Octave (math)

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The (real) octaves , also octonions or Cayley numbers , are an extension of the quaternions and have the set symbol . They arise through the application of the doubling process from the quaternions and form an alternative body . As a coordinate range, they thus provide an example of a real, i.e. non- Desarguean Moufang plane in synthetic geometry . ${\ displaystyle \ mathbb {O}}$

history

The octonions were first described in 1843 by John Thomas Graves in a letter to William Rowan Hamilton . Separately, they were published (first) by Arthur Cayley in 1845 .

Multiplication table

The octonions are an 8-dimensional algebra over the real numbers . A possible multiplication is - with the base  - given as follows: ${\ displaystyle (1, \ mathrm {i}, \ mathrm {j}, \ mathrm {k}, \ mathrm {l}, \ mathrm {m}, \ mathrm {n}, \ mathrm {o})}$

${\ displaystyle {\ begin {matrix} \ mathrm {i} ^ {2} = \ mathrm {j} ^ {2} = \ mathrm {k} ^ {2} = \ mathrm {l} ^ {2} = \ mathrm {m} ^ {2} = \ mathrm {n} ^ {2} = \ mathrm {o} ^ {2} = - 1 \\\ mathrm {i} = \ mathrm {jk} = \ mathrm {lm} = \ mathrm {on} = - \ mathrm {kj} = - \ mathrm {ml} = - \ mathrm {no} \\\ mathrm {j} = \ mathrm {ki} = \ mathrm {ln} = \ mathrm { mo} = - \ mathrm {ik} = - \ mathrm {nl} = - \ mathrm {om} \\\ mathrm {k} = \ mathrm {ij} = \ mathrm {lo} = \ mathrm {nm} = - \ mathrm {ji} = - \ mathrm {ol} = - \ mathrm {mn} \\\ mathrm {l} = \ mathrm {mi} = \ mathrm {nj} = \ mathrm {ok} = - \ mathrm {im } = - \ mathrm {jn} = - \ mathrm {ko} \\\ mathrm {m} = \ mathrm {il} = \ mathrm {oj} = \ mathrm {kn} = - \ mathrm {li} = - \ mathrm {jo} = - \ mathrm {nk} \\\ mathrm {n} = \ mathrm {jl} = \ mathrm {io} = \ mathrm {mk} = - \ mathrm {lj} = - \ mathrm {oi} = - \ mathrm {km} \\\ mathrm {o} = \ mathrm {ni} = \ mathrm {jm} = \ mathrm {kl} = - \ mathrm {in} = - \ mathrm {mj} = - \ mathrm {lk} \ end {matrix}}}$

This can be used to calculate the units:

${\ displaystyle \ mathrm {((((((ij) k) l) m) n) o} = -1}$

Instead of real numbers , the elements can also be adjoint to another (commutative) field - with the given multiplication table and as the center . The result is called the Cayley algebra over (which, however, is not for everyone without a zero divisor ). ${\ displaystyle (\ mathrm {i, j, k, l, m, n, o})}$ ${\ displaystyle K}$ ${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle K}$

With the notation of the unit octonions in the form

${\ displaystyle \ {e_ {0}, e_ {1}, e_ {2}, e_ {3}, e_ {4}, e_ {5}, e_ {6}, e_ {7} \},}$

where the scalar element is designated and can be identified with the real number 1, the multiplication matrix is ​​written: ${\ displaystyle e_ {0}}$

