Tensor products of positive definite quadratic forms. II.

1978 ◽  
Vol 1978 (299-300) ◽  
pp. 161-170
Keyword(s):  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite

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, 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.

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?

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

Local deformations of positive-definite quadratic forms

2012 ◽  
Vol 64 (7) ◽  
pp. 1019-1035
Author(s):  
V. M. Bondarenko ◽  
V. V. Bondarenko ◽  
Yu. N. Pereguda
Keyword(s):  
Quadratic Forms ◽  
Positive Definite ◽  
Local Deformations

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.

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