Journal of Mathematics of Kyoto University

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

Masaki Nakagawa

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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 $x_{1},\ldots , x_{n}$.

Article information

J. Math. Kyoto Univ., Volume 44, Number 1 (2004), 43-53.

First available in Project Euclid: 14 August 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57T10: Homology and cohomology of Lie groups
Secondary: 17B55: Homological methods in Lie (super)algebras


Nakagawa, Masaki. Cohomology operations in the space of loops on the exceptional Lie group $E_6$. J. Math. Kyoto Univ. 44 (2004), no. 1, 43--53. doi:10.1215/kjm/1250283582.

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