We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing of commutators and anticommutators of Clifford algebra elements. This method allows us to find out and prove a number of new properties of Clifford algebra elements.
We show that complex representations of Clifford algebra can always be reduced either to a real or to a quaternionic algebra depending on signature of complex space thus showing that complex spinors are unavoidably either real Majorana spinors or quaternionic spinors. We use this result to support (1,3) signature for Minkowski space.
Closed form expressions for a multivector exponential and logarithm are presented in real Clifford geometric algebras Cl(p,q)when n=p+q=1 (complex and hyperbolic numbers) and n=2 (Hamilton, split and conectorine quaternions). Starting from Cl(0,1) and Cl(1,0) algebras wherein square of a basis vector is either -1 or +1, we have generalized exponential and logarithm formulas to 2D quaternionic algebras, Cl(0,2), Cl(1,1), and Cl(2,0). The sectors in the multivector coefficient space where 2D logarithm exists are found. They are related with a square root of the multivector.
Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups
秩为3和4的Coxeter（反射）群以及E_8的Coxeter（反射）群的四元数表示在文献中得到了广泛的应用。本文分析了这样的考克斯特群的克利福德几何代数框架，它提供了一个简单的方法来执行反射和旋转，同时更清楚地揭示了底层的几何。克利福德方法表明，四元数表示实际上有非常简单的几何解释。秩为3的群A_1xA_1xA_1，A_3，B_3和H_3用纯四元数表示为确定Coxeter群反射面的Hodge对偶根向量。两个连续的反射导致旋转，由两个反射向量的几何积描述，给出Clifford旋量。秩为3的群A_1xA_1xA_1，A_3，B_3和H_3的旋量给出了二元多面体群的一个新的简单构造。这些又生成秩为4的群A_1 x A_1 x A_1 x A_1，D_4，F_4和H_4，它们广泛使用的四元数表示被证明是伪装的旋量。因此，Clifford几何积实际上是从秩3群导出秩4群。特别是D_4，F_4和H_4群是特殊的结构，我们的研究对它们有新的认识。
In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension $n$. The formulas involve only the operations of multiplication, summation, and operations of conjugation without explicit use of matrix representation. We use methods of Clifford algebras (including the method of quaternion typification …
The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, we present an introduction to the main ideas of Clifford algebra, with the main goal to develop a package for Clifford algebra calculations for the computer algebra program Mathematica. The Clifford algebra package is thus a powerful tool since it allows the manipulation of all Clifford mathematical objects. The package also provides a visualization tool for elements of Clifford Algebra in the 3-di…
In these lectures, we discuss some well-known facts about Clifford algebras: matrix representations, Cartan's periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in $n$ dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.
This paper is dedicated to the memory of Zbigniew Oziewicz, to his generosity, intelligence and intensity in the search that is science and mathematics. The paper begins with a basic construction that produces Clifford algebras inductively, starting with a base algebra A that is associative and has an involution. This construction is an analog of the Cayley-Dickson Construction that produces the complex numbers, quaternions and octonions starting from the real numbers. Our basic construction always produces associative algebras and can be iterated an indefinite number of times. We generalize t…
Real Clifford algebras for arbitrary number of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real, complex or quaternionic type. Spinors are defined as elements of minimal or quasi-minimal left ideals within the Clifford algebra and as representations of the pin and spin groups. Two types of Dirac adjoint spinors are introduced carefully. The relation between mathematical structures and applications to describe relativistic fermions is emphasized throughou…
On the Transposition Anti-Involution in Real Clifford Algebras III: The Automorphism Group of the Transposition Scalar Product on Spinor Spaces
在[arXiv.1005.3554v1]中引入了实Clifford代数Cl_{p,q}对非退化二次型的一个依赖于（p，q）的转置反对合T.在[arXiv.1005.3558v1]中，我们证明了，依赖于（p-q）mod 8的值，映射T产生旋量表示中表示矩阵的转置、复厄米特或四元数厄米特共轭。由此产生的标量积一般不同于两个已知的标准标量积[Lounesto，Clifford algebras and Spinors 2001]。我们提供了一个完整的签名（p，q）相关的分类的不变性组Gpq_{p,q}的这个产品的p+q <= 9。映射T被认为是由（扭）群代数理论[Passmann，The Algebraic Structure of Group Rings 1985]所知的“星”映射，其中Clifford代数Cl_{p,q}被看作是扭群环k^t[（Z_2）^n]，n=p+q。讨论并列出了稳定群Gpq（f）_{p，q}的重要子群及其横截与旋量空间生成元的关系.
