The adjoint action of a Lie group on the space of loops

Akira KONO; Kazumoto KOZIMA

doi:10.2969/jmsj/04530495

Akira KONO, Kazumoto KOZIMA

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ジャーナルフリー

1993 年 45 巻 3 号 p. 495-510

DOI https://doi.org/10.2969/jmsj/04530495

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Published: 1993 Received: 1992/03/18 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) S. Araki, Differential Hopf algebra and the cohomology mod 3 of the compact exceptional groups E₇ and E₈, Ann. of Math., 73 (1961), 404-436.
2) S. Araki, Cohomology modulo 2 of the compact exceptional groups E₆ and E₇, J, Math. Osaka City Univ., 12 (1961), 43-65.
3) S. Araki and Y. Shikata, Cohomology mod 2 of the compact exceptional group E₈, Proc. Japan Acad., 37 (1961), 619-622.
4) M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London, 308A (1982), 523-615.
5) A. Borel, Sur l'homologie et la cohomologie des groupes de Lie compacts connexes, Amer. J. Math., 76 (1954), 273-342.
6) A. Borel and J. P. Serre, Groupes de Lie et puissances reduites de Steenrod, Amer. J. Math., 73 (1953), 409-448.
7) R. Bott, An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. France, 84 (1956), 251-281.
8) R. Bott, The space of loops on a Lie group, Michigan Math. J., 5 (1958), 35-61.
9) P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland, 1975.
10) K. Ishitoya, A. Kono and H. Toda, Hopf algebra structure of mod 2 cohomology of simple Lie group, Publ. Res. Inst. Math. Sci., Kyoto Univ., 12 (1976), 141-167.
11) A. Kono, Hopf algebra structure of simple Lie groups, J. Math. Kyoto Univ., 17 (1977), 259-298.
12) A. Kono, On the cohomology of the 2-connected cover of the loop space of simple Lie groups, Publ. Res. Inst. Math. Sci., Kyoto Univ., 22 (1986), 537-541.
13) A. Kono and K. Kozima, The mod 2 homology of the space of loops on the exceptional Lie group, Proc. Roy. Soc. Edinburgh, 112A (1989), 187-202.
14) A. Kono and K. Kozima, Homology of the Kac-Moody groups III, J. Math. Kyoto Univ., 31-4 (1991), 1115-1120.
15) A. Kono and M. Mimura, Cohomology mod 2 of the classifying space of the compact connected Lie group of type E₆, J. Pure Appl. Algebra, 6 (1975), 61-81.
16) A. Kono and M. Mimura, Cohomology operations and Hopf algebra structure of the compact, exceptional Lie groups E₇ and E₈, Proc. London Math. Soc., III 35 (1977), 345-358.
17) J. Milnor and J. Moore, On the structure of Hopf algebra, Ann. of Math., 81 (1965), 211-264.
18) M. Mimura, Homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ., 6 (1967), 131-176.
19) M. Mimura and H. Toda, Cohomology operations and homotopy of compact Lie groups I, Topology, 9 (1970), 317-336.
20) J. P. Serre, Groupes d'homotopie et classes de groupes abéliens, Ann. of Math., 58 (1953), 258-294.
21) H. Toda, Cohomology of classifying space of exceptional Lie group, Conference on manifolds, Tokyo, 1973, pp. 265-271.
22) H. Toda, On the cohomology rings of some homogeneous spaces, J. Math. Kyoto Univ., 15 (1975), 185-199.

Right : [1] S. Araki, Differential Hopf algebra and the cohomology mod 3 of the compact exceptional groups E₇ and E₈, Ann. of Math., 73 (1961), 404-436.
[2] S. Araki, Cohomology modulo 2 of the compact exceptional groups E₆ and E₇, J, Math. Osaka City Univ., 12 (1961), 43-65.
[3] S. Araki and Y. Shikata, Cohomology mod 2 of the compact exceptional group E₈, Proc. Japan Acad., 37 (1961), 619-622.
[4] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London, 308A (1982), 523-615.
[5] A. Borel, Sur l'homologie et la cohomologie des groupes de Lie compacts connexes, Amer. J. Math., 76 (1954), 273-342.
[6] A. Borel and J. P. Serre, Groupes de Lie et puissances reduites de Steenrod, Amer. J. Math., 73 (1953), 409-448.
[7] R. Bott, An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. France, 84 (1956), 251-281.
[8] R. Bott, The space of loops on a Lie group, Michigan Math. J., 5 (1958), 35-61.
[9] P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland, 1975.
[10] K. Ishitoya, A. Kono and H. Toda, Hopf algebra structure of mod 2 cohomology of simple Lie group, Publ. Res. Inst. Math. Sci., Kyoto Univ., 12 (1976), 141-167.
[11] A. Kono, Hopf algebra structure of simple Lie groups, J. Math. Kyoto Univ., 17 (1977), 259-298.
[12] A. Kono, On the cohomology of the 2-connected cover of the loop space of simple Lie groups, Publ. Res. Inst. Math. Sci., Kyoto Univ., 22 (1986), 537-541.
[13] A. Kono and K. Kozima, The mod 2 homology of the space of loops on the exceptional Lie group, Proc. Roy. Soc. Edinburgh, 112A (1989), 187-202.
[14] A. Kono and K. Kozima, Homology of the Kac-Moody groups III, J. Math. Kyoto Univ., 31-4 (1991), 1115-1120.
[15] A. Kono and M. Mimura, Cohomology mod 2 of the classifying space of the compact connected Lie group of type E₆, J. Pure Appl. Algebra, 6 (1975), 61-81.
[16] A. Kono and M. Mimura, Cohomology operations and Hopf algebra structure of the compact, exceptional Lie groups E₇ and E₈, Proc. London Math. Soc., III 35 (1977), 345-358.
[17] J. Milnor and J. Moore, On the structure of Hopf algebra, Ann. of Math., 81 (1965), 211-264.
[18] M. Mimura, Homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ., 6 (1967), 131-176.
[19] M. Mimura and H. Toda, Cohomology operations and homotopy of compact Lie groups I, Topology, 9 (1970), 317-336.
[20] J. P. Serre, Groupes d'homotopie et classes de groupes abéliens, Ann. of Math., 58 (1953), 258-294.
[21] H. Toda, Cohomology of classifying space of exceptional Lie group, Conference on manifolds, Tokyo, 1973, pp. 265-271.
[22] H. Toda, On the cohomology rings of some homogeneous spaces, J. Math. Kyoto Univ., 15 (1975), 185-199.

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