In abstract algebra, sedenions form a 16dimensional nonassociative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions. The set of sedenions is denoted by .
The term "sedenion" is also used for other 16dimensional algebraic structures, such as a tensor product of 2 copies of the quaternions, or the algebra of 4 by 4 matrices over the reals, or that studied by Smith (1995).
Contents
Cayley–Dickson Sedenions
Arithmetic
Like (Cayley–Dickson) octonions, multiplication of Cayley–Dickson 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 , the power x^{n} is welldefined.
Every sedenion is a real linear combination of the unit sedenions 1, e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7}, e_{8}, e_{9}, e_{10}, e_{11}, e_{12}, e_{13}, e_{14} and e_{15}, which form a basis of the vector space of sedenions.
The sedenions have a multiplicative identity element 1 and multiplicative inverses, but they are not a division algebra. This is because they have zero divisors; this means that two nonzero numbers can be multiplied to obtain a zero result: a trivial example is (e_{3} + e_{10})×(e_{6} − e_{15}). All hypercomplex number systems based on the Cayley–Dickson construction from sedenions on contain zero divisors.
The multiplication table of these unit sedenions follows:
Applications
Moreno (1998) showed that the space of norm 1 zerodivisors of the sedenions is homeomorphic to the compact form of the exceptional Lie group G_{2}.
See also
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