Nelson, Cintya, Sueli Lattices Lattices from Octonion Algebras Octonion Algebras Lattices via Octonion Algebras Nelson G. Brasil Junior 1 Cintya W. O. Benedito 2 Examples Sueli I. R. Costa 1 1 Institute of Mathematics, Statistics and Computer Science (IMECC), University of Campinas (Unicamp) 2 S˜ ao Paulo State University (Unesp), Campus of S˜ ao Jo˜ ao da Boa Vista ... 1 / 6
Lattices A full-rank lattice Λ is a set of R n composed by all integer linear Nelson, Cintya, combination of n linearly independent vectors v 1 , · · · , v n ∈ R n . Sueli Lattices 4 Octonion Algebras Lattices via Octonion Algebras 2 Examples - 4 - 2 2 4 - 2 - 4 Figure: Hexagonal Lattice 2 / 6
Octonion Algebras An octonion algebra C = ( a, b, c ) K over a number field K is an Nelson, Cintya, algebra of dimension 8 over K , with basis { e 0 , . . . , e 7 } such Sueli that e 0 = 1 , e 2 1 = a, e 2 2 = b, e 2 4 = c , and a, b, c ∈ K \{ 0 } . Lattices Let C = ( a, b, c ) K an octonion algebra over K . If Octonion Algebras 7 � Lattices via x = x 0 + x i e i ∈ C and x 0 , . . . , x 7 ∈ K then Octonion Algebras i =1 Examples 7 � x = x 0 − x i e i is the conjugate of x . i =1 Reduced Trace and Reduced Norm of x is Trd ( x ) = x + x e Nrd ( x ) = x · x . e 1 e 2 e 3 e 4 e 5 e 6 e 7 1 · 1 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 1 e 1 a − e 4 ae 7 e 2 ae 6 e 5 − ae 3 e 2 e 2 e 4 b − be 5 − e 1 e 3 be 7 − be 6 e 3 e 3 − ae 7 be 5 ab − e 6 ae 2 − be 4 abe 1 e 4 e 4 − e 2 e 1 e 6 c ce 7 ce 3 − ce 5 e 5 e 5 − ae 6 − e 3 − ae 2 − ce 7 ac ce 1 ace 4 e 6 e 6 − e 5 − be 7 be 4 − ce 3 − ce 1 bc bce 2 e 7 e 7 ae 3 be 6 − abe 1 ce 5 − ace 4 − bce 2 abc 3 / 6
Lattices via Octonion Algebras I = O an order of C with basis B = { v 1 , . . . , v 8 } ; Nelson, Cintya, Sueli Z -basis of o K : { u 1 , . . . , u n } Lattices ( I , α ) , α ∈ K totally positive; Octonion Algebras B ′ = { v i u j } = { w 1 , . . . , w 8 n } , i = 1 , . . . , 8 e j = 1 , . . . , n. Lattices via Octonion Algebras Gram matrix: Examples G = tr K / Q ( αTrd ( w i w j )) , (1) Theorem Let Λ = ( I , α ) be the ideal lattice with a Gram matrix G as in (1). Then the determinant of Λ can be written as det (Λ) = d 8 K N ( α ) 8 N K / Q (det( B )) (2) where B = Trd ( v ℓ v ℓ ′ ) 8 ℓ,ℓ ′ =1 and v ℓ , v ℓ ′ ∈ B basis of I . 4 / 6
Examples Example Nelson, Cintya, Sueli K = Q ; C = ( − 1 , − 1 , − 1) K ; Lattices √ √ x = (1 + e 1 ) / 2 , y = (1 + e 2 ) / 2 e z = ( e 1 + e 2 + e 3 + e 4 ) / 2 ; Octonion Algebras O with basis A = { x, y, xy, z, xz, yz, ( xy ) z } , α = 1 ; Lattices via Octonion Algebras The resultant Gram matrix using (1) is unimodular which main Examples diagonal is even. Therefore, the lattice Λ = ( O , α ) is an ideal lattice congruent to the E 8 lattice. Example √ K = Q ( 2) ; C = ( − 1 , − 1 , − 1) K ; O with basis √ √ √ √ B = { 2 e, 2 x, 2 y, 2 xy, 2 2 z, 2 2 xz, 2 2 yz, 2 2( xy ) z } . √ Λ 2 = ( O , 2 + 2) has Gram matrix (after the LLL reduction) with determinant det ( G ) = 2 8 and squared minimum distance equals to 4 . 5 / 6
References I Nelson, Cintya, Sueli F.-T. Tu and Y. Yang, “Lattice packing from quaternion Lattices algebras,” Algebraic Number Theory and Related Topics , 2012. Octonion Algebras N. G. B. Brasil Jr., C. W. O. Benedito, and S. I. R. Costa, Lattices via Octonion “Lattices associated with octonion algebras,” To appear , 2018. Algebras Examples J. Baez, “The octonions,” Bulletin of the American Mathematical Society , vol. 39, no. 2, pp. 145–205, 2002. C. Waldner, “Cycles and the cohomology of arithmetic subgroups of the exceptional group g 2 ,” Ph.D. dissertation, uniwien, 2008. F. Van der Blij and T. Springer, “The arithmetics of octaves and of the group g2,” in Indagationes Mathematicae (Proceedings) , vol. 62. Elsevier, 1959, pp. 406–418. 6 / 6
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