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Identities For Algebras Obtained From The Cayley-Dickson Process
Abstract
The Cayley-Dickson process gives a recursive method of constructing a nonassociative algebra of dimension 2 n for all n 0, beginning with any ring of scalars. The algebras in this sequence are known to be flexible quadratic algebras; it follows that they are noncommutative Jordan algebras: they satisfy the flexible identity in degree 3 and the Jordan identity in degree 4. For the integral sedenion algebra (the double of the octonions) we determine a complete set of generators for the multilinear identities in degrees 5. Since these identities are satisfied by all flexible quadratic algebras, it follows that a multilinear identity of degree 5 is satisfied by all the algebras obtained from the Cayley-Dickson process if and only if it is satisfied by the sedenions.
... For this reason, the study of all kinds of identities on these algebras is very useful for obtaining new properties and relations. Several papers are devoted to the study of these identities ( [1][2][3][4] etc.). Therefore, it is very interesting to continue the study of these identities in algebras obtained by the Cayley-Dickson process, since these relations can be helpful to replace the missing commutativity, associativity, and alternativity. ...
... To obtain identities in these algebras, in [1] the linearization method was presented, a method which we will use in the next sentence. Proposition 1. ...
Due to the computational aspects which appear in the study of algebras obtained by the Cayley–Dickson process, it is difficult to obtain nice properties for these algebras. For this reason, finding some identities in such algebras plays an important role in obtaining new properties of these algebras and facilitates computations. In this regard, in the first part of this paper, we present some new identities and properties in algebras obtained by the Cayley–Dickson process. As another remark regarding the computational aspects in these algebras, in the last part of this paper, we solve some quadratic equations in the real division quaternion algebra when their coefficients are some special elements. These special coefficients allowed us to solve interesting quadratic equations, providing solutions directly, without using specialized softs.
... Theorem (Th. 4 ...
... Proof. From the results of the first author and Hentzel [4], it follows that all ternary identity of sedenions follow from the flexible law (a 1 , a 2 , a 3 ) + (a 3 , a 2 , a 1 ) = 0, and that each such identity is satisfied by all the algebras obtained from the Cayley-Dickson process if and only if it is satisfied by the sedenions. Thus, it remains to determine which of the operads from the previous theorem admit a nontrivial map to the operad of flexible algebras. ...
We resolve a ten year old open question of Loday of describing Koszul operads that act on the algebra of octonions. In fact, we obtain the answer by solving a more general classification problem: we find all Koszul operads among those encoding associator dependent algebras.
... These algebras extend the real numbers to more complex structures, incorporating properties such as addition, multiplication, and conjugation. Originally introduced in 1845 by the mathematician Arthur Cayley [1], and later analyzed by Leonard Eugene Dickson in 1919 [2], the Cayley-Dickson construction has found significant applications in various branches of mathematics, including algebra, analysis, and geometry [3][4][5][6][7][8][9][10][11][12][13][14] along with mathematical physical applications [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. ...
A generalized construction procedure for algebraic number systems is hereby presented. This procedure offers an efficient representation and computation method for complex numbers, quaternions, and other algebraic structures. The construction method is then illustrated across a range of examples. In particular, the novel developments reported herein provide a generalized form of the Cayley–Dickson construction through involutive dimagmas, thereby allowing for the treatment of more general spaces other than vector spaces, which underlie the associated algebra structure.
... (1) the roots of f ′ (x) lie in the convex hull of the roots of f (x), (2) the roots of f ′ (x) lie in the convex hull of the roots of C f (x), the companion polynomial of f (x), (3) the roots of f ′ (x) lie in the "snail" (see Section 4) of f (x). ...
We study the roots of polynomials over Cayley--Dickson algebras over an arbitrary field and of arbitrary dimension. For this purpose we generalize the concept of spherical roots from quaternion and octonion polynomials to this setting, and demonstrate their basic properties. We show that the spherical roots (but not all roots) of a polynomial f(x) are also roots of its companion polynomial (defined to be the norm of f(x)). For locally-complex Cayley--Dickson algebras, we show that the spherical roots of (defined formally) belong to the convex hull of the roots of , and we also prove that all roots of are contained in the snail of f(x), as defined by Ghiloni and Perotti for quaternions. The latter two results generalize the classical Gauss--Lucas theorem to the locally-complex Cayley--Dickson algebras, and we also generalize Jensen's classical theorem on real polynomials to this setting.
... The higher Cayley-Dickson algebras only add additional trialities, i.e. copies of G(2), and reasonably no new physics beyond sedenions. Futhermore, sedenion algebra might represent the archetype of all non-associative and non-division flexible algebras, if n > 3 Cayley-Dickson algebras do not differ from sedenions for what concerns the multilinear identities (or algebraic properties) content, as suggested in [71]. ...
In this article I propose a new criterion to extend the Standard Model of particle physics from a straightforward algebraic conjecture: the symmetries of physical microscopic forces originate from the automorphism groups of main Cayley-Dickson algebras, from complex numbers to octonions and sedenions. This correspondence leads to a natural enlargement of the Standard Model color sector, from a SU(3) gauge group to an exceptional Higgs-broken G(2) group, following the octonionic automorphism relation guideline. In this picture, an additional ensemble of massive G(2)-gluons emerges, which is separated from the particle dynamics of the Standard Model.
