L^A_\mu\subset A$. We observe that $\dim L^A_\mu=1$ if $A$ is a Frobenius algebra and we conclude this equality for finite dimensional left coideal subalgebras of a weakly finite Hopf algebra. In general we prove that if $\dim L^A_\mu>0$ then $\dim A <\infty$. Given a group-like element $g\in H$ we define the space $L^g_{ A}\subset A'$ of $g$-cointegrals on $A$ and linking this concept with the theory of $\mu$-integrals we observe that: - every semisimple left coideal subalgebra $A\subset H$ which is preserved by the antipode squared admits a faithful $1$-cointegral; - every unimodular finite dimensional left coideal subalgebra $A\subset H$ admitting a faithful $1$-cointegral is preserved by the antipode square; - every non-degenerate right group-like projection in a cosemisimple Hopf algebra is a two sided group-like projection. Finally we list all $\varepsilon$-integrals for left coideals subalgebras in Taft algebras and we list all $g$-cointegrals on them.