Is it permissible for a husband to marry another woman without his wife's knowledge? And what is the ruling on not forgiving the husband for his action while accepting to live with him?

For a man to marry a second wife, it is not a condition that the first wife be informed, or that she give permission, or that she consent, because this is his right that Allah has made permissible for him, provided he acts justly. It is not permissible for the first wife to seek divorce for this reason, as a woman's request for divorce without need is forbidden. What the husband did is not betrayal, and the wife should fear Allah and distance herself from the whisperings of Satan and jealousy, which may displease Allah.