... The higher Cayley-Dickson algebras only add additional trialities, i.e. copies of G (2), and reasonably no new physics beyond sedenions. Futhermore, sedenion algebra might represent the archetype of all non-associative and nondivision flexible algebras, if n > 3 Cayley-Dickson algebras do not differ from sedenions for what concerns the multilinear identities (or algebraic properties) content, as suggested in 71 . ...
In this article I propose a new criterion to extend the Standard Model of particle physics from a straightforward algebraic conjecture: the symmetries of physical microscopic forces originate from the automorphism groups of main Cayley–Dickson algebras, from complex numbers to octonions and sedenions. This correspondence leads to a natural enlargement of the Standard Model color sector, from a SU(3) gauge group to an exceptional Higgs-broken G(2) group, following the octonionic automorphism relation guideline. In this picture, an additional ensemble of massive G(2)-gluons
emerges, which is separated from the particle dynamics of the Standard Model.
... The higher Cayley-Dickson algebras only add additional trialities, i.e. copies of G(2), and reasonably no new physics beyond sedenions. Futhermore, sedenion algebra might represent the archetype of all non-associative and non-division flexible algebras, if n > 3 Cayley-Dickson algebras do not differ from sedenions for what concerns the multilinear identities (or algebraic properties) content, as suggested in [71]. ...
In this article I propose a new criterion to individuate the origin and the properties of the dark matter particle sector. The emerging candidates come from a straightforward algebraic conjecture: the symmetries of physical microscopic forces originate from the automorphism groups of main Cayley–Dickson algebras, from complex numbers to octonions and sedenions. This correspondence leads to a natural enlargement of the Standard Model color sector, from a SU(3) gauge group to an exceptional Higgs-broken G(2) group, following the octonionic automorphism relation guideline. In this picture, dark matter is a relic heavy G(2)-gluons ensemble, separated from the particle dynamics of the Standard Model due to the high mass scale of its constituents.
The roots of polynomials over Cayley–Dickson algebras over an arbitrary field and of arbitrary dimension are studied. It is shown that the spherical roots of a polynomial f(x) are also roots of its companion polynomial Cf(x). We generalize the classical theorems for complex and real polynomials by Gauss–Lucas and Jensen to locally-complex Cayley–Dickson algebras: it is proved that the spherical roots of f′(x) belong to the convex hull of the roots of Cf(x), and we also show that all roots of f′(x) are contained in the snail of f(x), as defined by Ghiloni and Perotti.
We resolve a 10-year-old open question of Loday of describing Koszul operads that act on the algebra of octonions. In fact, we obtain the answer by solving a more general classification problem: we find all Koszul operads among those encoding associator dependent algebras.
This text is intended as an introduction to abstract algebra for advanced
undergraduates. Much of it is accessible to advanced third year students,
and the first nine chapters can be covered even earlier. There are many
introductions to abstract algebra, so a new one should have some
distinguishing characteristics. The main distinguishing characteristics
of this text are the coverage of universal algebra and category theory, and
their use in the introductory presentation of topics in algebra; and the
coverage of various topics outside the main line of classical abstract
algebra, which are of interest. These include topological spaces,
model theory, computability theory, point lattices, and some algorithms.
The sedenions, elements of a 16-dimensional real division semialgebra with multiplicative Euclidean norm, are presented. The induced multiplication on the 15-sphere forms a left loop that is a loop almost everywhere. Suitable left multiplications in the left loop provide an 8-dimensional non-vanishing vector field on the 15-sphere.
The Cayley-Dickson doubling procedure of algebras is modified at the fourth step where the transition from octonions to sedenions occurs. To account for nonassociativity of octonions a special kind of ternary operation is introduced. Some properties of the ternary sedenion algebra are discussed.
Among those algebras whose multiplication does not satisfy the associative law is a particular family of noncommutative Jordan algebras, the generalized Cayley-Dickson algebras. These are certain central simple algebras whose dimensions are all powers of two. Most of this paper is concerned with giving the classification up to isomorphism of those of dimensions 16, 32, and 64 and determining the automorphism groups. In addition to this some generalized Cayley-Dicksondivision algebras are constructed. Precise criteria for when the 16-dimensional algebras are division algebras are formulated and applied to algebras over some common fields. For higher dimensions no such criteria are given. However, specific examples of division algebras for each dimension 2t are constructed over power-series fields.
This chapter presents a technique that enables a computer to process identities. The examples used are from nonassociative algebras, but the method is general and can be applied to many other situations. Identities are expressed as elements of the group ring on the symmetric group and then are transferred to matrices by the well-known isomorphisms of group rings. This approach has several advantages over the standard approach, which uses a system of linear equations. The system indicates whether a desirable identity is a consequence of other assumed identities. If the identity is not a consequence of the assumed identities, it tells how close it is to being true. This lets the investigator know the routes that are worth pursuing. Second, it uses very little array area. Identities involving four unknowns are processed, with the largest array being a 3 × 3 matrix.
In an algebraic investigation of isoparametric hypersurfaces, J. Dorfmeister and E. Neher encountered a nondegerate quadratic form which permitted composition with a trilinear product, . In this paper we give a complete description of such composition triples: they are all obtained as isotopes of permutations of standard triples or x(yz) determined by a composition algebra, with the quadratic form Q the usual norm form. For any fixed Q this leads to 1 isotopy class in dimensions 1 and 2, 3 classes in the dimension 4 quaternion case, and 6 classes in the dimension 8 octonion case.