New PDF release: A survey of the spherical space form problem

By J. F. Davis

ISBN-10: 3718602504

ISBN-13: 9783718602506

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36 Let us first recall the following well-known results given in the above reference: Cr,s ⊗C1,1 Cr+1,s+1 (one can even use a tensor product of Z2 -graded algebras); Cr,s ⊗ C0,2 Cs,r+2 ; C1,0 R ⊕ R, C0,1 C; Cr,s ⊗ C0,2 Cs,r+2 ; C2,0 C1,1 m(2, R), C0,2 H; m(n, R) ⊗ m(n, R) m(nm, R); C ⊗ C C ⊕ C; m(n, R) ⊗ R m(n, R); m(n, C) ⊗ C m(n, C);37 m(n, R) ⊗ H m(n, H); H ⊗ C m(2, C), H ⊗ H m(4, R); 35 C. T. C. Wall, Graded algebras anti-involutions, simple groups and symmetric spaces, op. cit. , for example, T.

Albert, op. cit. p. 51, for example, or J. P. Serre, Seminaire H. S. 1950–1951, 2e exposé 7-01: W. A. Benjamin, Inc, 1967, New York, Amsterdam). Theorem: Let A be a central simple algebra finite over K, and let f and g be two K-isomorphisms from a simple algebra B into A. Then, there exists an invertible element x ∈ A such that for any b ∈ B, f (b) = xg(b)x −1 . Corollary: Any K-automorphism of a central simple algebra finite over K is an inner automorphism. 26 1 Classic Groups: Clifford Algebras, Projective Quadrics, and Spin Groups quantity y = g = −g T has the property we desire.

C(E, q) is isomorphic to a total matrix algebra of degree 2r over C. C(E, q) possesses, up to an equivalence, a unique irreducible representation ρ of degree 2r . We call the corresponding space S of this representation the space of spinors for C. The representation of C + , ρ + induced by ρ, is called the ˜ of the spin representation of C + . ρ induces a representation of the Clifford group G + + ˜ ˜ even Clifford group G+ = G , and of the reduced Clifford group G0 , which are respectively denoted by ρ, ρ + , and ρ0+ and are also called spin representations.

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A survey of the spherical space form problem by J. F. Davis


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