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Tensor products of positive definite quadratic forms, V

Published online by Cambridge University Press:  22 January 2016

Yoshiyuki Kitaoka
Yoshiyuki Kitaoka*
Affiliation:
Department of Mathematics, Faculty of Science, Nagoya University, Chikusa-ku, Nagoya, 464, Japan
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Our aim is to prove

THEOREM. Let L be a positive lattice of E-type such that [L: L̃] < ∞ and L̃ is indecomposable.

  • (i) If L ≅ L1 ⊗ L2 for positive lattices L1, L2, then L1, L2 are of E-type and [L1: L̃1], [L2:L̃2] < ∞ and L̃1, L̃2 are indecomposable.

  • (ii) If L is indecomposable with respect to tensor product, then for each indecomposable positive lattice X we have

  • (1) L ⊗ X ≅ L ⊗ Y implies X ≅ Y for a positive lattice Y,

  • (2) If X= ⊗t L ⊗ X′ where X′ is not divided by L, then O(L ⊗ X) is generated by O(L), O(X′) and interchanges of L’s, and

  • (3) L ⊗ X is indecomposable.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 82 , June 1981 , pp. 99 - 111
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[1] Kitaoka, Y., Scalar extension of quadratic lattices II, Nagoya Math. J., 67 (1977), 159–164.Google Scholar
[2] Kitaoka, Y., Tensor products of positive definite quadratic forms, Göttingen Nachr. Nr., 4 (1977).Google Scholar
[3] Kitaoka, Y., Tensor products of positive definite quadratic forms II, J. reine angew. Math., 299/300 (1978), 161–170.Google Scholar
[4] Kitaoka, Y., Tensor products of positive definite quadratic forms IV, Nagoya Math. J., 73 (1979), 149–156.Google Scholar
[5] O’Meara, O. T., Introduction to quadratic forms, Berlin-Heidelberg-New York, 1963.Google Scholar