The Turkish Alp Arslan has no ancestry among the Arab tribes relating to the Prophet Muhammed's kin, or even among the Quraysh tribe (the confederation which ruled Mecca prior to the rise of Islam). Plus, Seljuq's were Sunnis, and the purpose of their "empire" was officially to re-unify the fractured Abbasid Caliphate in Asia and to support the spiritual authority of the Abbasid clan within Sunni Islam. The Seljuq Sultans were more akin to Japanese Shoguns and were content with complete power in military matters. If the Seljuq Empire endured longer, it might be possible for a daughter of the Seljuq dynasty to marry into the Abbasid clan, and a future Caliph would have maternal Turkish ancestry. But, he would still identify as an Abbasid, and the Seljuqs would just prefer the normal arrangement anyway.