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Sedenion

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Title: Sedenion  
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Subject: Octonion, Hypercomplex number, Non-associative algebra, Number systems, Irrational number
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Sedenion

In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions. The set of sedenions is denoted by \mathbb{S}.

The term "sedenion" is also used for other 16-dimensional algebraic structures, such as a tensor product of 2 copies of the biquaternions, or the algebra of 4 by 4 matrices over the reals, or that studied by Smith (1995).

Contents

  • Arithmetic 1
  • Applications 2
  • See also 3
  • References 4

Arithmetic

Like octonions, multiplication of sedenions is neither commutative nor associative. But in contrast to the octonions, the sedenions do not even have the property of being alternative. They do, however, have the property of power associativity, which can be stated as for any element x of \mathbb{S}, the power x^n is well-defined. They are also flexible.

Every sedenion is a linear combination of the unit sedenions e0, e1, e2, e3, ...,e15, which form a basis of the vector space of sedenions. Every sedenion can be represented in the form

x = x_0e_0 + x_1e_1 + x_2e_2 + \ldots + x_{14}e_{14} + x_{15}e_{15},\,.

Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is distributive over addition.

Like other algebras based on the Cayley–Dickson construction, the sedenions contain the algebra it was constructed from. So they contain the octonions (e0 to e7 in the table below), and therefore also the quaternions (e0 to e3), complex numbers (e0 and e1) and reals (e0).

The sedenions have a multiplicative identity element e0 and multiplicative inverses but they are not a division algebra because they have zero divisors. This means that two non-zero sedenions can be multiplied to obtain zero: an example is (e3 + e10)×(e6e15). All hypercomplex number systems based on the Cayley–Dickson construction after sedenions contain zero divisors.

The multiplication table of these unit sedenions follows:

 ×  e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15
e0 e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15
e1 e1 e0 e3 e2 e5 e4 e7 e6 e9 e8 e11 e10 e13 e12 e15 e14
e2 e2 e3 e0 e1 e6 e7 e4 e5 e10 e11 e8 e9 e14 e15 e12 e13
e3 e3 e2 e1 e0 e7 e6 e5 e4 e11 e10 e9 e8 e15 e14 e13 e12
e4 e4 e5 e6 e7 e0 e1 e2 e3 e12 e13 e14 e15 e8 e9 e10 e11
e5 e5 e4 e7 e6 e1 e0 e3 e2 e13 e12 e15 e14 e9 e8 e11 e10
e6 e6 e7 e4 e5 e2 e3 e0 e1 e14 e15 e12 e13 e10 e11 e8 e9
e7 e7 e6 e5 e4 e3 e2 e1 e0 e15 e14 e13 e12 e11 e10 e9 e8
e8 e8 e9 e10 e11 e12 e13 e14 e15 e0 e1 e2 e3 e4 e5 e6 e7
e9 e9 e8 e11 e10 e13 e12 e15 e14 e1 e0 e3 e2 e5 e4 e7 e6
e10 e10 e11 e8 e9 e14 e15 e12 e13 e2 e3 e0 e1 e6 e7 e4 e5
e11 e11 e10 e9 e8 e15 e14 e13 e12 e3 e2 e1 e0 e7 e6 e5 e4
e12 e12 e13 e14 e15 e8 e9 e10 e11 e4 e5 e6 e7 e0 e1 e2 e3
e13 e13 e12 e15 e14 e9 e8 e11 e10 e5 e4 e7 e6 e1 e0 e3 e2
e14 e14 e15 e12 e13 e10 e11 e8 e9 e6 e7 e4 e5 e2 e3 e0 e1
e15 e15 e14 e13 e12 e11 e10 e9 e8 e7 e6 e5 e4 e3 e2 e1 e0

From the above table, we can see that:

e_0e_i = e_ie_0 = e_i,
e_ie_i = -e_0 \,\, \text{for}\,\, i \neq 0,
e_ie_j = -e_je_i \,\, \text{for}\,\, i \neq j \,\,\text{and}\,\, i,j \neq 0.
e_i(e_je_k) = -(e_ie_j)e_k \,\,\text{for}\,\, i \neq j,\,\, i,j \neq 0\,\,\text{and}\,\, e_ie_j\neq \plusmn e_k.

Applications

Moreno (1998) showed that the space of norm 1 zero-divisors of the sedenions is homeomorphic to the compact form of the exceptional Lie group G2.

See also

References

  • Imaeda, K.; Imaeda, M. (2000), "Sedenions: algebra and analysis", Applied mathematics and computation 115 (2): 77–88,  
  • Kinyon, M.K., Phillips, J.D., Vojtěchovský, P.: C-loops: Extensions and constructions, Journal of Algebra and its Applications 6 (2007), no. 1, 1–20. [1]
  • Kivunge, Benard M. and Smith, Jonathan D. H: "Subloops of sedenions", Comment.Math.Univ.Carolinae 45,2 (2004)295–302.
  • Moreno, Guillermo (1998), "The zero divisors of the Cayley–Dickson algebras over the real numbers", Sociedad Matemática Mexicana. Boletí n. Tercera Serie 4 (1): 13–28,  
  • Smith, Jonathan D. H. (1995), "A left loop on the 15-sphere",  
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