Split Algebra Research Articles

Subalgebras of octonion algebras

For an arbitrary unitary octonion algebra, we determine all subalgebras. It turns out that every subalgebra of dimension less than four is associative, while every subalgebra of dimension greater than four is not associative. In any split octonion algebra, there are both associative and non-associative subalgebras of dimension four. Except for one-dimensional subalgebras spanned by idempotents, any two isomorphic subalgebras are in the same orbit under automorphisms.

On Noetherian algebras, Schur functors and Hemmer–Nakano dimensions

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander’s correspondence, classical Schur–Weyl duality and Soergel’s Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B B , and module categories of endomorphism algebras of generator-relative cogenerators over B B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n n so that the endomorphism algebra of a generator-cogenerator provides an n n -faithful cover, in the sense of Rouquier, of B B . The degree n n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B B is bounded above by the number of non-isomorphic simple modules of B B . We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category O \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category O \mathcal {O} establishing an integral version of Soergel’s Struktursatz. We show that deformations of the combinatorial Soergel’s functor have better homological properties than the classical one.

An extension of quaternionic metrics to octonions in a non-Riemannian spacetime: Including Yang–Mills-like fields within a division octonion algebra

In this paper, we develop an extended space-time geometry that includes an internal space built with division octonions with realization via Pauli matrices and Zorn (vectorial) matrices. Here, we extend a former field theory that used split octonion algebra to a division octonion algebra on a non-Riemannian manifold. The octonionic algebra is the last possible algebra allowed by the Hurwitz theorem. The interpretation of the “octonionic field” in this division octonionic algebra is not straightforward, however it behaves in a similar way to the quaternionic Yang–Mills fields. Former work suggests that this Yang–Mills-like octonionic field is associated with the field governing quarks within nucleons. In this work, we discover that the inclusion of octonionic fields in the geometry of an internal space necessarily excludes the quaternionic (Yang–Mills) fields in an extended non-Riemannian geometry. This is not what is expected from the standpoint of a “unified” field theory, which leads us to propose a different approach to Einstein’s unified field theory.

The (anti‐)η$$ \eta $$‐Hermitian solution to a novel system of matrix equations over the split quaternion algebra

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti‐) ‐Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti‐) ‐Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented.

Generalized sharped cubic form and split spin factor algebra

There is a well-known construction of a Jordan algebra via a sharped cubic form. We introduce a generalized sharped cubic form and prove that the split spin factor algebra is induced by this construction, and it satisfies the identity ( ( a , b , c ) , d , b ) + ( ( c , b , d ) , a , b ) + ( ( d , b , a ) , c , b ) = 0 . The split spin factor algebras have recently appeared in the classification of 2-generated axial algebras of Monster type fulfilled by T. Yabe; their properties were studied by J. McInroy and S. Shpectorov.

Octonions as Clifford-like algebras

The associative Cayley-Dickson algebras over the field of real numbers are also Clifford algebras. The alternative but nonassociative real Cayley-Dickson algebras, notably the octonions and split octonions, share with Clifford algebras an involutary anti-automorphism and a set of mutually anticommutative generators. On the basis of these similarities, we introduce Kingdon algebras: alternative Clifford-like algebras over vector spaces equipped with a symmetric bilinear form. Over three-dimensional vector spaces, our construction quantizes an alternative non-associative analogue of the exterior algebra. The octonions and split octonions, along with other real generalized Cayley-Dickson algebras in Albert's sense, arise as Kingdon algebras. Our construction gives natural characterizations of the octonion and split octonion algebras by a universality property endowing them with a selected superalgebra structure.

Trialitarian triples

Trialitarian triples are triples of central simple algebras of degree 8 with orthogonal involution that provide a convenient structure for the representation of trialitarian algebraic groups as automorphism groups. This paper explicitly describes the canonical “trialitarian” isomorphisms between the spin groups of the algebras with involution involved in a trialitarian triple, using a rationally defined shift operator that cyclically permutes the algebras. The construction relies on compositions of quadratic spaces of dimension 8, which yield all the trialitarian triples of split algebras. No restriction on the characteristic of the base field is needed.

