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A lattice analogue of a theorem of Chow
Published online by Cambridge University Press: 09 April 2009
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Let K be a skewfield, E a left vector space over K, r an integer ≧ 1 and Gr(E) the set of all r-dimensional subspaces of E, called the Grassmannian of index r. The function d(A, B) = r — dim (A∩B) is a distance on Gr(E). If K′ is a skewfield and E′ a left vector space over K1, then any semilinear ismorphism u: E → E1(relative to an isomorphism K → K') induces a distance preserving bijection Gr(u):Gr(E) → Gr(E′). When E has finite dimension n and 2r = n, another example of such a mapping is obtained by taking K′ = Kop, E′ = E* and defining wr: Gr(E) → Gr(E*) to be wr(A) = {f ∈ E*&f(A) = 0}.
- Type
- Research Article
- Information
- Journal of the Australian Mathematical Society , Volume 16 , Issue 2 , September 1973 , pp. 216 - 219
- Copyright
- Copyright © Australian Mathematical Society 1973
References
[1]Birkhoff, G., Lattice Theory, (3rd edition Colloq. Publ. vol. 25, Amer. Math. Soc., 1967).Google Scholar
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[3]Dieudonné, J., La géométrie des groups classiques, (3ème edition Springer Verlag, Berlin, 1971).Google Scholar
[4]Schreier, J. and Ulam, S., ‘Über die Automorphismengruppe der Permutationsgruppe der natürlichen Zahlenfolge’, Fund. Math. 28 (1937), 258–260.CrossRefGoogle Scholar
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