 ${\ displaystyle e_ {i} e_ {j}}$ ${\ displaystyle e_ {0}}$ ${\ displaystyle e_ {1}}$ ${\ displaystyle e_ {2}}$ ${\ displaystyle e_ {3}}$ ${\ displaystyle e_ {4}}$ ${\ displaystyle e_ {5}}$ ${\ displaystyle e_ {6}}$ ${\ displaystyle e_ {7}}$ ${\ displaystyle e_ {j}}$ ${\ displaystyle e_ {i}}$ ${\ displaystyle e_ {0}}$ ${\ displaystyle e_ {1}}$ ${\ displaystyle e_ {2}}$ ${\ displaystyle e_ {3}}$ ${\ displaystyle e_ {4}}$ ${\ displaystyle e_ {5}}$ ${\ displaystyle e_ {6}}$ ${\ displaystyle e_ {7}}$ ${\ displaystyle e_ {1}}$ ${\ displaystyle -e_ {0}}$ ${\ displaystyle e_ {3}}$ ${\ displaystyle -e_ {2}}$ ${\ displaystyle e_ {5}}$ ${\ displaystyle -e_ {4}}$ ${\ displaystyle -e_ {7}}$ ${\ displaystyle e_ {6}}$ ${\ displaystyle e_ {2}}$ ${\ displaystyle -e_ {3}}$ ${\ displaystyle -e_ {0}}$ ${\ displaystyle e_ {1}}$ ${\ displaystyle e_ {6}}$ ${\ displaystyle e_ {7}}$ ${\ displaystyle -e_ {4}}$ ${\ displaystyle -e_ {5}}$ ${\ displaystyle e_ {3}}$ ${\ displaystyle e_ {2}}$ ${\ displaystyle -e_ {1}}$ ${\ displaystyle -e_ {0}}$ ${\ displaystyle e_ {7}}$ ${\ displaystyle -e_ {6}}$ ${\ displaystyle e_ {5}}$ ${\ displaystyle -e_ {4}}$ ${\ displaystyle e_ {4}}$ ${\ displaystyle -e_ {5}}$ ${\ displaystyle -e_ {6}}$ ${\ displaystyle -e_ {7}}$ ${\ displaystyle -e_ {0}}$ ${\ displaystyle e_ {1}}$ ${\ displaystyle e_ {2}}$ ${\ displaystyle e_ {3}}$ ${\ displaystyle e_ {5}}$ ${\ displaystyle e_ {4}}$ ${\ displaystyle -e_ {7}}$ ${\ displaystyle e_ {6}}$ ${\ displaystyle -e_ {1}}$ ${\ displaystyle -e_ {0}}$ ${\ displaystyle -e_ {3}}$ ${\ displaystyle e_ {2}}$ ${\ displaystyle e_ {6}}$ ${\ displaystyle e_ {7}}$ ${\ displaystyle e_ {4}}$ ${\ displaystyle -e_ {5}}$ ${\ displaystyle -e_ {2}}$ ${\ displaystyle e_ {3}}$ ${\ displaystyle -e_ {0}}$ ${\ displaystyle -e_ {1}}$ ${\ displaystyle e_ {7}}$ ${\ displaystyle -e_ {6}}$ ${\ displaystyle e_ {5}}$ ${\ displaystyle e_ {4}}$ ${\ displaystyle -e_ {3}}$ ${\ displaystyle -e_ {2}}$ ${\ displaystyle e_ {1}}$ ${\ displaystyle -e_ {0}}$

Except for the elements in the column and row, the matrix is ​​skew symmetrical. The multiplication can also be written: ${\ displaystyle e_ {0}}$

${\ displaystyle e_ {i} e_ {j} = {\ begin {cases} e_ {j}, & {\ text {for}} i = 0 \\ e_ {i}, & {\ text {for}} j = 0 \\ - \ delta _ {ij} e_ {0} + \ varepsilon _ {ijk} e_ {k}, & {\ text {otherwise}} \ end {cases}}}$

with the Kronecker delta and the completely antisymmetric tensor with the value +1 for ijk = 123, 145, 176, 246, 257, 347, 365. ${\ displaystyle \ delta _ {ij}}$${\ displaystyle \ varepsilon _ {ijk}}$

This is not the only choice of the multiplication table, there are 480 other possibilities, generated by permutation of the associated with sign changes, but all of which lead to isomorphic algebras. ${\ displaystyle e_ {i}}$

Cayley-Dickson construction

Octonions can be understood as pairs (a, b) of quaternions and the octonion multiplication of the pairs (a, b) and (c, d) over

${\ displaystyle (a, b) (c, d) = (ac-d ^ {*} b, da + bc ^ {*}),}$

define, where the conjugation is a quaternion. ${\ displaystyle a ^ {*}}$

Multiplication of the octonions with the help of the Fano level

Fano level

The multiplication of the octonions can be shown in the Fano plane (see figure on the right). The points correspond to the seven unit octonions in the imaginary part of the octonions (i.e. without ). ${\ displaystyle e_ {0} = 1}$

An order is given by the arrows and the multiplication of two neighboring elements on a straight line results in the third element on the straight line when proceeding in the direction of the arrow (some of the straight lines are circular in the figure). The straight line is cycled, i.e. the straight line can be imagined as virtually closed: (a, b, c) = (c, a, b) = (b, c, a). If you move against the direction of the arrow, you get a minus sign. For example results . So if (a, b, c) is a straight line in the Fano diagram (with order according to the direction of the arrow), then ab = c and ba = -c. As above applies and for . The diagram is easy to convince yourself that multiplication is non-associative. ${\ displaystyle e_ {5} \ cdot e_ {3} = e_ {6}}$${\ displaystyle e_ {0} = 1}$${\ displaystyle e_ {i} ^ {2} = - 1}$${\ displaystyle i = 1, \ cdots, 7}$

Each “straight line” in the Fano diagram forms a sub-algebra of octonions with the one element, which is isomorphic to the quaternions. Each point forms a sub-algebra with the unit element that is isomorphic to the complex numbers. A straight line and a point outside the line already produce the whole diagram (that is, any two imaginary unit octonions , connected to form a straight line, and an additional imaginary unit Oktonion ). ${\ displaystyle e_ {i}}$${\ displaystyle e_ {j}}$${\ displaystyle e_ {i} \ cdot e_ {j}}$${\ displaystyle e_ {k}}$

properties

The octonions are a division algebra with one element.

