The Hopf algebra structure of \(MU_*(\Omega Sp(n))\) (Q789054)

Let \(\Omega Sp(n)\) be the loop space of the \(n\)-th symplectic group. The author determines \(MU_*(\Omega Sp(n))\) as a Hopf algebra over \(MU_*=MU_*(point)\). Let \(C\) be an \(MU_*\)-algebra, and let \(f,g\in C[[x]]\). Define \(f\square g\in(C\otimes_{MU_*}C)[[x]]\) to be the product power series \(\sum_{i\geq 0}(\sum_{j+k=i}f_ j\otimes g_ k)\quad x^ i\). The main result of the paper is: there are elements \(r_{2i-1}\in MU_*(\Omega Sp(n))\) such that \(MU_*(\Omega Sp(n))=MU_*[r_ 1,r_ 3,\dots,r_{2n-1}]\) as an algebra and there exists \(P(x)\in MU_*[[x]]\) such that the diagonal \(\phi\) is given by \[ \phi(r_{2k-1})=[\frac{1\square R_ n+R_ n\square 1+P\cdot(R_ n\square R_ n)}{1\otimes 1+R_ n\square R_ n}]_{2k-1} \] where \(R_ n=\sum^{n}_{i=1}r_{2i-1}\quad x^{2i-1}\) and \([f]_ j\) denotes the coefficient of \(x^ j\) in the power series \(f\).