{"title":"On the Representations of Clifford and SO(1,9) Algebras for 8-Component Dirac Equation","authors":"V. M. Simulik, I. I. Vyikon","doi":"10.1007/s00006-023-01295-7","DOIUrl":"10.1007/s00006-023-01295-7","url":null,"abstract":"<div><p>Extended gamma matrix Clifford–Dirac and SO(1,9) algebras in the terms of <span>(8 times 8)</span> matrices have been considered. The 256-dimensional gamma matrix representation of Clifford algebra for 8-component Dirac equation is suggested. Two isomorphic realizations <span>(textit{C}ell ^{texttt {R}})</span>(0,8) and <span>(textit{C}ell ^{texttt {R}})</span>(1,7) are considered. The corresponding gamma matrix representations of 45-dimensional SO(10) and SO(1,9) algebras, which contain standard and additional spin operators, are introduced as well. The SO(10), SO(1,9) and the corresponding <span>(textit{C}ell ^{texttt {R}})</span>(0,8)<span>(, textit{C}ell ^{texttt {R}})</span>(1,7) representations are determined as algebras over the field of real numbers. The suggested gamma matrix representations of the Lie algebras SO(10), SO(1,9) are constructed on the basis of the Clifford algebras <span>(textit{C}ell ^{texttt {R}})</span>(0,8)<span>(, textit{C}ell ^{texttt {R}})</span>(1,7) representations. Comparison with the corresponded algebras in the space of standard 4-component Dirac spinors is demonstrated. The proposed mathematical objects allow generalization of our results, obtained earlier for the standard Dirac equation, for equations of higher spin and, especially, for equations, describing particles with spin 3/2. The maximal 84-dimensional pure matrix algebra of invariance of the 8-component Dirac equation in the Foldy–Wouthuysen representation is found. The corresponding symmetry of the Dirac equation in ordinary representation is found as well. The possible generalizations of considered Lie algebras to the arbitrary dimensional SO(n) and SO(m,n) are discussed briefly.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 5","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48923275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Series Representation of Solutions of Polynomial Dirac Equations","authors":"Doan Cong Dinh","doi":"10.1007/s00006-023-01297-5","DOIUrl":"10.1007/s00006-023-01297-5","url":null,"abstract":"<div><p>In this paper, we consider the polynomial Dirac equation <span>( left( D^m+sum _{i=0}^{m-1}a_iD^iright) u=0, (a_iin {mathbb {C}}))</span>, where <i>D</i> is the Dirac operator in <span>({mathbb {R}}^n)</span>. We introduce a method of using series to represent explicit solutions of the polynomial Dirac equations.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 5","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45557662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A New Type of Quaternionic Regularity","authors":"A. Vajiac","doi":"10.1007/s00006-023-01292-w","DOIUrl":"10.1007/s00006-023-01292-w","url":null,"abstract":"<div><p>I introduce a notion of quaternionic regularity using techniques based on hypertwined analysis, a refined version of general hypercomplex theory. In the quaternionic and biquaternionic cases, I show that hypertwined holomorphic (regular) functions admit a decomposition in a hypertwined sum of regular functions in certain subalgebras. The hypertwined quaternionic regularity lies in between slice regularity and the modified Cauchy–Fueter theories, and proves to have a direct impact on reformulations of quaternionic and spacetime algebra quantum theories.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 5","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-023-01292-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42111341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some Estimates for the Cauchy Transform in Higher Dimensions","authors":"Longfei Gu","doi":"10.1007/s00006-023-01294-8","DOIUrl":"10.1007/s00006-023-01294-8","url":null,"abstract":"<div><p>We give estimates of the Cauchy transform in Lebesgue integral norms in Clifford analysis framework which are the generalizations of Cauchy transform in complex plane, and mainly establish the <span>((L^{p}, L^{q}))</span>-boundedness of the Clifford Cauchy transform in Euclidean space <span>({mathbb {R}^{n+1}})</span> using the Clifford algebra and the Hardy–Littlewood maximal function. Furthermore, we prove Hedberg estimate and Kolmogorov’s inequality related to Clifford Cauchy transform. As applications, some respective results in complex plane are directly obtained. Based on the properties of the Clifford Cauchy transform and the principle of uniform boundedness, we solve existence of solutions to integral equations with Cauchy kernel in quaternionic analysis.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 5","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48118925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Explicit Twisted Group Algebra Structure of the Cayley–Dickson Algebra","authors":"Guangbin Ren, Xin Zhao","doi":"10.1007/s00006-023-01296-6","DOIUrl":"10.1007/s00006-023-01296-6","url":null,"abstract":"<div><p>The Cayley–Dickson algebra has long been a challenge due to the lack of an explicit multiplication table. Despite being constructible through inductive construction, its explicit structure has remained elusive until now. In this article, we propose a solution to this long-standing problem by revealing the Cayley–Dickson algebra as a twisted group algebra with an explicit twist function <span>(sigma (A,B))</span>. We show that this function satisfies the equation </p><div><div><span>$$begin{aligned} e_Ae_B=(-1)^{sigma (A,B)}e_{Aoplus B} end{aligned}$$</span></div></div><p>and provide a formula for the relationship between the Cayley–Dickson algebra and split Cayley–Dickson algebra, thereby giving an explicit expression for the twist function of the split Cayley–Dickson algebra. Our approach not only resolves the lack of explicit structure for the Cayley–Dickson algebra and split Cayley–Dickson algebra but also sheds light on the algebraic structure underlying this fundamental mathematical object.