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Advances in Applied Clifford Algebras最新文献

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A Generalized Eigenvector–Eigenvalue Identity from the Viewpoint of Exterior Algebra
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-02-26 DOI: 10.1007/s00006-025-01375-w
Małgorzata Stawiska
{"title":"A Generalized Eigenvector–Eigenvalue Identity from the Viewpoint of Exterior Algebra","authors":"Małgorzata Stawiska","doi":"10.1007/s00006-025-01375-w","DOIUrl":"10.1007/s00006-025-01375-w","url":null,"abstract":"<div><p>We consider square matrices over <span>(mathbb {C})</span> satisfying an identity relating their eigenvalues and the corresponding eigenvectors re-proved and discussed by Denton, Parker, Tao and Zhang, called the eigenvector-eigenvalue identity. We prove that for an eigenvalue <span>(lambda )</span> of a given matrix, the identity holds if and only if the geometric multiplicity of <span>(lambda )</span> equals its algebraic multiplicity. We do not make any other assumptions on the matrix and allow the multiplicity of the eigenvalue to be greater than 1, which provides a substantial generalization of the identity. In the proof, we use exterior algebra, particularly the properties of higher adjugates of a matrix.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143489423","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}
引用次数: 0
General Aspects of Jackson Calculus in Clifford Analysis
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-02-25 DOI: 10.1007/s00006-025-01374-x
Martha Lina Zimmermann, Swanhild Bernstein, Baruch Schneider
{"title":"General Aspects of Jackson Calculus in Clifford Analysis","authors":"Martha Lina Zimmermann,&nbsp;Swanhild Bernstein,&nbsp;Baruch Schneider","doi":"10.1007/s00006-025-01374-x","DOIUrl":"10.1007/s00006-025-01374-x","url":null,"abstract":"<div><p>We consider an extension of Jackson calculus into higher dimensions and specifically into Clifford analysis for the case of commuting variables. In this case, Dirac is the operator of the first <i>q</i>-partial derivatives (or <i>q</i>-differences) <span>({_{q}}mathbf {mathcal {D}}= sum _{i=1}^n e_i,{_{q}}partial _i)</span>, where <span>({_{q}}partial _i)</span> denotes the <i>q</i>-partial derivative with respect to <span>(x_i)</span>. This Dirac operator factorizes the <i>q</i>-deformed Laplace operator. Similar to the case of classical Clifford analysis, we then consider the <i>q</i>-deformed Euler and Gamma operators and their relations to each other. Nullsolutions of this <i>q</i>-Dirac equation are called <i>q</i>-monogenic. Using the Fischer decomposition, we can decompose the space of homogeneous polynomials into spaces of <i>q</i>-monogenic polynomials. Using the <i>q</i>-deformed Cauchy–Kovalevskaya extension theorem, we can construct <i>q</i>-monogenic functions. Overall, we show the analogies and the differences between classical Clifford and Jackson-Clifford analysis. In particular, <i>q</i>-monogenic functions need not be monogenic and vice versa.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-025-01374-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480923","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}
引用次数: 0
Branching of Weil Representation for (G_2)
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-01-29 DOI: 10.1007/s00006-025-01370-1
Zhiqiang Wang, Xingya Fan
{"title":"Branching of Weil Representation for (G_2)","authors":"Zhiqiang Wang,&nbsp;Xingya Fan","doi":"10.1007/s00006-025-01370-1","DOIUrl":"10.1007/s00006-025-01370-1","url":null,"abstract":"<div><p>This paper presents a discussion on the branching problem that arises in the Weil representation of the exceptional Lie group of type <span>(G_2)</span>. The focus is on its decomposition under the threefold cover of <span>(SL(2,, {mathbb {R}}))</span> associated with the short root of <span>(G_2)</span>.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143056622","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}
引用次数: 0
Cubic Dirac operator for (U_q({mathfrak {sl}}_2)) 的三次狄拉克算子 $$U_q({mathfrak {sl}}_2)$$
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-01-22 DOI: 10.1007/s00006-025-01372-z
Andrey Krutov, Pavle Pandžić
