Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free and concentrated in even degree.

Bott has also, using generating varieties, calculated this Hopf algebra structure for $G = SO(n)$, $Spin(n)$, $SU(n)$, $G_2$ in The space of loops on a Lie group. Michigan Math. J., 5:35-61, 1958. The same technique was used by Watanabe in The homology of the loop space of the exceptional group F4. Osaka J. Math., 15:463-474, 1978 for $F_4$ and by Nakagawa in The space of loops on the exceptional Lie group E6. Osaka J. Math., 40:429-448, 2003 for $E_6$. Using Bott periodicity, Kono and Kozima calculated this Hopf algebra structure on the homology for $Sp(n)$ in The space of loops on a symplectic group. Japanese J. Math., 4:461-480, 1978.

However, I haven't been able to locate similar calculations for the exceptional Lie groups $E_7$ and $E_8$. Does anyone know where these can be found? Seeing that it is already quite involved to do these calculations for $F_4$ and $E_6$, I'd rather not try to them myself. However, if some simple method exists, answers explaining it are also welcome.

Also note that Bott's result for $G_2$ in The space of loops on a Lie group seems to be incorrect. Watanabe writes in The homology of the loop space of the exceptional group F4:

There is a misprint in Bott's result on $H_\ast(\Omega G_2)$ [5;p. 60]. The coproduct formula for $w \in H_{10}(ΩG_2)$ is an error. It is corrected by exchanging 2 for 3.

This can also be found in Clarke's On the K -theory of the loop space of a Lie group, Proc. Camb. Phil. Soc. (1974), 76, 1.

  • 1
    $\begingroup$ Probably this is not what you are looking for, but the mod $2$ homology of $\Omega E_8$ is determined by Hamanaka as a Hopf algebra in the following paper: Hamanaka, Hiroaki. Adjoint action on homology mod $2$ of $E\sb 8$ on its loop space. J. Math. Kyoto Univ. 36 (1996), no. 4, 779--787. MR1443747 (98f:57062). But I guess you want the integral homology... $\endgroup$ – Dai Tamaki Aug 9 '10 at 13:50
  • $\begingroup$ Yes, I'd prefer the integral homology, but mod p is also appreciated. $\endgroup$ – skupers Aug 9 '10 at 14:11
  • $\begingroup$ Then you might be interested in two more papers by Hamanaka and coauthors. They are also in J. Math. Kyoto Univ. Both of them are in vol. 37. One of them is pp.169--176. The other one is pp.441--453. They study the mod $5$ and mod $3$ homology, respectively. $\endgroup$ – Dai Tamaki Aug 9 '10 at 15:19
  • $\begingroup$ As you noted, the homology is free as a Z-module. So the various p-primary computations can be used to recover the integral homology (actually, knowing that the homology is finitely generated as a Z-module would be enough for that). Having the mod p answer is not quite equivalent to having the p-primary answer, but it's a useful first step. [terminology: "p-primary" means "with coefficients in Z localized at p"] $\endgroup$ – André Henriques Aug 15 '10 at 6:46

This is more of a suggestion than an answer. Let $K$ be a compact Lie group and $G$ its complexification. It is known (see Pressley-Segal) that the based loop space of $K$ and the affine Grassmannian $Gr_G$ of $G$ have the same (co)homology groups. It is a theorem of Ginzburg that the the cohomology ring of $Gr_G$ may be (canonically) identified with the enveloping algebra of the centraliser of a regular nilpotent element $e \in \mathfrak{g}^{\vee}$ (here $\mathfrak{g}^{\vee}$ denotes the Lie algebra of the group $G^{\vee}$ Langlands dual to $G$).

One can dualise this to get that the homology of $Gr_G$ can be identified with the functions on $B^{\vee}_e$, where $B^{\vee}$ denotes a Borel subgroup of $G^{\vee}$ and $e$ is a regular nilpotent element in the Lie algebra of $B^{\vee}$. This is explained in a very nice paper by Yun and Zhu called "Integral homology of loop groups via Langlands dual groups". They prove that this induces an isomorphism of the corresponding group schemes (with the Hopf algebra structure on the homology of $Gr_G$ inducing the group scheme structure on one side).

So this doesn't provide explicit formulas, but it does give you a way to work out the Hopf algebra structure (even over the integers in the simply laced case -- again see Yun-Zhu for this statement) in terms of combinatorics of root systems.

If you are interested in torsion phenomena the paper "Some arithmetical results on semi-simple Lie algebras" by Springer might be useful.


Have you looked at 'The Homotopy Theory of Lie Groups' by M. Mimura (Chapter 19 in the Handbook of Algebraic Topology, Edited by I.M. James)? Mimura's chapter also has an extensive list of references; see especially [10], [11].

  • $\begingroup$ There's also a two-volume set by Mimura and Toda, called Topology of Lie groups with extensive calculations. $\endgroup$ – José Figueroa-O'Farrill Aug 7 '10 at 0:57
  • $\begingroup$ The survey in the Handbook of Algebraic Topology doesn't cover the Hopf algebra structure on the homology of the loop space, although I did use it to locate the article by Kono and Kozima. Mimura and Toda also doesn't seem to deal with this, though I only looked at the Google Books preview. $\endgroup$ – skupers Aug 7 '10 at 8:03

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