TENSOR PRODUCTS OF POSITIVE DEFINITE ... - Project Euclid
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TENSOR PRODUCTS OF POSITIVE DEFINITE ... - Project Euclid
lattices M, N with rk M > 1, rk JV > 1, L is called indecomposable with respect to tensor products. Let A be a finite set and [ , ] a mapping from A X A to {ί|0 < t < 1}.
Y. Kitaoka Nagoya Math. J . Vol. 98 (1984), 133-137
TENSOR PRODUCTS OF POSITIVE DEFINITE QUADRATIC FORMS, VII YOSHIYUKI KITAOKA In this paper we generalize results of the third paper of this series. As a corollary we can show the following: Let Lt (1 < ί < 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 2 and Mi = K®L implies rk if or rk L = 1. Then we have n — m and Li is equivalent to a constant multiple of Ms(ί) for some permutation s. Therefore we get the uniqueness of decompositions with respect to tensor products in this case. We explain notations and terminology. By a positive lattice we mean a lattice on a positive definite quadratic space over the rational number field. Let L be a positive lattice. We put m(L) = min Q(x), xφO
where Q( ) is a quadratic form associated with L. Put SJl(L) = {xeL\ Q(x) = m(L)} and denote by L a submodule of L spanned by 9K(L). If 2ft(L(x) M) c {x®y\xeL,y e l } holds for every positive lattice M, then L is called of S-type and then Ίfl(L (x) M) = SK(L) ® SK(Λf), m ( ^ ® -M) = m(L)m(M) hold. Unless L is isometric to the tensor product of positive lattices M, N with rk M > 1, rk JV > 1, L is called indecomposable with respect to tensor products. Let A be a finite set and [ , ] a mapping from A X A to {ί|0 < t < 1} satisfying ( i ) [α, a'] = 1 if and only if a =
