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Tensor products of positive definite quadratic forms, VII

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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In this paper we generalize results of the third paper of this series. As a corollary we can show the following: Let Li (1 ≤ i ≤ n) be a positive definite quadratic form which is equivalent to one of Cartan matrices of Lie algebras of type An (n ≥ 2), Dn (n ≥ 4), E6, E7, E8 and assume that is positive definite quadratic forms and satisfies that rk Mt ≥ 2 and implies rk K or rk L = 1. Then we have n = m and Lt is equivalent to a constant multiple of Ms(i) for some permutation s. Therefore we get the uniqueness of decompositions with respect to tensor products in this case.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 96 , December 1984 , pp. 133 - 137
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1984

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
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[ 4 ] Kitaoka, Y., Tensor products of positive definite quadratic forms, V, Nagoya Math. J., 82 (1981), 99–111.CrossRefGoogle Scholar
[ 5 ] O’Meara, O.T., Introduction to quadratic forms, Berlin-Heidelberg-New York, 1963.Google Scholar