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Let G be the pro-algebraic group attached to the tannakian category ofpolarizable rational Hodge structures. We show that the quotient of G by itsderived group is the Serre group, the derived group of G is the simplyconnected covering of the adjoint group of G, and that the adjoint group G is aproduct of specific simple algebraic groups. As the Mumford--Tate groups areexactly the algebraic quotients of G, this also describes them.

Classification of the Mumford--Tate Groups of Rational Polarizable Hodge Structures