scholarly journals Tensor products of positive definite quadratic forms IV

1979 ◽  
Vol 73 ◽  
pp. 149-156 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
Orthogonal Groups ◽  
Quadratic Lattices

Let L, M, N be positive definite quadratic lattices over Z. We treated the following problem in [5], [6]:If L ⊗ M is isometric to L ⊗ N, then is M isometric to N?We gave a condition (**) in [6] such that the answer is affirmative for an indecomposable lattice L satisfying (**), and we gave some examples. In this paper we introduce a certain apparently weaker condition (A) than the condition (**), and we show that the condition (A) implies the condition (**) and more on integral orthogonal groups than a result in [6].

scholarly journals Tensor Products of Positive definite Quadratic forms III

1978 ◽  
Vol 70 ◽  
pp. 173-181 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
List Type ◽  
Quadratic Lattices

In the previous papers [2], [3] we treated the following two questions. Let L,M,N be positive definite quadratic lattices over Z: (i) If L, M are indecomposable, then is L⊗M indecomposable?(ii) Does L ⊗ M ⋍ L ⊗ N imply M ⋍ N?

scholarly journals The minimum and the primitive representation of positive definite quadratic forms

1994 ◽  
Vol 133 ◽  
pp. 127-153 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Forms ◽  
Positive Definite ◽  
Primitive Representation ◽  
Quadratic Lattices

Let M, N be positive definite quadratic lattices over Z with rank(M) = m and rank(N) = n respectively. When there is an isometry from M to N, we say that M is represented by N (even in the local cases). In the following, we assume that the localization Mp is represented by Np for every prime p. Let us consider the following assertion Am,n(N):Am,n(N): There exists a constant c(N) dependent only on N so that M is represented by N if min(M) > c(N), where min(M) denotes the least positive number represented by M.

scholarly journals The minimum and the primitive representation of positive definite quadratic forms II

1996 ◽  
Vol 141 ◽  
pp. 1-27 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Primitive Representation ◽  
Quadratic Lattices

We are concerned with representation of positive definite quadratic forms by a positive definite quadratic form. Let us consider the following assertion Am, n : Let M, N be positive definite quadratic lattices over Z with rank(M) = m and rank(N) = n respectively. We assume that the localization Mp is represented by Np for every prime p, that is there is an isometry from Mp to Np. Then there exists a constant c(N) dependent only on N so that M is represented by N if min(M) > c(N), where min(M) denotes the least positive number represented by M.

scholarly journals Tensor products of positive definite quadratic forms, V

1981 ◽  
Vol 82 ◽  
pp. 99-111 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Tensor Product ◽  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
List Type

Our aim is to proveTHEOREM. Let L be a positive lattice of E-type such that [L: L̃] < ∞ and L̃ is indecomposable. (i)If L ≅ L1 ⊗ L2for 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.

scholarly journals Tensor products of positive definite quadratic forms, VII

1984 ◽  
Vol 96 ◽  
pp. 133-137 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Lie Algebras ◽  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Constant Multiple ◽  
The Third ◽  
Cartan Matrices

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.

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