Is it permissible to open a shop in the name of one's brother in exchange for a sum of money paid to him, knowing that he is a student and agrees to that?

There is no harm in opening a commercial shop in your brother's name if the state permits it and your brother agrees to it. There is no obligation to pay him a specific amount at the beginning of the project, and it is not permissible to stipulate that, because it would be a recompense for sponsorship, which is a contract of gratuitous assistance ( 'aqd irfaq) that does not involve compensation. However, if the license holder has incurred fees and expenses, he may take a fee in return for his efforts, not for his sponsorship. Similarly, it is not permissible for him to receive a fixed monthly compensation.