Cohomology operations in the space of loops on the exceptional Lie group $E_6$

Abstract

Let $E_{6}$ be the compact 1-connected exceptional Lie group of rank 6. In [9] we determined the Hopf algebra structure of $H_{*}(\Omega E_{6}; \mathbb{Z})$ by the generating variety approach of R. Bott [1]. In this case, as a generating variety we can take $EIII$, the irreducible Hermitian symmetric space of exceptional type. Then as Bott pointed out in [1], §6, we can determine the action of the mod $p$ Steenrod algebra $\mathcal{A}_{p}$ on $H^{*}(\Omega E_{6}; \mathbb{Z}_{p})$ from that on $H^{*}(EIII; \mathbb{Z}_{p})$ for all primes $p$.

In this paper, for ease of algebraic description, we compute the action of $\mathcal{A}_{p_{*}}$, the dual of $\mathcal{A}_{p}$, on $H^{*}(\Omega E_{6}; \mathbb{Z}_{p})$ for $p = 2, 3$ (For larger primes the description is easy). In the course of computation we also determine the action of $\mathcal{A}_{3}$ on $H^{*}(E_{6}/T; \mathbb{Z}_{3})$, where T is a maximal torus of $E_{6}$.

The paper is constructed as follows: In Section 2 we recall some results concerning the cohomology of some homogeneous spaces of $E_{6}$. In Section 3 by considering the action of the Weyl group on $E_{6}/T$, we determine the cohomology operations in $EIII$. Using the results obtained, in Section 4 we shall determine the cohomology operations in $\Omega E_{6}$.

Throughout this paper $\sigma _{i}(x_{1},\ldots , x_{n})$ denotes the $i$-th elementary symmetric function in the variables $x_{1},\ldots , x_{n}$.