Objection 1. It would seem that it does not belong to Christ as God to sit at the right hand of the Father. For, as God, Christ is the Father's right hand. But it does not appear to be the same thing to be the right hand of anyone and to sit on his right hand. Therefore, as God, Christ does not sit at the right hand of the Father.
Objection 2. Further, in the last chapter of Mark (16:19) it is said that "the Lord Jesus was taken up into heaven, and sitteth on the right hand of God." But it was not as God that Christ was taken up to heaven. Therefore neither does He, as God, sit at the right hand of God.
Objection 3. Further, Christ as God is the equal of the Father and of the Holy Ghost. Consequently, if Christ sits as God at the right hand of the Father, with equal reason the Holy Ghost sits at the right hand of the Father and of the Son, and the Father Himself on the right hand of the Son; which no one is found to say.
On the contrary, Damascene says (De Fide Orth. iv): that "what we style as the Father's right hand, is the glory and honor of the Godhead, wherein the Son of God existed before ages as God and as consubstantial with the Father."
I answer that, As may be gathered from what has been said (1) three things can be understood under the expression "right hand." First of all, as Damascene takes it, "the glory of the Godhead": secondly, according to Augustine "the beatitude of the Father": thirdly, according to the same authority, "judiciary power." Now as we observed (1) "sitting denotes" either abiding, or royal or judiciary dignity. Hence, to sit on the right hand of the Father is nothing else than to share in the glory of the Godhead with the Father, and to possess beatitude and judiciary power, and that unchangeably and royally. But this belongs to the Son as God. Hence it is manifest that Christ as God sits at the right hand of the Father; yet so that this preposition "at," which is a transitive one, implies merely personal distinction and order of origin, but not degree of nature or dignity, for there is no such thing in the Divine Persons, as was shown in I, 42, 3,4.
Reply to Objection 1. The Son of God is called the Father's "right hand" by appropriation, just as He is called the "Power" of the Father (1 Corinthians 1:24). But "right hand of the Father," in its three meanings given above, is something common to the three Persons.
Reply to Objection 2. Christ as man is exalted to Divine honor; and this is signified in the aforesaid sitting; nevertheless such honor belongs to Him as God, not through any assumption, but through His origin from eternity.
Reply to Objection 3. In no way can it be said that the Father is seated at the right hand of the Son or of the Holy Ghost; because the Son and the Holy Ghost derive their origin from the Father, and not conversely. The Holy Ghost, however, can be said properly to sit at the right hand of the Father or of the Son, in the aforesaid sense, although by a kind of appropriation it is attributed to the Son, to whom equality is appropriated; thus Augustine says (De Doctr. Christ. i) that "in the Father there is unity, in the Son equality, in the Holy Ghost the connection of unity with equality."
Source: Summa Theologica (New Advent)