The term sedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the
biquaternions, or the algebra of 4 × 4
matrices over the real numbers, or that studied by
Smith (1995).
Arithmetic
A visualization of a 4D extension to the cubic
octonion,[2] showing the 35 triads as
hyperplanes through the real vertex of the sedenion example given.
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 that, for any element x of , the power is well defined. They are also
flexible.
Every sedenion is a
linear combination of the unit sedenions , , , , ..., ,
which form a
basis of the
vector space of sedenions. Every sedenion can be represented in the form
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 they were constructed from. So, they contain the octonions (generated by to in the table below), and therefore also the
quaternions (generated by to ),
complex numbers (generated by and ) and real numbers (generated by ).
The sedenions have a multiplicative
identity element and multiplicative inverses, but they are not a
division algebra because they have
zero divisors. This means that two nonzero sedenions can be multiplied to obtain zero: an example is . All
hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.
A sedenion multiplication table is shown below:
Sedenion properties
From the above table, we can see that:
and
Anti-associative
The sedenions are not fully anti-associative. Choose any four generators, and . The following 5-cycle shows that these five relations cannot all be anti-associative.
In particular, in the table above, using and the last expression associates.
Quaternionic subalgebras
The 35 triads that make up this specific sedenion multiplication table with the 7 triads of the
octonions used in creating the sedenion through the
Cayley–Dickson construction shown in bold:
The binary representations of the indices of these triples
bitwise XOR to 0.
Moreno (1998) showed that the space of pairs of norm-one sedenions that multiply to zero is
homeomorphic to the compact form of the exceptional
Lie groupG2. (Note that in his paper, a "zero divisor" means a pair of elements that multiply to zero.)
Guillard & Gresnigt (2019) demonstrated that the three generations of
leptons and
quarks that are associated with unbroken
gauge symmetry can be represented using the algebra of the complexified sedenions . Their reasoning follows that a primitive
idempotentprojector — where is chosen as an
imaginary unit akin to for in the
Fano plane — that
acts on the
standard basis of the sedenions uniquely divides the algebra into three sets of
split basis elements for , whose adjoint
left actionson themselves generate three copies of the
Clifford algebra which in-turn contain
minimal left ideals that describe a single generation of
fermions with unbroken gauge symmetry. In particular, they note that
tensor products between normed division algebras generate zero divisors akin to those inside , where for the lack of alternativity and associativity does not affect the construction of minimal left ideals since their underlying split basis requires only two basis elements to be multiplied together, in-which associativity or alternativity are uninvolved. Still, these ideals constructed from an adjoint algebra of left actions of the algebra on itself remain associative, alternative, and
isomorphic to a Clifford algebra. Altogether, this permits three copies of to exist inside . Furthermore, these three complexified octonion subalgebras are not independent; they share a common subalgebra, which the authors note could form a theoretical basis for
CKM and
PMNS matrices that, respectively, describe
quark mixing and
neutrino oscillations.
Sedenion neural networks provide [further explanation needed] a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.[3][4]
Biss, Daniel K.; Christensen, J. Daniel; Dugger, Daniel; Isaksen, Daniel C. (2007). "Large annihilators in Cayley-Dickson algebras II". Boletin de la Sociedad Matematica Mexicana. 3: 269–292.
arXiv:math/0702075.
Bibcode:
2007math......2075B.
L. S. Saoud and H. Al-Marzouqi, "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm," in IEEE Access, vol. 8, pp. 144823-144838, 2020,
doi:10.1109/ACCESS.2020.3014690.
The term sedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the
biquaternions, or the algebra of 4 × 4
matrices over the real numbers, or that studied by
Smith (1995).
Arithmetic
A visualization of a 4D extension to the cubic
octonion,[2] showing the 35 triads as
hyperplanes through the real vertex of the sedenion example given.
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 that, for any element x of , the power is well defined. They are also
flexible.
Every sedenion is a
linear combination of the unit sedenions , , , , ..., ,
which form a
basis of the
vector space of sedenions. Every sedenion can be represented in the form
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 they were constructed from. So, they contain the octonions (generated by to in the table below), and therefore also the
quaternions (generated by to ),
complex numbers (generated by and ) and real numbers (generated by ).
The sedenions have a multiplicative
identity element and multiplicative inverses, but they are not a
division algebra because they have
zero divisors. This means that two nonzero sedenions can be multiplied to obtain zero: an example is . All
hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.
A sedenion multiplication table is shown below:
Sedenion properties
From the above table, we can see that:
and
Anti-associative
The sedenions are not fully anti-associative. Choose any four generators, and . The following 5-cycle shows that these five relations cannot all be anti-associative.
In particular, in the table above, using and the last expression associates.
Quaternionic subalgebras
The 35 triads that make up this specific sedenion multiplication table with the 7 triads of the
octonions used in creating the sedenion through the
Cayley–Dickson construction shown in bold:
The binary representations of the indices of these triples
bitwise XOR to 0.
Moreno (1998) showed that the space of pairs of norm-one sedenions that multiply to zero is
homeomorphic to the compact form of the exceptional
Lie groupG2. (Note that in his paper, a "zero divisor" means a pair of elements that multiply to zero.)
Guillard & Gresnigt (2019) demonstrated that the three generations of
leptons and
quarks that are associated with unbroken
gauge symmetry can be represented using the algebra of the complexified sedenions . Their reasoning follows that a primitive
idempotentprojector — where is chosen as an
imaginary unit akin to for in the
Fano plane — that
acts on the
standard basis of the sedenions uniquely divides the algebra into three sets of
split basis elements for , whose adjoint
left actionson themselves generate three copies of the
Clifford algebra which in-turn contain
minimal left ideals that describe a single generation of
fermions with unbroken gauge symmetry. In particular, they note that
tensor products between normed division algebras generate zero divisors akin to those inside , where for the lack of alternativity and associativity does not affect the construction of minimal left ideals since their underlying split basis requires only two basis elements to be multiplied together, in-which associativity or alternativity are uninvolved. Still, these ideals constructed from an adjoint algebra of left actions of the algebra on itself remain associative, alternative, and
isomorphic to a Clifford algebra. Altogether, this permits three copies of to exist inside . Furthermore, these three complexified octonion subalgebras are not independent; they share a common subalgebra, which the authors note could form a theoretical basis for
CKM and
PMNS matrices that, respectively, describe
quark mixing and
neutrino oscillations.
Sedenion neural networks provide [further explanation needed] a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.[3][4]
Biss, Daniel K.; Christensen, J. Daniel; Dugger, Daniel; Isaksen, Daniel C. (2007). "Large annihilators in Cayley-Dickson algebras II". Boletin de la Sociedad Matematica Mexicana. 3: 269–292.
arXiv:math/0702075.
Bibcode:
2007math......2075B.
L. S. Saoud and H. Al-Marzouqi, "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm," in IEEE Access, vol. 8, pp. 144823-144838, 2020,
doi:10.1109/ACCESS.2020.3014690.