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-023-01296-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42909053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Repeated Cayley–Dickson Processes and Subalgebras of Dimension 8","authors":"Jacques Helmstetter","doi":"10.1007/s00006-023-01289-5","DOIUrl":"10.1007/s00006-023-01289-5","url":null,"abstract":"<div><p>Let <i>K</i> be a field of characteristic other than 2, and let <span>(mathcal {A}_n)</span> be the algebra deduced from <span>(mathcal {A}_1=K)</span> by <i>n</i> successive Cayley–Dickson processes. Thus <span>(mathcal {A}_n)</span> is provided with a natural basis <span>((f_E))</span> indexed by the subsets <i>E</i> of <span>({1,2,ldots ,n})</span>. Two questions have motivated this paper. If a subalgebra of dimension 4 in <span>(mathcal {A}_n)</span> is spanned by 4 elements of this basis, is it a quaternion algebra? The answer is always “yes”. If a subalgebra of dimension 8 in <span>(mathcal {A}_n)</span> is spanned by 8 elements of this basis, is it an octonion algebra? The answer is more often “no” than “yes”. The present article establishes the properties and the formulas that justify these two answers, and describes the fake octonion algebras.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49542285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The General Solution to a System of Linear Coupled Quaternion Matrix Equations with an Application","authors":"Long-Sheng Liu","doi":"10.1007/s00006-023-01283-x","DOIUrl":"10.1007/s00006-023-01283-x","url":null,"abstract":"<div><p>Linear coupled matrix equations are widely utilized in applications, including stability analysis of control systems and robust control. In this paper, we establish the necessary and sufficient conditions for the consistency of the system of linear coupled matrix equations and derive an expression of the corresponding general solution (where it is solvable) over quaternion. Additionally, we investigate the necessary and sufficient conditions for the system of linear coupled matrix equations with construct to have a solution and derive a formula of its general solution (where it is solvable). Finally, an algorithm and an example were provided in order to further illustrate the primary outcomes of this paper.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45406355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bicomplex Weighted Bergman Spaces and Composition Operators","authors":"Stanzin Dolkar, Sanjay Kumar","doi":"10.1007/s00006-023-01291-x","DOIUrl":"10.1007/s00006-023-01291-x","url":null,"abstract":"<div><p>In this paper, we study the bicomplex version of weighted Bergman spaces and the composition operators acting on them. We also investigate the Bergman kernel, duality properties and Berezin transform. This paper is essentially based on the work of Zhu (Operator Theory in Function Spaces of Math. Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence, 2007).</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46018306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Oscar González Cervantes, J. Emilio Paz Cordero, Daniel González Campos
{"title":"On Some Quaternionic Series","authors":"J. Oscar González Cervantes, J. Emilio Paz Cordero, Daniel González Campos","doi":"10.1007/s00006-023-01293-9","DOIUrl":"10.1007/s00006-023-01293-9","url":null,"abstract":"<div><p>The aim of this work is to show that given <span>(uin {mathbb {H}}{setminus }{mathbb {R}})</span>, there exists a differential operator <span>(G^{-u})</span> whose solutions expand in quaternionic power series expansion <span>( sum _{n=0}^infty (x-u)^n a_n)</span> in a neighborhood of <span>(uin {mathbb {H}})</span>. This paper also presents Stokes and Borel-Pompeiu formulas induced by <span>(G^{-u})</span> and as consequence we give some versions of Cauchy’s Theorem and Cauchy’s Formula associated to these kind of regular functions.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-023-01293-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42452252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Some Lie Groups in Degenerate Clifford Geometric Algebras","authors":"Ekaterina Filimoshina, Dmitry Shirokov","doi":"10.1007/s00006-023-01290-y","DOIUrl":"10.1007/s00006-023-01290-y","url":null,"abstract":"<div><p>In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups contain degenerate spin groups, Lipschitz groups, and Clifford groups as subgroups in the case of arbitrary dimension and signature. The considered Lie groups can be of interest for various applications in physics, engineering, and computer science.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-023-01290-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46698511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}