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Nonassociative algebra over the real numbers
In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero
Split-octonion
example of an octonion algebra is the classical octonions, which are an octonion algebra over R, the field of real numbers. The split-octonions also form
Octonion_algebra
Hypercomplex number system
mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented
Octonion
Reals with an extra square root of +1 adjoined
origin are proportional to j exp(aj). In 1933 Max Zorn was using the split-octonions and noted the composition algebra property. He realized that the Cayley–Dickson
Split-complex_number
Method for producing composition algebras
split-octonions, which are isomorphic to Zorn(R). Applying the original Cayley–Dickson construction to the split-complexes also results in the split-quaternions
Cayley–Dickson_construction
Algebraic structure
16. The set of invertible split-octonions forms a nonassociative Moufang loop, as does the set of unit norm split-octonions. More generally, the set of
Moufang_loop
defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution
Albert_algebra
Element of a unital algebra over the field of real numbers
{1,i1} satisfying i12=+1, split-quaternions with basis {1,i1,i2,i3} satisfying i12=−1,i22=i32=+1, and split-octonions with basis {1,i1,…,i7} satisfying
Hypercomplex_number
Type of algebras, possibly non associative
quadratic form x2 − y2, quaternions and split-quaternions, octonions and split-octonions. Every composition algebra has an associated bilinear form B(x
Composition_algebra
Four-dimensional associative algebra over the reals
(Yaglom 1968 Rosenfeld 1988) Dual quaternions Pauli matrices Split-biquaternions Split-octonions Karzel, Helmut & Günter Kist (1985) "Kinematic Algebras and
Split-quaternion
Vector on which a quadratic form is zero
hi is a null vector. The real subalgebras, split complex numbers, split quaternions, and split-octonions, with their null cones representing the light
Null_vector
Relation between Lie algebras depicted as a square
exceptional Lie groups all exist because of the octonions": G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical
Freudenthal_magic_square
Simple Lie group; the automorphism group of the octonions
The compact form of G2 can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen
G2_(mathematics)
Element in a ring whose some power is 0
contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions ℂ⊗ℍ, and complex octonions ℂ⊗𝕆. If a nilpotent infinitesimal
Nilpotent
Four-dimensional number system
largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. The
Quaternion
field. Image: Description: Algebraic group: the automorphism group of a split octonion algebra. Special fields: This type exists over any field. (Tits 1966)
List of irreducible Tits indices
List_of_irreducible_Tits_indices
Element of an algebra using quaternions and split-complex numbers
the direct sum of two quaternion algebras – van der Waerden 1985 Split-octonions Clifford, W. K. (1873). "Preliminary Sketch of Biquaternions". In Tucker
Split-biquaternion
Hypercomplex number system
Cayley–Dickson construction to the octonions, which can be mathematically expressed as 𝕊=𝒞𝒟(𝕆,1). As such, the octonions are isomorphic to a subalgebra
Sedenion
Generalization of affine connections
5-manifolds: Here G = Aut(Os) is the automorphism group of the algebra Os of split octonions, a closed subgroup of SO(3,4), and P is the intersection of G with
Cartan_connection
Comprehensive physical model
generation of 16 fermions can be put into the form of an octonion with each element of the octonion being an 8-vector. If the 3 generations are then put in
Grand_Unified_Theory
Mathematics book
The Geometry of the Octonions is a mathematics book on the octonions, a system of numbers generalizing the complex numbers and quaternions, presenting
The_Geometry_of_the_Octonions
52-dimensional exceptional simple Lie group
Y, Z are octonion valued. Another way of writing these invariants is as (combinations of) Tr(M), Tr(M2) and Tr(M3) of the hermitian octonion matrix: M=[xZ‾YZyX‾Y‾Xz]
F4_(mathematics)
German mathematician (1906–1993)
dem Mathematischen Seminar der Universität Hamburg. Zorn showed that split-octonions could be represented by a mixed-style of matrices called Zorn's vector-matrix
Max_August_Zorn
Setting of relativistic physics in geometric algebra
location (link) Lasenby, A. (2022), "Some recent results for SU(3) and Octonions within the Geometric Algebra approach to the fundamental forces of nature"
Spacetime_algebra
nevertheless, octonions are known by the name Cayley gave them – or as Cayley numbers. The major deduction from the existence of octonions was the eight
History_of_quaternions
type G2(q) is usually constructed as the automorphism groups of the split octonions. Hence, it has a natural representation as a subgroup of the 7-dimensional
List of transitive finite linear groups
List_of_transitive_finite_linear_groups
Generalization of quaternions to other fields
ramifies is called the discriminant of B. Composition algebra Cyclic algebra Octonion algebra Hurwitz quaternion order Hurwitz quaternion See Milies & Sehgal
Quaternion_algebra
Mathematical operation on vectors in 3D space
for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. The nonexistence of nontrivial vector-valued
Cross_product
Concept in mathematics
E. Dickson. For example, the group G2 is the automorphism group of an octonion algebra over k. By contrast, the Chevalley groups of type F4, E7, E8 over
Reductive_group
133-dimensional exceptional simple Lie group
using an algebra that is the tensor product of the quaternions and the octonions, and is also known as a Rosenfeld projective plane, though it does not
E7_(mathematics)
Functions of complex quaternions
(PDF). p. 3. Conway, John H; Smith, Derek A (2003). On Quaternions and Octonions. Boca Raton, Florida: CRC Press. ISBN 978-1-56881-134-5. Viro, Oleg (2021)
Biquaternion_functions
Length in a vector space
real numbers ℝ, the complex numbers ℂ, the quaternions ℍ, and lastly the octonions 𝕆, where the dimensions of these spaces over the real numbers are 1,2
Norm_(mathematics)
78-dimensional exceptional simple Lie group
2E6: see below. The algebraic group G2 is the automorphism group of the octonions. The exceptional Jordan algebra A of dimension 27, known as the Albert
E6_(mathematics)
Generalized sphere of dimension n (mathematics)
Possesses an almost complex structure coming from the set of pure unit octonions. SO(7)/SO(6)=G2/SU(3). The question of whether it has a complex structure
N-sphere
Number with a real and an imaginary part
while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all normed
Complex_number
Mathematical structure in abstract algebra
Eric W. (2015). "C-Star Algebra". Wolfram MathWorld. Baez, John (2015). "Octonions". Department of Mathematics. University of California, Riverside. Archived
*-algebra
Nonabelian group of order 120
polyhedral groups, p. 68 Conway, John H.; Smith, Derek A. (2003). On Quaternions and Octonions. Natick, Massachusetts: AK Peters, Ltd. ISBN 1-56881-134-9.
