Will the Imam be sinful because his wife does not want his mother, whom his brother gives nothing from the deceased father's retirement money, which the brother is responsible for?

You are not permitted to mistreat your mother or expel her on the pretext of your brother's shortcomings. You must honor her and treat her kindly. Your wife bears no sin unless she wrongs your mother, and she too should treat your mother with kindness. The person most deserving of your good companionship is your mother. As for the retirement pension granted to heirs, there is a detailed explanation: if it is an inheritance, then each heir has their legally stipulated share, and it is not permissible to allocate it to only one of the heirs. If it is a grant from the awarding entity, then one must adhere to the conditions stipulated by that entity. If the entity decided it is for the mother and she authorized her son to receive it, there is no objection, and he must manage it well and observe his mother's right.