Is zakat obligatory on the half of the house I own in partnership with my brother, and how much is it, and when does the period of zakat obligation on it begin, is it from the date of paying the first installments?

If you bought half of the house with the intention of selling it, then you owe zakat on its value every year from the day you acquire ownership. This is done by appraising half of it and paying two and a half percent (2.5%). However, if your intention was to live in it or to rent it out, then there is no zakat due on it, based on the Prophet's (peace be upon him) saying: "A man does not owe sadaqa (zakat) on his slave or his horse."