Lepton flavor violation in the $\mu \nu$SSM with slepton flavor mixing

The $\mu \nu$SSM, one of supersymmetric extensions of the Standard Model, introduces three right-handed neutrino superfields to solve the $\mu$ problem and violates lepton number. Within framwork of the $\mu \nu$SSM, we investigate the lepton flavor violating (LFV) processes $Z\to l_i^{\pm }{l}_j^{\mp }$ with slepton flavor mixing. Simultaneously, we consider the LFV processes $l_j^{-}\to l_i^{-}\gamma$, $l_j^{-}\to l_i^{-}{l}_i^{-}{l}_i^{+}$ and muon conversion to electron in nuclei.