Hello Raddi...
I am so sorry for what you, your mom, and your brother are going through. This is a tough situation indeed and one that I fully recognize. I lived with my mom alone...for 22 years. She would do many of the things you are describing. As a child I didn't know any better...and wanted to believe what she was saying. She would not allow me to talk with friends anymore that would formerly come to play. She cut us off from family, neighbors..and everybody before she would have her psychotic breakdowns.
It is hard to get a parent who is going through this...into a hospital for treatment. I had to get very creative sometimes to convince her to go. Sometimes I was successful and sometimes not. The advice given to you already is correct...the person has to either be homicidal or suicidal to be committed.
Does your mom have a social worker or any links to the mental health system? I would keep calling people within your local mental health system until someone listens.
Please do let us know what happens....I am just so sorry you and your family are going through this. I do feel for you.