  In this paper we study some generalization of the notion of a formal group
over ring, which may be called a formal group over Hopf algebra (FGoHA). The
first example of FGoHA was found under the study of cobordism's ring of some
$H$-space $\hat{Gr}$. The results, which are represented in this paper, show
that some constructions of the theory of formal group may be generalized to
FGoHA. For example, if ${\frak F}(x\otimes 1,1\otimes x) \in
(H{\mathop{\hat{\otimes}}\limits_R}H)[[x\otimes 1,1\otimes x]]$ is a FGoHA over
a Hopf algebra $(H,\mu,\nu, \Delta,\epsilon, S)$ over a ring $R$ without
torsion, then there exists a logarithm, i.e. the formal series ${\frak g}(x)\in
H_\mathbb{Q}[[x]]$ such that $(\Delta {\frak g})({\frak F}(x\otimes 1,1\otimes
x))= {\frak c}+{\frak g}(x)\otimes 1+1\otimes {\frak g}(x),$ where ${\frak
c}\in H_\mathbb{Q}{\mathop{\hat{\otimes}}\limits_{R_ \mathbb{Q}}}H_\mathbb{Q},
(\id \otimes \epsilon){\frak c}=0=(\epsilon \otimes \id){\frak c}$ and $(\id
\otimes \Delta){\frak c}+1\otimes {\frak c}-(\Delta \otimes \id){\frak
c}-{\frak c}\otimes 1=0$ (recall that the last condition means that ${\frak c}$
is a cocycle in the cobar complex of the Hopf algebra $H_\mathbb{\mathbb{Q}}$).
On the other hand, FGoHA have series of new properties. For example, the
convolution on a Hopf algebra allows us to get new FGoHA from given.