On singular value decomposition for split quaternion matrices and applications in split quaternionic mechanics

In the theoretical studies and numerical computations of split quaternionic quantum mechanics, the singular value decomposition theorem of split quaternion matrices plays an essential role. In general the problem of singular value decomposition over split quaternion algebra has hitherto remained tangential for split quaternion matrices. In this paper, by means of a real representation matrix of a split quaternion matrix, the singular value decomposition of split quaternion matrices is studied, the singular value decomposition theorem and the corresponding algorithm for split quaternion matrices are given. Finally, three applications for solving the split quaternion least squares problem and the wave separation problem are given based on the singular value decomposition over split quaternion algebra.

Semidefinite programming bounds for binary codes from a split Terwilliger algebra

We study the upper bounds for $A(n,d)$, the maximum size of codewords with length $n$ and Hamming distance at least $d$. Schrijver studied the Terwilliger algebra of the Hamming scheme and proposed a semidefinite program to bound $A(n, d)$. We derive more sophisticated matrix inequalities based on a split Terwilliger algebra to improve Schrijver's semidefinite programming bounds on $A(n, d)$. In particular, we improve the semidefinite programming bounds on $A(18,4)$ to $6551$.

Characteristic of Quaternion Algebra Over Fields

Quaternion is an extension of the complex number system. Quaternion are discovered by formulating 4 points in 4-dimensional vector space using the cross product between two standard vectors. Quaternion algebra over a field is a 4-dimensional vector space with bases and the elements of the algebra are members of the field. Each element in quaternion algebra has an inverse, despite the fact that the ring is not commutative. Based on this, the purpose of this study is to obtain the characteristics of split quaternion algebra and determine how it interacts with central simple algebra. The research method used in this paper is literature study on quaternion algebra, field and central simple algebra. The results of this study establish the equivalence of split quaternion algebra as well as the theorem relating central simple algebra and quaternion algebra. The conclusion obtained from this study is that split quaternion algebra has five different characteristics and quaternion algebra is a central simple algebra with dimensions less than equal to four.

Cellular Noetherian algebras with finite global dimension are split quasi-hereditary

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.

A necessary and sufficient condition for the existence of nontrivial Sn-invariants in the splitting algebra

For a monic polynomial [Formula: see text] over a commutative, unitary ring [Formula: see text] the splitting algebra [Formula: see text] is the universal [Formula: see text]-algebra such that [Formula: see text] splits in [Formula: see text]. The symmetric group acts on the splitting algebra by permuting the roots of [Formula: see text]. It is known that if the intersection of the annihilators of the elements [Formula: see text] and [Formula: see text] (where [Formula: see text] depends on [Formula: see text]) in [Formula: see text] is zero, then the invariants under the group action are exactly equal to [Formula: see text]. We show that the converse holds.

From octonions to composition superalgebras via tensor categories

The nontrivial unital composition superalgebras, of dimension 3 and 6, which exist only in characteristic 3, are obtained from the split Cayley algebra and its order 3 automorphisms, by means of the process of semisimplification of the symmetric tensor category of representations of the cyclic group of order 3. Connections with the extended Freudenthal magic square in characteristic 3, that contains some exceptional Lie superalgebras specific of this characteristic are discussed too. In the process, precise recipes to go from (nonassociative) algebras in this tensor category to the corresponding superalgebras are given.

The Lorentz Group Using Conformal Geometric Algebra and Split Quaternions for Color Image Processing: Theory and Practice

The processing of color images is of great interest, because the human perception of color is a very complex process, still not well understood. In this article, firstly the authors present an analysis of the well-known mathematical methods used to model color as a phenomenon and to process color images. We propose an adequate metric to process color images using the Minkowski or space-time metric. We propose that the processing of HSV images can be done using the HSV cone in the quaternion split algebra or the conformal geometric algebra frameworks. We show that the processing of RGB images using the Euclidean metric in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {R}^{3}$ </tex-math></inline-formula> doesn’t yield good results. The colors of images change by daylight, we understand this phenomenon as the color change follows the action of the Lorentz Lie group. Consequently, we formulate a new model for representing and processing color using split quaternions and space-time conformal geometric algebra. We propose new algorithms for practical color image processing: we formulate the novel Split Quaternion Fourier Transform for color image processing and we interpolate color images using Split Motors which belong to the space-time conformal geometric algebra. The experimental part proves that real color images have to be processed in the HSV cone. We show successful applications of the Quaternion Split Fourier Transform and the interpolation processing. We compare these computations using the RGB metric in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {R}^{3}$ </tex-math></inline-formula> space and using the Minkowski metric in the HSV cone. This comparison shows clearly that the color image processing using the Minkowski metric in the HSV cone performs better.

