Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschitz groups). Geometric (algebra based) calculus allows, e.g., to optimize learning algorithms of Clifford neurons, etc.
Keywords: Hypercomplex algebra, hypercomplex analysis, geometry, science, engineering.
We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincar\' e-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the $\mathbb{Z}_2$-graded PBW deformations of quantum exterior algebras. We also determine the pos…
We compute point schemes of some regular algebras using (Wolfram) Mathematica. These algebras are Ore extensions of regular graded skew Clifford algebras of global dimension 3.
For some monoids, we give a method of composing invertibility preserving maps associated to "partial involutions." Also, we define the notion of "determinants for finite dimensional algebras over a field." As examples, we give invertibility preserving maps for Clifford algebras into a field and determinants for Clifford algebras into a field, where we assume that the algebras are generated by less than or equal to 5 generators over the field. On the other hand, "determinant formulas for Clifford algebras" are known. We understand these formulas as an expression that connects invertibility pres…
We prove that the alternative Clifford algebra of a nondegenerate ternary quadratic form is an octonion algebra over the ring of polynomials in one variable over the field of definition.
All 7-dimensional subalgebras of the 8-dimensional Clifford algebra over the field C of complex numbers are found. Canonical bases are used throughout the determination. It is found that the 8-dimensional Clifford algebra over C has exactly eight 7-dimensional subalgebras and each of these eight is over the complex numbers. It follows that the 8-dimensional Clifford algebra over C has no 7-dimensional subalgebra over the real numbers. Another consequence is that the eight 7-dimensional subalgebras of the 8-dimentional Clifford over C are in fact 7-dimensional subalgebras of all greater dimensi…
Some connections between quadratic forms over the field of two elements, Clifford algebras of quadratic forms over the real numbers, real graded division algebras, and twisted group algebras will be highlighted. This allows to revisit real Clifford algebras in terms of the Arf invariant of the associated quadratic forms over the field of two elements, and give new proofs of some classical results.
It is shown that the 16-dimensional Clifford algebra over any field has no one 15-dimensional subalgebra. Canonical bases are used throughout the determination.
We study the representation theoretic results of the binary cubic generic Clifford algebra $\mathcal C$, which is an Artin-Schelter regular algebra of global dimension five. In particular, we show that $\mathcal C$ is a PI algebra of PI degree three and compute its point variety and discriminant ideals. As a consequence, we give a necessary and sufficient condition on a binary cubic form $f$ for the associated Clifford algebra $\mathcal C_f$ to be an Azumaya algebra.
We define a canonical quadratic pair on the Clifford algebra of an algebra with quadratic pair over a field. This allows us to extend to the characteristic 2 case the notion of trialitarian triples, from which we derive a characterization of totally decomposable quadratic pairs in degree 8. We also describe trialitarian triples involving algebras of small Schur index.
在“A note on generalized Clifford algebras and representations”（Caenepeel，S.; Van Oystaeyen，F.，Comm. Algebra 17（1989）no.1，93- 102.）广义Clifford代数是通过Clifford表示引入的;它们对应于有限群（阿贝尔群）的射影表示，比如说$G$，使得相应的扭群环具有最小中心。后者则转化为这样的事实，即相应的2-上循环允许最小（无！）射线类别的数量，并且这迫使以适当的方式将$G$分解为循环分量，参见图10。Zmud，M.，辛几何和有限阿贝尔群的射影表示，（俄文）Mat. Sb.（N.S.）87（129）（1972），3- 17.在这篇小论文中，我将提供一种方法来表示一个阿贝尔群克利福德代数使用一个矩阵，然后给出一种方法来计算是否中心是平凡的。
In this paper, we introduce a ring isomorphism between the Clifford algebra $C\ell_{1,2}$ and a ring of matrices. By such a ring isomorphism, we introduce the concept of the Moore-Penrose inverse in Clifford algebra $C\ell_{1,2}$. Using the Moore-Penrose inverse, we solve the linear equation $axb=d$ in $C\ell_{1,2}$. We also obtain necessary and sufficient conditions for two numbers in $C\ell_{1,2}$ to be similar.
In this paper, we introduce a ring isomorphism between the Clifford algebra $C\ell_2$ and a ring of matrices, and represent the elements in $C\ell_2$ by real matrices. By such a ring isomorphism, we introduce the concept of the Moore-Penrose inverse in Clifford algebra $C\ell_2$. we solve the linear equation $axb=d$, $ax=xb$ and $ax=\bar{x}b$. We also obtain necessary and sufficient conditions for two numbers in $C\ell_2$ to be similar and pseudosimilar.
If $V$ is a finite-dimensional unital commutative (maybe nonassociative) algebra carrying an associative positive definite bilinear form then there exist a nonzero idempotent $c\ne e$ ($e$ being the algebra unit) of the shortest possible length $|c|^2$. In particular, $|c|^2\le \frac12|e|^2$. We prove that the equality holds exactly when $V$ is a Jordan algebra of Clifford type.
We give another version of Huang's proof that an induced subgraph of the n-dimensional cube graph containing over half the vertices has maximal degree at least $\sqrt{n}$, which implies the Sensitivity Conjecture. This argument uses Clifford algebras of positive definite signature in a natural way. We also prove a weighted version of the result.
In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras -- subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras -- graded central simple algebras or graded central simple alge…
