Formal groups over Hopf algebras

Abstract

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 $(H,μ,ν, Δ,ε, S)$ 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 $(Δ{\frak g})({\frak F}(x\otimes 1,1\otimes x))= {\frak c}+{\frak g}(x)\otimes 1+1\otimes {\frak g}(x),$ where ${\frak c}\in H_\mathbb{Q}{\mathop{\hat{\otimes}}\limits_{R_ \mathbb{Q}}}H_\mathbb{Q}, (\id \otimes ε){\frak c}=0=(ε\otimes \id){\frak c}$ and $(\id \otimes Δ){\frak c}+1\otimes {\frak c}-(Δ\otimes \id){\frak c}-{\frak c}\otimes 1=0$ (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.