Authors:P. D. Beites; A. P. Nicolás Pages: 955 - 964 Abstract: Some identities satisfied by certain standard composition algebras, of types II and III, are studied and become candidates for the characterization of the mentioned types. Composition algebras of arbitrary dimension, over a field F with char \({(F) \neq 2}\) and satisfying the identity \({x^{2}y = n(x)y}\) are shown to be standard composition algebras of type II. As a consequence, the identity \({yx^{2} = n(x)y}\) characterizes the type III. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0668-8 Issue No:Vol. 27, No. 2 (2017)

Authors:Adam Chapman; Casey Machen Pages: 1065 - 1072 Abstract: Given a central division algebra D of degree d over a field F, we associate to any standard polynomial \(\phi (z)=z^n+c_{n-1} z^{n-1}+\cdots +c_0\) over D a “companion polynomial” \(\Phi (z)\) of degree nd with coefficients in F. The roots of \(\Phi (z)\) in D are exactly the set of conjugacy classes of the roots of \(\phi (z)\) . When D is a quaternion algebra, we explain how all the roots of \(\phi (z)\) can be recovered from the roots of \(\Phi (z)\) . Along the way, we are able to generalize a few known facts from \(\mathbb {H}\) to any division algebra. The first is the connection between the right eigenvalues of a matrix and the roots of its characteristic polynomial. The second is the connection between the roots of a standard polynomial and left eigenvalues of the companion matrix. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0740-4 Issue No:Vol. 27, No. 2 (2017)

Authors:Arsenio Moreno García; Tania Moreno García; Ricardo Abreu Blaya; Juan Bory Reyes Pages: 1147 - 1159 Abstract: In this paper we derive a Cauchy integral representation formula for the solutions of the sandwich equation \(\partial _{\underline{x}}f\partial _{\underline{x}}=0\) , where \(\partial _{\underline{x}}\) stands for the first-order vector-valued rotation invariant differential operator in the Euclidean space \({\mathbb R}^m\) , called Dirac operator. Such a solutions are referred in the literature as inframonogenic functions and represent an extension of the monogenic functions, i.e., null solutions of \(\partial _{\underline{x}}\) , which represent higher-dimensional generalizations of the classic Cauchy–Riemann operator. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0745-z Issue No:Vol. 27, No. 2 (2017)

Authors:Abdullah Inalcik Pages: 1329 - 1341 Abstract: In this paper, we establish the concept of similarity and semi-similarity for elements of set of degenerate quaternions, pseudodegenerate quaternions and doubly degenerate quaternions by solving \({ax=xb}\) and \({xay=b, ybx=a}\) , respectively. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0674-x Issue No:Vol. 27, No. 2 (2017)

Authors:Artur Kobus; Jan L. Cieśliński Pages: 1369 - 1386 Abstract: We consider the scator space in 1+2 dimensions—a hypercomplex, non-distributive hyperbolic algebra introduced by Fernández-Guasti and Zaldívar. We find a method for treating scators algebraically by embedding them into a distributive and commutative algebra. A notion of dual scators is introduced and discussed. We also study isometries of the scator space. It turns out that zero divisors cannot be avoided while dealing with these isometries. The scator algebra may be endowed with a nice physical interpretation, although it suffers from lack of some physically demanded important features. Despite that, there arise some open questions, e.g., whether hypothetical tachyons can be considered as usual particles possessing time-like trajectories. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0658-x Issue No:Vol. 27, No. 2 (2017)

Authors:John Leventides; George Petroulakis Pages: 1503 - 1515 Abstract: In this article we define a number of varieties and sets in the projective space \({\mathbb{P} \left(\wedge^{2}\mathbb{R}^{n}\right)}\) , which are obtained from the spectral analysis of a 2-tensor in \({\wedge^2\mathbb{R}^n}\) . We refer to these sets as \({\mathcal{G}_{\mathcal{V}}}\) , special cases of which are the Grassmannian and its extremal variety, as well as other sets. We study the problem of finding the projective variety whose 2-tensors in \({\wedge^{2}\mathbb{R}^{n}}\) have the maximum distance from these varieties, via the use of a special mapping which we call Extr \({(\cdot)}\) . It is shown that successive applications of this mapping lead either to the Grassmannian or its Extremal variety. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0721-7 Issue No:Vol. 27, No. 2 (2017)