Local derivations and automorphisms of Cayley algebras

The present paper is devoted to the description of local and 2-local derivations and automorphisms on Cayley algebras over an arbitrary field F. Given a Cayley algebra C with norm n, let C0 be its subspace of trace 0 elements. We prove that the space of all local derivations of C coincides with the Lie algebra {d∈so(C,n)|d(1)=0} which is isomorphic to the orthogonal Lie algebra so(C0,n). Surprisingly, the behavior of 2-local derivations depends on the Cayley algebra being split or division. Every 2-local derivation on the split Cayley algebra is a derivation, so they are isomorphic to the exceptional Lie algebra g2(F) if charF≠2,3. On the other hand, on division Cayley algebras over a field F, the sets of 2-local derivations and local derivations coincide. As a corollary we obtain descriptions of local and 2-local derivations of the seven-dimensional simple non-Lie Malcev algebras over fields of characteristic ≠2,3. Further, we prove that the group of all local automorphisms of C coincides with the group {φ∈O(C,n)|φ(1)=1}. As in the case of 2-local derivations, the behavior of 2-local automorphisms depends on the Cayley algebra being split or division. Every 2-local automorphism on the split Cayley algebra is an automorphism, so they form the exceptional Lie group G2(F) if charF≠2,3. On the other hand, on division Cayley algebras over a field F, the groups of 2-local automorphisms and local automorphisms coincide.

RETRACTED ARTICLE: Split Malcev-Poisson-Jordan algebras

We introduce the class of split Malcev-Poisson-Jordan algebras as the natural extension of the one of split Malcev Poisson algebras, and therefore split (non-commutative) Poisson algebras. We show that a split Malcev-Poisson-Jordan algebra P can be written as a direct sum with any Ij a non-zero ideal of P in such a way that satisfies for . Under certain conditions, it is shown that the above decomposition of P is by means of the family of its simple ideals.

Quadratic Equation in Split Quaternions

Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split quaternions and two real nonlinear systems, we derive explicit formulas for computing the roots of x2+bx+c=0 in split quaternion algebra.

Composition color algebras

This work aims to define the composition color algebras and classify them. For this, we use the Cayley-Dickson process and the construction of the “canonical” basis of a split Cayley algebra as a reference.

BPS black hole entropy and attractors in very special geometry. Cubic forms, gradient maps and their inversion

We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of mathcal{N} = 2, D = 4 ungauged supergravity obtained as reduction of minimal, matter-coupled D = 5 supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named BPS system, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous non-symmetric scalar manifolds (whose classification is known in terms of L(q, P, Ṗ) models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an invertibility condition for the gradient map of the corresponding cubic form (to have a birational inverse map which is given by homogeneous polynomials of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with Ṗ = 0, such as L(q, P), 1 ⩽ q ⩽ 3 and P ⩾ 1, or L(q, 1), 4 ⩽ q ⩽ 9, and one model with Ṗ = 1, namely L(4, 1, 1). We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of L(1, P) P ⩾ 2 non-symmetric models of mathcal{N} = 2, D = 4 supergravity.

Electromagnetic curves and Rytov curves based on the hyperbolic split quaternion algebra

This paper aims to investigate the hyperbolically the motion of the polarization plane traveling along the linearly polarized light wave in the optical fiber via the hyperbolic split quaternion algebra. The motion of the electric field (polarization vector) about an axis on the general hyperboloid is described by the Lorentzian scalar product space Ra1,a2,a32,1. The hyperbolic geometric (Berry) phase models are generated through the pseudo-spheres of Ra1,a2,a32,1. Then, the parametric representations of the Rytov curves are obtained via the hyperbolic split quaternion product and one-parameter homothetic motion. Then, the hyperbolically motion of the electric field is expressed by the Fermi–Walker parallel transportation law. Moreover, the electromagnetic curves (EM-curves) associated with the electric field E are determined via the hyperbolic geometric phase models in the optical fiber. Furthermore, some motivating examples are presented by using the MAPLE program.