Is it permissible for a brother to give his sister zakat from his money, when she lives with their mother, whom the brother supports?

It is not permissible for a brother to give his Zakat to his sister if he is responsible for her upkeep and provides for her. However, if she is poor and he does not provide for her, then she is among those eligible to receive Zakat, and giving Zakat to her is better because it combines charity and kinship.