Authors:Yuan-Min Li Pages: 1517 - 1530 Abstract: Quadratic representations are very useful in the study of Euclidean Jordan algebras and complementarity problems. In this paper, we provide some characterizations of the complementarity properties for the quadratic representation P a . For example, P a has the E0-property; P a is monotone iff \({\pm a \in {\mathcal K}}\) . In addition, the algebra and cone automorphism invariance of some E-properties are studied. By use of the quadratic representations, the Jordan quad E-property is proved to keep cone automorphism invariant in simple Jordan algebras. The pseudomonotone property is shown to be cone automorphism invariant. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0678-6 Issue No:Vol. 27, No. 2 (2017)

Authors:Vishnu Narayan Mishra; Shikha Pandey Pages: 1633 - 1646 Abstract: In the present paper, we introduce the generalized form of (p, q) Baskakov–Durrmeyer Operators with Stancu type parameters. We derived the local and global approximation properties of these operators and obtained the convergence rate and behaviour for the Lipschitz functions. Moreover, we give comparisons and some illustrative graphics for the convergence of operators to some function. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0738-y Issue No:Vol. 27, No. 2 (2017)

Authors:Craig A. Nolder; Guanghong Wang Pages: 1647 - 1657 Abstract: We use Fourier multipliers of the Dirac operator and Cauchy transform to obtain composition theorems and integral representations. In particular we calculate the multiplier of the \(\Pi \) -operator. This operator is the hypercomplex version of the Beurling-Ahlfors transform in the plane. The hypercomplex Beurling-Ahlfors transform is a direct generalization of the Beurling-Ahlfors transform and reduces to this operator in the plane. We give an integral representation for iterations of the hypercomplex Buerling-Ahlfors transform and we present here a bound for the \(L^p\) -norm. Such \(L^p\) -bounds are essential for applications of the Beurling-Ahlfors transformation in the plane. The upper bound presented here is \(m(p^*-1)\) where m is the dimension of the Euclidean space on which the functions are defined, \(1<p<\infty \) and \(p^*=\max (p,p/(p-1))\) . We use recent estimates on second order Riesz transforms to obtain this result. Using the Fourier multiplier of the \(\Pi \) -operator we express this operator as a hypercomplex linear combination of second order Riesz transforms. PubDate: 2017-06-01 DOI: 10.1007/s00006-016-0752-0 Issue No:Vol. 27, No. 2 (2017)

Authors:Juan Du; Ron Goldman; Stephen Mann Abstract: We flesh out the affine geometry of \({{\mathbb {R}}^3}\) represented inside the Clifford algebra \({\mathbb {R}}(4,4)\) . We show how lines and planes as well as conic sections and quadric surfaces are represented in this model. We also investigate duality between different representations of points, lines, and planes, and we show how to represent intersections between these geometric elements. Formulas for lengths, areas, and volumes are also provided. PubDate: 2017-07-20 DOI: 10.1007/s00006-017-0798-7

Authors:Armando Reyes; Héctor Suárez Abstract: The aim of this paper is to investigate a general notion of \(\sigma \) -PBW extensions over Armendariz rings. As an application, the properties of Baer, quasi-Baer, p.p., and p.q.-Baer are established for these extensions. We generalize several results in the literature for Ore extensions of injective type and skew PBW extensions. PubDate: 2017-07-17 DOI: 10.1007/s00006-017-0800-4

Authors:Baoling Guan; Liangyun Chen; Bing Sun Abstract: The purpose of this paper is to study the relationships between a Hom-Lie superalgebra and its induced 3-ary-Hom-Lie superalgebra. We provide an overview of the theory and explore the structure properties such as ideals, center, derived series, solvability, nilpotency, central extensions, and the cohomology. PubDate: 2017-07-15 DOI: 10.1007/s00006-017-0801-3

