Shadow and Quasinormal Modes of a Rotating Loop Quantum Black Hole

In this paper, we construct an effective rotating loop quantum black hole (LQBH) solution, starting from the spherical symmetric LQBH by applying the Newman-Janis algorithm modified by Azreg-A\"{i}nou's non-complexification procedure, and study the effects of loop quantum gravity { (LQG) on its shadow}. Given the rotating {LQBH}, we discuss its horizon, ergosurface, and regularity {as} $r \to 0$. Depending on the values of the specific angular momentum $a$ and the polymeric function $P$ arising from {LQG}, we {find} that the rotating solution we obtained can represent a regular black hole, a regular extreme black hole, or a regular spacetime {without horizon (a non-black-hole solution)}. We also {study} the effects of {LQG} and rotation, and {show} that, in addition to the specific angular momentum, the polymeric function {also} causes deformations in the size and shape of the black hole shadow. Interestingly, for a given value of $a$ and inclination angle $\theta_0$, the apparent size of the shadow monotonically decreases, and the shadow gets more distorted with increasing $P$. We also {consider the effects of $P$ on the deviations from the circularity of the shadow, and find} that the deviation from circularity increases with increasing $P$ for fixed values of $a$ and $\theta_0$. Additionally, we explore the observational implications of $P$ in comparison with the latest Event Horizon Telescope (EHT) observation of the supermassive black hole, M$87$. The connection between the shadow radius and quasinormal modes in the eikonal limit as well as {the} deflection of massive particles are also considered.