Is it permissible for a wife to seek divorce due to her husband's lying, injustice, and falsely accusing her honor, and what are the rights of each of them in this case?

A woman is not permitted to seek a divorce without a legitimate Shari (Islamic legal) justification. What was mentioned about the husband—lying, accusations against one's honor and chastity, injustice, and emotional abuse—if true, contradicts kind treatment and good companionship. It is recommended to offer sincere advice to the husband with gentleness and leniency, and to seek help from some virtuous people, along with frequent supplication. If the situation persists and you are being harmed, then you have the right to seek a divorce, but do not hasten until you clearly discern where the benefit lies.