Authors:Ali Atasoy; Erhan Ata; Yusuf Yayli; Yasemin Kemer Abstract: We present a new different polar representation of split and dual split quaternions inspired by the Cayley–Dickson representation. In this new polar form representation, a split quaternion is represented by a pair of complex numbers, and a dual split quaternion is represented by a pair of dual complex numbers as in the Cayley–Dickson form. Here, in a split quaternion these two complex numbers are a complex modulus and a complex argument while in a dual split quaternion two dual complex numbers are a dual complex modulus and a dual complex argument. The modulus and argument are calculated from an arbitrary split quaternion in Cayley–Dickson form. Also, the dual modulus and dual argument are calculated from an arbitrary dual split quaternion in Cayley–Dickson form. By the help of polar representation for a dual split quaternion, we show that a Lorentzian screw operator can be written as product of two Lorentzian screw operators. One of these operators is in the two-dimensional space produced by 1 and i vectors. The other is in the three-dimensional space generated by 1, j and k vectors. Thus, an operator in a four-dimensional space is expressed by means of two operators in two and three-dimensional spaces. Here, vector 1 is in the intersection of these spaces. PubDate: 2017-07-03 DOI: 10.1007/s00006-017-0797-8

Authors:R. Serôdio; P. D. Beites; José Vitória Abstract: This is a work on an application of the real split-quaternions to Spatial Analytic Geometry. Concretely, the intersection of a double cone and a line, which can be the empty set, a point, two points or a line, is studied in the real split-quaternions setting. PubDate: 2017-06-16 DOI: 10.1007/s00006-017-0796-9

Authors:Gerhard Opfer Abstract: This paper will contain an extension of Niven’s algorithm of 1941, which in its original form is designed for finding zeros of unilateral polynomials p over quaternions \({\mathbb {H}}\) . The extensions will cover the algebra \({{\mathbb {H}}_{\mathrm{coq}}}\) of coquaternions, the algebra \({{\mathbb {H}}_{\mathrm{nec}}}\) of nectarines and the algebra \({{\mathbb {H}}_{\mathrm{con}}}\) of conectarines. These are nondivision algebras in \({{\mathbb {R}}}^4\) . In addition, it is also shown that in all algebras the most difficult part of Niven’s algorithm can easily be solved by inserting the roots of the companion polynomial c of p, with the result, that all zeros of all unilateral polynomials over all noncommutative \({{\mathbb {R}}}^4\) algebras can be found. In addition, for all four algebras the maximal number of zeros can be given. For the three nondivision algebras besides the known types of zeros: isolated, spherical, hyperbolic, a new type of zero will appear, which will be called unexpected zero of p. PubDate: 2017-06-13 DOI: 10.1007/s00006-017-0786-y

Authors:Padmini Veerapen Abstract: This paper is a survey of work done on \(\mathbb {Z}\) -graded Clifford algebras (GCAs) and \(\mathbb {Z}\) -graded skew Clifford algebras (GSCAs) by Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998), Stephenson and Vancliff (J Algebra 312(1): 86–110, 2007), Cassidy and Vancliff (J Lond Math Soc 81:91–112, 2010), Nafari et al. (J Algebra 346(1): 152–164, 2011), Vancliff and Veerapen (Contemp Math 592: 241–250, 2013), (J Algebra 420:54–64, 2014). In particular, we discuss the hypotheses necessary for these algebras to be Artin Schelter-regular (Adv Math 66:171–216, 1987), (The Grothendieck Festschrift. Birkhäuser, Boston, 1990) and show how certain ‘points’ called, point modules, can be associated to them. We may view an AS-regular algebra as a noncommutative analog of the polynomial ring. We begin our survey with a fundamental result in Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998) that is essential to subsequent results discussed here: the connection between point modules and rank-two quadrics. Using, in part, this connection the authors in Stephenson and Vancliff (J Algebra 312(1): 86–110, 2007) provide a method to construct GCAs with finitely many distinct isomorphism classes of point modules. In Cassidy and Vancliff (J Lond Math Soc 81:91–112, 2010), Cassidy and Vancliff introduce a quantized analog of a GCA, called a graded skew Clifford algebra and Nafari et al. (J Algebra 346(1): 152–164, 2011) show that most Artin Schelter-regular algebras of global dimension three are either twists of graded skew Clifford algebras of global dimension three or Ore extensions of graded Clifford algebras of global dimension two. Vancliff and Veerapen (Contemp Math 592: 241–250, 2013), (J Algebra 420:54–64, 2014) go a step further and generalize the result of Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998), between point modules and rank-two quadrics, by showing that point modules over GSCAs are determined by (noncommutative) quadrics of \(\mu \) -rank at most two. PubDate: 2017-06-12 DOI: 10.1007/s00006-017-0795-x