  In the present notes, we study a generalization of the Peterson subalgebra to
an oriented (generalized) cohomology theory which we call the formal Peterson
subalgebra. Observe that by recent results of Zhong the dual of the formal
Peterson algebra provides an algebraic model for the oriented cohomology of the
affine Grassmannian.
  Our first result shows that the centre of the formal affine Demazure algebra
generates the formal Peterson subalgebra. Our second observation is motivated
by the Peterson conjecture. We show that a certain localization of the formal
Peterson subalgebra for the extended Dynkin diagram of type $\hat A_1$ provides
an algebraic model for `quantum' oriented cohomology of the projective line.
Our last result can be viewed as an extension of the previous results on Hopf
algebroids of structure algebras of moment graphs to the case of affine root
systems. We prove that the dual of the formal Peterson subalgebra (an oriented
cohomology of the affine Grassmannian) is the $0$th Hochshild homology of the
formal affine Demazure algebra.
