Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Octonion
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==Properties== Octonionic multiplication is neither [[commutative]]: :{{math|''e{{sub|i}} e{{sub|j}}'' {{=}} β''e{{sub|j}} e{{sub|i}}'' β ''e{{sub|j}} e{{sub|i}}''}} if {{mvar|i}}, {{mvar|j}} are distinct and non-zero, nor [[associative]]: :{{math|(''e{{sub|i}} e{{sub|j}}'') ''e{{sub|k}}'' {{=}} β''e{{sub|i}}'' (''e{{sub|j}} e{{sub|k}}'') β ''e{{sub|i}}''(''e{{sub|j}} e{{sub|k}}'')}} if {{mvar|i}}, {{mvar|j}}, {{mvar|k}} are distinct, non-zero and {{math|''e{{sub|i}} e{{sub|j}}'' β Β±''e{{sub|k}}''}}. The octonions do satisfy a weaker form of associativity: they are alternative. This means that the subalgebra generated by any two elements is associative. Actually, one can show that the subalgebra generated by any two elements of <math>\ \mathbb{O}\ </math> is [[isomorphic]] to [[real numbers|{{math|β}}]], [[complex numbers|{{math|β}}]], or [[quaternions|{{math|β}}]], all of which are associative. Because of their non-associativity, octonions cannot be represented by a subalgebra of a [[matrix ring]] over {{math|β}}, unlike the real numbers, complex numbers, and quaternions. The octonions do retain one important property shared by {{math|β}}, {{math|β}}, and {{math|β}}: the norm on <math>\ \mathbb{O}\ </math> satisfies :<math> \| x y \| = \| x \|\ \| y \| ~.</math> This equation means that the octonions form a [[composition algebra]]. The higher-dimensional algebras defined by the CayleyβDickson construction (starting with the [[sedenion]]s) all fail to satisfy this property. They all have [[zero divisor]]s. Wider number systems exist which have a multiplicative modulus (for example, 16 dimensional conic sedenions). Their modulus is defined differently from their norm, and they also contain zero divisors. As shown by [[Adolf Hurwitz|Hurwitz]], {{math|β}}, {{math|β}}, or {{math|β}}, and <math>\ \mathbb{O}\ </math> are the only normed division algebras over the real numbers. These four algebras also form the only alternative, finite-dimensional [[division algebra]]s over the real numbers ([[up to]] an isomorphism). Not being associative, the nonzero elements of <math>\ \mathbb{O}\ </math> do not form a [[Group (mathematics)|group]]. They do, however, form a [[loop (algebra)|loop]], specifically a [[Moufang loop]]. ===Commutator and cross product=== The [[commutator]] of two octonions {{mvar|x}} and {{mvar|y}} is given by :<math>[x, y] = xy - yx ~.</math> This is antisymmetric and imaginary. If it is considered only as a product on the imaginary subspace <math>\ \operatorname\mathcal{I_m}\bigl[\ \mathbb{O}\ \bigr]\ </math> it defines a product on that space, the [[seven-dimensional cross product]], given by :<math>x \times y = \tfrac{\ 1\ }{ 2 }\ (xy - yx) ~.</math> Like the [[cross product]] in three dimensions this is a vector orthogonal to {{mvar|x}} and {{mvar|y}} with magnitude :<math>\|x \times y\| = \|x\|\ \|y\|\ \sin \theta ~.</math> But like the octonion product it is not uniquely defined. Instead there are many different cross products, each one dependent on the choice of octonion product.<ref>{{harvp|Baez|2002|pp=37β38}}</ref> ===Automorphisms=== An [[automorphism]], {{mvar|A}}, of the octonions is an invertible [[linear transformation]] of <math>\ \mathbb{O}\ </math> which satisfies :<math>A(xy) = A(x)\ A(y) ~.</math> The set of all automorphisms of <math>\ \mathbb{O}\ </math> forms a group called {{nobr|{{math|[[G2 (mathematics)|''G''{{sub|2}}]]}} .}}<ref>{{harv|Conway|Smith|2003|loc=ch 8.6}}</ref> The group {{math|''G''{{sub|2}} }} is a [[simply connected]], [[Compact group|compact]], real [[Lie group]] of dimension 14. This group is the smallest of the exceptional Lie groups and is isomorphic to the [[subgroup]] of {{math|Spin(7)}} that preserves any chosen particular vector in its 8 dimensional real spinor representation. The group {{math|Spin(7)}} is in turn a subgroup of the group of isotopies described below. ''See also'': {{math|[[PSL(2,7)]]}} β the [[automorphism group]] of the Fano plane. ===Isotopies=== An [[isotopy of an algebra]] is a triple of [[bijection|bijective]] linear maps {{mvar|a}}, {{mvar|b}}, {{mvar|c}} such that if {{math|''xy'' {{=}} ''z''}} then {{math|''a''(''x'')''b''(''y'') {{=}} ''c''(''z'')}}. For {{math|''a'' {{=}} ''b'' {{=}} ''c''}} this is the same as an automorphism. The isotopy group of an algebra is the group of all isotopies, which contains the group of automorphisms as a subgroup. The isotopy group of the octonions is the group {{math|Spin<sub>8</sub>(β)}}, with {{mvar|a}}, {{mvar|b}}, {{mvar|c}} acting as the three 8 dimensional representations.<ref>{{harv|Conway|Smith|2003|loc=ch 8}}</ref> The subgroup of elements where {{mvar|c}} fixes the identity is the subgroup {{math|Spin<sub>7</sub>(β)}}, and the subgroup where {{mvar|a}}, {{mvar|b}}, {{mvar|c}} all fix the identity is the automorphism group {{nobr|{{math|''G''{{sub|2}} }} .}} ===Matrix representation=== Just as quaternions can be [[Quaternion#Matrix_representations|represented as matrices]], octonions can be represented as tables of quaternions. Specifically, because any octonion can be defined a pair of quaternions, we represent the octonion <math> ( q_0, q_1 )</math> as: <math display=block>\begin{bmatrix} q_0 & q_1 \\ -q_1^* & q_0^* \end{bmatrix}</math> Using a slightly modified (non-associative) quaternionic matrix multiplication: <math display=block>\begin{bmatrix} \alpha_0 & \alpha_1 \\ \alpha_2 & \alpha_3 \end{bmatrix}\circ\begin{bmatrix} \beta_0 & \beta_1 \\ \beta_2 & \beta_3 \end{bmatrix}=\begin{bmatrix} \alpha_0\beta_0+\beta_2\alpha_1 & \beta_1\alpha_0+\alpha_1\beta_3\\ \beta_0\alpha_2+\alpha_3\beta_2 & \alpha_2\beta_1+\alpha_3\beta_3 \end{bmatrix}</math> we can translate octonion addition and multiplication to the respective operations on quaternionic matrices.<ref name="Ensembles"></ref>
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)