  We prove the semisimplicity conjecture for A-motives over finitely generated
fields K. This conjecture states that the rational Tate modules V_p(M) of a
semisimple A-motive M are semisimple as representations of the absolute Galois
group of K. This theorem is in analogy with known results for abelian varieties
and Drinfeld modules, and has been sketched previously by Akio Tamagawa.
  We deduce two consequences of the theorem for the algebraic monodromy groups
G_p(M) associated to an A-motive M by Tannakian duality. The first requires no
semisimplicity condition on M and states that G_p(M) may be identified
naturally with the Zariski closure of the image of the absolute Galois group of
K in the automorphism group of V_p(M). The second states that the connected
component of G_p(M) is reductive if M is semisimple and has a separable
endomorphism algebra.