They do not form an oblique body (and no body) because they violate that

Associative law of multiplication: .${\ displaystyle a \ cdot (b \ cdot c) = (a \ cdot b) \ cdot c}$

However, it applies to all octaves a and b :

${\ displaystyle a \ cdot (a \ cdot b) = (a \ cdot a) \ cdot b}$and .${\ displaystyle a \ cdot (b \ cdot b) = (a \ cdot b) \ cdot b}$

This property is called alternative and can be viewed as a weakened form of associativity (a sub-algebra formed from any two octonions is associative). The octon ions form an alternative body .

The relationship follows from the alternative

${\ displaystyle a \ cdot (b \ cdot a) = (a \ cdot b) \ cdot a}$.

This relationship is also called the flexibility law .

The octonions also fulfill the Moufang identities

${\ displaystyle [a \ cdot (b \ cdot a)] \ cdot c = a \ cdot [b \ cdot (a \ cdot c)]}$

and

${\ displaystyle (a \ cdot b) \ cdot (c \ cdot a) = a \ cdot [(b \ cdot c) \ cdot a]}$

Applying the doubling process to the octaves yields the sedions . However, they are no longer free of zero divisors (and also no longer alternative). In the context of the doubling process, the algebras under consideration lose increasingly important properties starting from the real numbers, first the order property for the complex numbers, then the commutativity for the quaternions and the associativity for the octonions. All four together form the only finite-dimensional, normalized division algebras with one element over the real numbers (theorem of Hurwitz).

The automorphism group of octonions is the smallest exceptional simple Lie group . It is of dimension 14 and can be understood as a subgroup of which, in its 8-dimensional real spinor representation, leaves any given vector fixed. It has two fundamental representations of 14 dimensions (the adjoint representation ) and 7 dimensions (this is given precisely by its operations on the seven-dimensional imaginary part of the octonion - understood as a vector space over the real numbers). ${\ displaystyle G_ {2}}$${\ displaystyle SO (7)}$

Representations

Every octave can be represented ...

... as 8-ary tuples of real numbers :${\ displaystyle (r_ {1}, r_ {2}, \ dotsc, r_ {8})}$
... as a 4-way tuple of complex numbers :${\ displaystyle (c_ {1}, c_ {2}, c_ {3}, c_ {4})}$
... as an ordered pair of quaternions :${\ displaystyle (h_ {1}, h_ {2})}$

The field of real numbers can be viewed as a substructure of : ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {O}}$

The following applies to all numbers from : corresponds to${\ displaystyle r}$${\ displaystyle \ mathbb {R}}$${\ displaystyle r}$${\ displaystyle (r, 0, \ dotsc, 0)}$

The body of complex numbers can be viewed as a substructure of : ${\ displaystyle \ mathbb {C}}$${\ displaystyle \ mathbb {O}}$

The following applies to all numbers from : corresponds to${\ displaystyle c}$${\ displaystyle \ mathbb {C}}$${\ displaystyle c}$${\ displaystyle (c, 0,0,0)}$

The oblique body of the quaternions can be viewed as a substructure of : ${\ displaystyle \ mathbb {H}}$${\ displaystyle \ mathbb {O}}$

The following applies to all numbers from : corresponds to${\ displaystyle h}$${\ displaystyle \ mathbb {H}}$${\ displaystyle h}$${\ displaystyle (h, 0)}$

For the octaves, addition and multiplication are defined in such a way that they are downward compatible, i.e. ...

... for all real numbers and the following applies: ${\ displaystyle r}$${\ displaystyle s}$
${\ displaystyle r + s = (r, 0, \ dotsc, 0) + (s, 0, \ dotsc, 0)}$
${\ displaystyle r \ cdot s = (r, 0, \ dotsc, 0) \ cdot (s, 0, \ dotsc, 0)}$
... for all complex numbers and the following applies: ${\ displaystyle c}$${\ displaystyle d}$
${\ displaystyle c + d = (c, 0,0,0) + (d, 0,0,0)}$
${\ displaystyle c \ cdot d = (c, 0,0,0) \ cdot (d, 0,0,0)}$
... for all quaternions and applies: ${\ displaystyle h}$${\ displaystyle i}$
${\ displaystyle h + i = (h, 0) + (i, 0)}$
${\ displaystyle h \ cdot i = (h, 0) \ cdot (i, 0)}$