Binary_icosahedral_group
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
ℂ,ℍ,𝕆 stand for the real numbers, complex numbers, quaternions, and octonions. In the symbols such as E6−26 for the exceptional groups, the exponent
Simple_Lie_group
ISBN 978-3-662-21943-0. Conway, John H.; Smith, Derek A. (2003). On Quaternions and Octonions. Natick, Massachusetts: AK Peters, Ltd. ISBN 1-56881-134-9.
Binary_octahedral_group
248-dimensional exceptional simple Lie group
because it can be built using an algebra that is the tensor product of the octonions with themselves, and is also known as a Rosenfeld projective plane, though
E8_(mathematics)
Theory of strings with supersymmetry
may involve the noncommutative geometry based on the quaternions and octonions, respectively. From the above discussion, it can be seen that physicists
Superstring_theory
Typeface style used in mathematics
natural numbers. May or may not include zero. 𝕆 U+1D546 𝕆 Represents the octonions. ℙ U+2119 ℙ Represents projective space, the probability of an event,
Blackboard_bold
Nonabelian group in algebraic group theory
S2CID 119272452. Conway, John H.; Smith, Derek A. (2003). On Quaternions and Octonions. Natick, Massachusetts: AK Peters, Ltd. ISBN 1-56881-134-9. Coxeter, H
Binary_tetrahedral_group
Polytope in 8-dimensional geometry
polytope can also be obtained by taking the 240 integral octonions of norm 1. Because the octonions are a nonassociative normed division algebra, these 240
4_21_polytope
Type of polygon
methods known before. The other is that the concept is an analogue to the octonion algebras, and quadratic Jordan division algebras of degree 3, that give
Moufang_polygon
(including prime numbers). It is denoted also by GF(q). 𝕆 Denotes the set of octonions. It is often denoted also by 𝐎. 𝕊 Denotes the set of sedenions. It is
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Algebra based on a vector space with a quadratic form
Geometric algebra Higher-dimensional gamma matrices Hypercomplex number Octonion Paravector Quaternion Spin group Spin structure Spinor Spinor bundle Also
Clifford_algebra
Eight-dimensional algebra over the real numbers
dual quaternion algebra. However, his terminology of "octonions" did not stick as today's octonions are another algebra. In 1891 Eduard Study realized that
Dual_quaternion
Vector space equipped with a bilinear product
Euclidean space R3 with multiplication given by the vector cross product Octonions Lie algebras Jordan algebras Alternative algebras Flexible algebras Power-associative
Algebra_over_a_field
(pseudo-)Riemannian manifold whose geodesics are reversible
Spin(9) 16 1 Cayley projective plane 𝕆P2 G G2 SO(4) 8 2 Space of subalgebras of the octonion algebra 𝕆 which are isomorphic to the quaternion algebra ℍ
Symmetric_space
How spheres of various dimensions can wrap around each other
or octonions instead of complex numbers. Here, too, π3(S7) and π7(S15) are zero. Thus the long exact sequences again break into families of split short
Homotopy_groups_of_spheres
Square matrices satisfy their characteristic equation
that the theorem holds. There is no such matrix representation for the octonions, since the multiplication operation is not associative in this case. However
Cayley–Hamilton_theorem
Geometric concept of a 2D space with "points at infinity" adjoined
The Cayley plane (OP2), a projective plane over the octonions, is one of these because the octonions do not form a division ring. Conversely, given a planar
Projective_plane
Algebraic structure with addition, multiplication, and division
non-commutative). This result is known as the Frobenius theorem. The octonions O, for which multiplication is neither commutative nor associative, is
Field_(mathematics)
Method for bounding the errors of numerical computations
manner, to other multidimensional number systems such as quaternions and octonions, but with the expense that we have to sacrifice other useful properties
Interval_arithmetic
Relativistic correction
ISBN 978-3-7643-7790-8. Conway, John H; Smith, Derek A (2003). On Quaternions and Octonions. Boca Raton, Florida: CRC Press. ISBN 978-1-56881-134-5. Girard, Patrick
Thomas_precession
Branch of mathematics
shortly. In 1844, Hamilton presented biquaternions, Cayley introduced octonions, and Grassmann introduced exterior algebras. James Cockle presented tessarines
Abstract_algebra
Special orthogonal group
to four dimensions. J. H. Conway and D. A. Smith: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A. K. Peters, 2003. Hathaway
Rotations in 4-dimensional Euclidean space
Rotations_in_4-dimensional_Euclidean_space
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