{"title":"Cubic Dirac operator for (U_q({mathfrak {sl}}_2))","authors":"Andrey Krutov,&nbsp;Pavle Pandžić","doi":"10.1007/s00006-025-01372-z","DOIUrl":"10.1007/s00006-025-01372-z","url":null,"abstract":"<div><p>We construct the <i>q</i>-deformed Clifford algebra of <span>(mathfrak {sl}_2)</span> and study its properties. This allows us to define the <i>q</i>-deformed noncommutative Weil algebra <span>(mathcal {W}_q(mathfrak {sl}_2))</span> for <span>(U_q(mathfrak {sl}_2))</span> and the corresponding cubic Dirac operator <span>(D_q)</span>. In the classical case this was done by Alekseev and Meinrenken in 2000. We show that the cubic Dirac operator <span>(D_q)</span> is invariant with respect to the <span>(U_q({mathfrak {sl}}_2))</span>-action and <span>(*)</span>-structures on <span>(mathcal {W}_q(mathfrak {sl}_2))</span>, moreover, the square of <span>(D_q)</span> is central in <span>(mathcal {W}_q(mathfrak {sl}_2))</span>. We compute the spectrum of the cubic element on finite-dimensional and Verma modules of <span>(U_q(mathfrak {sl}_2))</span> and the corresponding Dirac cohomology.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142991961","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}
引用次数: 0
The Wigner Little Group for Photons is a Projective Subalgebra 光子的Wigner小群是一个射影子代数
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-01-21 DOI: 10.1007/s00006-025-01369-8
Moab Croft, Hamish Todd, Edward Corbett
{"title":"The Wigner Little Group for Photons is a Projective Subalgebra","authors":"Moab Croft,&nbsp;Hamish Todd,&nbsp;Edward Corbett","doi":"10.1007/s00006-025-01369-8","DOIUrl":"10.1007/s00006-025-01369-8","url":null,"abstract":"<div><p>This paper presents the Geometric Algebra approach to the Wigner little group for photons using the Spacetime Algebra, incorporating a mirror-based view for physical interpretation. The shift from a <i>point-based view</i> to a <i>mirror-based view</i> is a modern movement that allows for a more intuitive representation of geometric and physical entities, with vectors and their higher-grade counterparts viewed as hyperplanes. This reinterpretation simplifies the implementation of homogeneous representations of geometric objects within the Spacetime Algebra and enables a <i>relative view</i> via projective geometry. Then, after utilizing the intrinsic properties of Geometric Algebra, the Wigner little group is seen to induce a projective geometric algebra as a subalgebra of the Spacetime Algebra. However, the dimension-agnostic nature of Geometric Algebra enables the generalization of induced subalgebras to <span>((1+n))</span>-dimensional Minkowski geometric algebras, termed <i>little photon algebras</i>. The lightlike transformations (translations) in these little photon algebras are seen to leave invariant the (pseudo)<i>canonical electromagetic field bivector</i>. Geometrically, this corresponds to Lorentz transformations that do not change the intersection of the spacelike polarization hyperplane with the lightlike wavevector hyperplane while simultaneously not affecting the lightlike wavevector hyperplane. This provides for a framework that unifies the analysis of symmetries and substructures of point-based Geometric Algebra with mirror-based Geometric Algebra.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142991449","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}
引用次数: 0
H-B Theorems of Cauchy Integral Operators in Clifford Analysis Clifford分析中Cauchy积分算子的H-B定理
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-01-18 DOI: 10.1007/s00006-025-01371-0
Yufeng Wang, Zhongxiang Zhang
{"title":"H-B Theorems of Cauchy Integral Operators in Clifford Analysis","authors":"Yufeng Wang,&nbsp;Zhongxiang Zhang","doi":"10.1007/s00006-025-01371-0","DOIUrl":"10.1007/s00006-025-01371-0","url":null,"abstract":"<div><p>In this article, we verify the boundedness of the Cauchy type integral operators under the generalized Hölder norm in Clifford analysis, which are called H-B theorems of the Cauchy integral operators in Clifford analysis. We first demonstrate the generalized 2P theorems and the generalized Muskhelishvili theorem in Clifford analysis by Du’s method derived from Du (J Math (PRC) 2(2):115–12, 1982) and Lu (Boundary value problems of analytic functions. World Scientific, Singapore, 1993), which greatly refines the coefficients estimate of inequality in Du et al. (Acta Math Sci 29B(1):210–224, 2009) and Zhang (Complex Var Elliptic Equ 52(6):455–473, 2007). Then, we obtain the H-B theorems which extend and improve the corresponding results in Du et al. (2009) and Wang and Du (Z Anal Anwend, 2024).</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142989238","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}
引用次数: 0
Multicomplex Ideals, Modules and Hilbert Spaces 多复理想、模与希尔伯特空间
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-01-17 DOI: 10.1007/s00006-025-01373-y
Derek Courchesne, Sébastien Tremblay