Conjugation, norm, inverse

The conjugate of an octonion

${\ displaystyle x = x_ {0} \, e_ {0} + x_ {1} \, e_ {1} + x_ {2} \, e_ {2} + x_ {3} \, e_ {3} + x_ {4} \, e_ {4} + x_ {5} \, e_ {5} + x_ {6} \, e_ {6} + x_ {7} \, e_ {7}}$

is defined as:

${\ displaystyle x ^ {*} = x_ {0} \, e_ {0} -x_ {1} \, e_ {1} -x_ {2} \, e_ {2} -x_ {3} \, e_ { 3} -x_ {4} \, e_ {4} -x_ {5} \, e_ {5} -x_ {6} \, e_ {6} -x_ {7} \, e_ {7}.}$

Conjugation is an involution and it applies

${\ displaystyle {(x \ cdot y)} ^ {*} = y ^ {*} \ cdot x ^ {*}}$

The scalar part of the octonion is given by:

${\ displaystyle {\ frac {x + x ^ {*}} {2}} = x_ {0} \, e_ {0}}$

and the remainder ( imaginary part , corresponding to a seven-dimensional sub-vector space)

${\ displaystyle {\ frac {xx ^ {*}} {2}} = x_ {1} \, e_ {1} + x_ {2} \, e_ {2} + x_ {3} \, e_ {3} + x_ {4} \, e_ {4} + x_ {5} \, e_ {5} + x_ {6} \, e_ {6} + x_ {7} \, e_ {7}.}$

The conjugation fulfills:

${\ displaystyle x ^ {*} = - {\ frac {1} {6}} (x + (e_ {1} x) e_ {1} + (e_ {2} x) e_ {2} + (e_ {3 } x) e_ {3} + (e_ {4} x) e_ {4} + (e_ {5} x) e_ {5} + (e_ {6} x) e_ {6} + (e_ {7} x ) e_ {7}).}$

The product of an octonion with its conjugate

${\ displaystyle x ^ {*} x = x_ {0} ^ {2} + x_ {1} ^ {2} + x_ {2} ^ {2} + x_ {3} ^ {2} + x_ {4} ^ {2} + x_ {5} ^ {2} + x_ {6} ^ {2} + x_ {7} ^ {2}.}$

returns a real number greater than or equal to zero and can be used to define a norm which corresponds to the usual Euclidean norm in the vector space representation of octonions:

${\ displaystyle \ | x \ | = {\ sqrt {x ^ {*} x}}.}$

The inverse element of a non-vanishing octonion can be written like this:

${\ displaystyle x ^ {- 1} = {\ frac {x ^ {*}} {\ | x \ | ^ {2}}}.}$

It applies ${\ displaystyle x ^ {- 1} \ cdot x = x \ cdot x ^ {- 1} = 1}$

The following applies to the product of two octonions:

${\ displaystyle \ | xy \ | = \ | x \ | \ | y \ |}$

This means that the octonions, like the real numbers, the complex numbers and the quaternions, form a composition algebra.

Applications

Exceptional Jordan algebras can be constructed using the Cayley algebras and exceptional Lie algebras can be specified using spaces of derivatives on such Jordan algebras .

Octonions can also be used to construct the almost complex structure on the 6-sphere.

In physics, octaves could be used to describe an eight-dimensional supersymmetry . This would also result in possible applications in connection with string theory and M-theory , since both are based on supersymmetry.

As early as 1973 there were attempts to use the subgroups SU (3) and SU (2) x SU (2) of the automorphism group of octonions to represent parts of the standard model (quarks) with octonions ( Murat Günaydin , Feza Gürsey ). The basic group structure of the interactions of the Standard Model is SU (3) x SU (2) x U (1) (1-2-3 symmetry for short). The group SU (3) results in the automorphism group of imaginary octonions by leaving one of the imaginary unit octonions fixed. From the mid-2010s, there were attempts by the physicist Cohl Furey to obtain further elements of the standard model from the octonion algebra. It starts from the tensor product of the four division algebras and regards particles as ideals in them. The space-time symmetries (Lorentz group) are in the part of the quaternions, the group structure of the Standard Model in the part of the octonions. In 2018, she implemented the full 1-2-3 symmetry group of the Standard Model with one generation of elementary particles. It corresponds to a version of GUT with calibration group SU (5) by Howard Georgi and Sheldon Glashow , but with a possible explanation for the prevention of proton decay. She also found an implementation of the unbroken symmetries SU (3) and U (1) of the standard three-generation model. ${\ displaystyle G_ {2}}$${\ displaystyle G_ {2}}$${\ displaystyle \ mathbb {R} \ otimes \ mathbb {C} \ otimes \ mathbb {H} \ otimes \ mathbb {O}}$${\ displaystyle \ mathbb {C} \ otimes \ mathbb {H}}$${\ displaystyle \ mathbb {C} \ otimes \ mathbb {O}}$