Lie symmetries of the Chow group of a Jacobian and the tautological   subring
  Let $J$ be the Jacobian of a smooth projective curve. We define a natural action of the Lie algebra of polynomial Hamiltonian vector fields on the plane, vanishing at the origin, on the Chow group of $J$ with rational coefficients. Using this action we obtain some relations between tautological cycles on $J$. 
Alexander Polishchuk