Lattices from Octonion Algebras Octonion Algebras Lattices via - - PowerPoint PPT Presentation

Nelson, Cintya, Sueli Lattices Octonion Algebras Lattices via Octonion Algebras Examples

Lattices from Octonion Algebras

Nelson G. Brasil Junior1 Cintya W. O. Benedito2 Sueli I. R. Costa1

1Institute of Mathematics, Statistics and Computer Science (IMECC), University of

Campinas (Unicamp)

2S˜

ao Paulo State University (Unesp), Campus of S˜ ao Jo˜ ao da Boa Vista

...

1 / 6

Nelson, Cintya, Sueli Lattices Octonion Algebras Lattices via Octonion Algebras Examples

Lattices

A full-rank lattice Λ is a set of Rn composed by all integer linear combination of n linearly independent vectors v1, · · · , vn ∈ Rn.

• 4
• 2

2 4

• 4
• 2

2 4

Figure: Hexagonal Lattice

2 / 6

Nelson, Cintya, Sueli Lattices Octonion Algebras Lattices via Octonion Algebras Examples

Octonion Algebras

An octonion algebra C = (a, b, c)K over a number field K is an algebra of dimension 8 over K, with basis {e0, . . . , e7} such that e0 = 1, e2

1 = a, e2 2 = b, e2 4 = c, and a, b, c ∈ K\{0}.

Let C = (a, b, c)K an octonion algebra over K. If x = x0 +

7

• i=1

xiei ∈ C and x0, . . . , x7 ∈ K then x = x0 −

7

• i=1

xiei is the conjugate of x. Reduced Trace and Reduced Norm of x is Trd(x) = x + x e Nrd(x) = x · x.

· 1 e1 e2 e3 e4 e5 e6 e7 1 1 e1 e2 e3 e4 e5 e6 e7 e1 e1 a −e4 ae7 e2 ae6 e5 −ae3 e2 e2 e4 b −be5 −e1 e3 be7 −be6 e3 e3 −ae7 be5 ab −e6 ae2 −be4 abe1 e4 e4 −e2 e1 e6 c ce7 ce3 −ce5 e5 e5 −ae6 −e3 −ae2 −ce7 ac ce1 ace4 e6 e6 −e5 −be7 be4 −ce3 −ce1 bc bce2 e7 e7 ae3 be6 −abe1 ce5 −ace4 −bce2 abc 3 / 6

Nelson, Cintya, Sueli Lattices Octonion Algebras Lattices via Octonion Algebras Examples

Lattices via Octonion Algebras

I = O an order of C with basis B = {v1, . . . , v8}; Z-basis of oK: {u1, . . . , un} (I, α), α ∈ K totally positive; B′ = {viuj} = {w1, . . . , w8n}, i = 1, . . . , 8 e j = 1, . . . , n. Gram matrix: G = trK/Q(αTrd(wiwj)), (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) = d8

KN(α)8NK/Q (det(B))

(2) where B = Trd(vℓvℓ′)8

ℓ,ℓ′=1 and vℓ, vℓ′ ∈ B basis of I. 4 / 6

Nelson, Cintya, Sueli Lattices Octonion Algebras Lattices via Octonion Algebras Examples

Examples

Example K = Q; C = (−1, −1, −1)K; x = (1 + e1)/ √ 2, y = (1 + e2)/ √ 2 e z = (e1 + e2 + e3 + e4)/2; O with basis A = {x, y, xy, z, xz, yz, (xy)z}, α = 1; The resultant Gram matrix using (1) is unimodular which main diagonal is even. Therefore, the lattice Λ = (O, α) is an ideal lattice congruent to the E8 lattice. Example K = Q( √ 2); C = (−1, −1, −1)K; O with basis B = {2e, 2x, 2y, 2xy, 2 √ 2z, 2 √ 2xz, 2 √ 2yz, 2 √ 2(xy)z}. Λ2 = (O, 2 + √ 2) has Gram matrix (after the LLL reduction) with determinant det (G) = 28 and squared minimum distance equals to 4.

5 / 6

Nelson, Cintya, Sueli Lattices Octonion Algebras Lattices via Octonion Algebras Examples

References I

F.-T. Tu and Y. Yang, “Lattice packing from quaternion algebras,” Algebraic Number Theory and Related Topics, 2012.

• N. G. B. Brasil Jr., C. W. O. Benedito, and S. I. R. Costa,

“Lattices associated with octonion algebras,” To appear, 2018.

• 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
• f the exceptional group g2,” Ph.D. dissertation, uniwien, 2008.
• F. Van der Blij and T. Springer, “The arithmetics of octaves and
• f the group g2,” in Indagationes Mathematicae (Proceedings),
• vol. 62.

Elsevier, 1959, pp. 406–418.

6 / 6