The Hopf algebra structure of $K_*(\Omega Sp (n))$

Abstract

Let G be a compact connected Lie group and QG the space of loops on G .R. Bott introduced an idea "the generating variety" for Q G and determined the bicommutative Hopf algebra H ( Q G ) for G= S U (n), S pin(n) and G , ([5]).Recently, F. Clarke determined the Hopf algebra structure of K ( Q G ) for G = S U(n), S pin(n) and G , where I( * ( ) is the Z/2Z-graded K-homology theory using the generating varieties ([8]).But the results for G= S p(n) is not known.In our recent paper [10], A. Kono and myself determined the Hopf algebrawhere h , ( ) is a complex oriented homology theory.However the method used there is not applicable for 1 -1,(S2Sp(n)) with h= K or MU.The purpose of this paper is to determine IC,(QSp(n)) as a Hopf algebra over Z .By the result of R. Bott, QS U and B U are homotopy equivalent as an H-space, and the Hopf algebraAs in proved in [10], we may consider K, k (S2Sp) a s a Hopf subalgebra of K (S 2 S U ) b y ( Q e ) * w h e re c: S p->S U is the complexification m a p .Moreover 1<",(S2Sp(n)) is a Hopf subalgebra of K (Q S p ) (cf.Theorem 1.1).Let R be a commutative ring with unit andthe diagonal 0 is given by Or2kj - (11=IT)(x)+ (P,, C 1 1 )(x) x • (F1=1F.)(x) 101 + (P"E P,i )(x)where F ( x ) = E7= , r2 i _1x 2 -1 and [E a i xi] 1 denotes the coefficient of x 5 in E a i xi.

Locations

• Kyoto journal of mathematics - View - PDF