{"title":"Multicomplex Ideals, Modules and Hilbert Spaces","authors":"Derek Courchesne,&nbsp;Sébastien Tremblay","doi":"10.1007/s00006-025-01373-y","DOIUrl":"10.1007/s00006-025-01373-y","url":null,"abstract":"<div><p>In this article we study some algebraic aspects of multicomplex numbers <span>({mathbb {M}}_n)</span>. For <span>(nge 2)</span> a canonical representation is defined in terms of the multiplication of <span>(n-1)</span> idempotent elements. This representation facilitates computations in this algebra and makes it possible to introduce a generalized conjugacy <span>(Lambda _n)</span>, i.e. a composition of the <i>n</i> multicomplex conjugates <span>(Lambda _n:=dagger _1cdots dagger _n)</span>, as well as a multicomplex norm. The ideals of the ring of multicomplex numbers are then studied in details, free <span>({mathbb {M}}_n)</span>-modules and their linear operators are considered and, finally, we develop Hilbert spaces on the multicomplex algebra.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142987799","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}
引用次数: 0
MiTopos MiTopos
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-12-19 DOI: 10.1007/s00006-024-01362-7
Bernd Schmeikal
{"title":"MiTopos","authors":"Bernd Schmeikal","doi":"10.1007/s00006-024-01362-7","DOIUrl":"10.1007/s00006-024-01362-7","url":null,"abstract":"<div><p>In the present article, the research work of many years is summarized in an interim report. This concerns the connection between logic, space, time and matter. The author always had in mind two things, namely 1. The discovery/construction of an interface between matter and mind, and 2. some entry points for the topos view that concern graphs, grade rotations and contravariant involutions in geometric Boolean lattices. In this part of the MiTopos theme I follow the historic approach to mathematical physics and remain with the Clifford algebra of the Minkowski space. It turns out that this interface is a basic morphogenetic structure inherent in both matter and thought. It resides in both oriented spaces and logic, and most surprisingly is closely linked to the symmetries of elementary particle physics.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142845127","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}
引用次数: 0
Self-Dual Maxwell Fields from Clifford Analysis 自对偶麦克斯韦场从克利福德分析
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-12-11 DOI: 10.1007/s00006-024-01368-1
C. J. Robson
{"title":"Self-Dual Maxwell Fields from Clifford Analysis","authors":"C. J. Robson","doi":"10.1007/s00006-024-01368-1","DOIUrl":"10.1007/s00006-024-01368-1","url":null,"abstract":"<div><p>The study of complex functions is based around the study of holomorphic functions, satisfying the Cauchy-Riemann equations. The relatively recent field of Clifford Analysis lets us extend many results from Complex Analysis to higher dimensions. In this paper, I decompose the Cauchy-Riemann equations for a general Clifford algebra into grades using the Geometric Algebra formalism, and show that for the Spacetime Algebra <i>Cl</i>(3, 1) these equations are the equations for a self-dual source free Maxwell field, and for a massless uncharged Spinor. This shows a deep link between fundamental physics and the Clifford geometry of Spacetime.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01368-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142809682","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}
引用次数: 0
STP Method for Solving the Least Squares Special Solutions of Quaternion Matrix Equations 求解四元数矩阵方程最小二乘特解的STP方法
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-12-02 DOI: 10.1007/s00006-024-01367-2
Weihua Chen, Caiqin Song
{"title":"STP Method for Solving the Least Squares Special Solutions of Quaternion Matrix Equations","authors":"Weihua Chen,&nbsp;Caiqin Song","doi":"10.1007/s00006-024-01367-2","DOIUrl":"10.1007/s00006-024-01367-2","url":null,"abstract":"<div><p>In this paper, we apply the semi-tensor product of matrices and the real vector representation of a quaternion matrix to find the least squares lower (upper) triangular Toeplitz solution of <span>(AX-XB=C)</span>, <span>(AXB-CX^{T}D=E)</span> and (anti)centrosymmetric solution of <span>(AXB-CYD=E)</span>. And the expressions of the least squares lower (upper) triangular Toeplitz and (anti)centrosymmetric solution for the studied equations are derived. Additionally, the necessary and sufficient conditions for the existence of solutions and general expression of the studied equations are given. Eventually, some numerical examples are provided for showing the validity and superiority of our method.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142757908","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}
引用次数: 0
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