Characterization of the mod 3 cohomology of $E_7$

Kono, Akira; Lin, James P.; Nishimura, Osamu

doi:10.1090/s0002-9939-03-07024-2

Cited by 3 publications

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“…Cherenousov [7] (2003) used Hasse principle to prove that the Rost invariant has trivial kernel for a quasi-split E 7 group. Kono, Lin and Nishimura [26] (2003) characterized the mod 3 cohomology of E 7 . Weiss [36] (2006) solved a fundamental question about the structure of the automorphism group of the Moufang quadrangles of type E 7 .…”

Section: Introductionmentioning

confidence: 99%

Polynomial Representation of $E_7$ and Its Combinatorial and PDE Implications

2008

Preprint

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In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of E 7 into a sum of irreducible submodules.Moreover, we obtain a combinatorial identity, saying that the dimensions of certain irreducible modules of E 7 are correlated by the binomial coefficients of fifty-five. Furthermore, we prove that two families of irreducible submodules with three integral parameters are solutions of the fundamental invariant differential operator corresponding to Cartan's unique quartic E 7 invariant.

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Section: Introductionmentioning

confidence: 99%

Polynomial Representation of $E_7$ and Its Combinatorial and PDE Implications

2008

Preprint

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H-spaces analogous to E8mod?3

Lin

2004

Math. Z.

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Let p be an odd prime. Let X 0 be a finite, p-local, simply connected homotopy associative H -space. Suppose H * (X 0 ; Z p ) contains the subalgebrasatisfying z 0 = P p x 0 = Q 0 P p P 1 r 0 , P p P 1 r 0 = P 1 y 0 for r 0 ∈ H 3 (X 0 ; Z p ). The only known examples occur for p = 3 and involve the Lie group E 8 . In this note we prove that if X 0 exists, then p must be 3. (2000): 55N22, 55P35, 55P45, 55Q25, 55R05, 55S05, 55S10, 55T10 Mathematics Subject Classification IntroductionIn this note we begin a study of the following questions. Question 1.For p an odd prime and X 0 a finite H -space when is QH even (X 0 ; Z p ) nontrivial?In the late 70's it was shown [21] that QH even (X 0 ; Z p ) could be nontrivial only in degrees of the form 2(p k + p k−1 +p i + · · · + 1). On the other hand, there are examples only for 2p + 2 for any p, and 2p 2 + 2 for p = 3. One of the major questions for finite H -space theory is to determine the possible degrees for QH even (X 0 ; Z p ).If X 0 admits the structure of a p-complete finite loop space, one can construct a generalized maximal torus and Weyl group [7], [8], [26]. Using these ideas, one can classify the mod p cohomology algebras of finite p-complete loop spaces [1]. It is known [1] that QH even (X 0 ; Z p ) can only be nontrivial in degrees 2p + 2 and

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Homology rings of homotopy associative H-spaces

Lin

2008

Topology and its Applications

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MSC:Let X be a homotopy associative mod p H-space for p an odd prime. The homology H * (X; F p ) is an associative ring, but not necessarily commutative. We study conditions when [x, y] = 0 for x, y elements of H * (X; F p ). Under certain conditions [x, y] = 0 imply ad l (x, y) = 0 for l = p − 2 or p − 1. These methods can be used to prove results about homology commutators that were previously obtained using the adjoint action [H. Hamanaka, S. Hara, A. Kono, Adjoint action of Lie groups on the loop spaces and cohomology of exceptional Lie groups, Transform. Group Theory (1996) 44-50, Korea Adv. Inst. Sci. Tech.; A. Kono, K. Kozima, The adjoint action of a Lie group on the space of loops, J. Math. Soc. Japan 45 (3) (1993) 495-509; A. Kono, J. Lin, O. Nishimura, Characterization of the mod 3 cohomology of E 7 , Proc. Amer. Math. Soc. 131 (10) (2003) 3289-3295]. We also generalize results of Kane [R. Kane, Torsion in homotopy associative H-spaces, Illinois J. Math. 20 (1976) 476-485] to nonfinite mod p homotopy associative H-spaces.

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Characterization of the mod 3 cohomology of $E_7$

Abstract: Abstract. It is shown that the mod 3 cohomology of a homotopy associative mod 3 H-space which is rationally equivalent to the Lie group E 7 and which has integral 3-torsion is isomorphic to that of E 7 as a Hopf algebra over the mod 3 Steenrod algebra.

Cited by 3 publications

References 20 publications

Polynomial Representation of $E_7$ and Its Combinatorial and PDE Implications

Polynomial Representation of $E_7$ and Its Combinatorial and PDE Implications

H-spaces analogous to E8mod?3

Homology rings of homotopy associative H-spaces

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