Lectures on Clifford (Geometric) Algebras and Applications

Abstract

Lecture 1: Introduction to Clifford Algebras Pertti Lounesto 1.1 Introduction • 1.2 Clifford algebra of the Euclidean plane • 1.3 Quaternions • 1.4 Clifford algebra of the Euclidean space 3 • 1.5 The electron spin in a magnetic field • 1.6 From column spinors to spinor operators • 1.7 In 4D: Clifford algebra Cℓ 4 of 4 • 1.8 Clifford algebra of Minkowski spacetime • 1.9 The exterior algebra and contractions • 1.10 The Grassmann-Cayley algebra and shuffle products • 1.11 Alternative definitions of the Clifford algebra • 1.12 References Lecture 2: Mathematical Structure of Clifford Algebras Ian Porteous 2.1 Clifford algebras • 2.2 Conjugation • 2.3 References Lecture 3: Clifford Analysis John Ryan 3.1 Introduction • 3.2 Foundations of Clifford analysis • 3.3 Other types of Clifford holomorphic functions • 3.4 The equation D k ƒ = 0 • 3.5 Conformal groups and Clifford analysis • 3.6 Conformally flat spin manifolds • 3.7 Boundary behavior and Hardy spaces • 3.8 More on Clifford analysis on the sphere • 3.9 The Fourier transform and Clifford analysis • 3.10 Complex Clifford analysis • 3.11 References Lecture 4: Applications of Clifford Algebras in Physics William E. Baylis 4.

Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4

“…• The further use of Clifford algebra computation software like CLIFFORD for MAPLE and other packages [24,22,25]. Of special interest in physics are the Clifford algebras of Minkowski spacetime, sometimes called [17] space-time algebras Cℓ 3,1 and Cℓ 1,3 .…”

Lie Algebra of Unit Tangent Bundle

“…The division algebra of real-quaternions H is isomorphic to the Clifford algebra C 0,2 = span R {1, e 1 , e 2 , e 1 e 2 }, i.e., H ∼ = C 0,2 , in dimension two when we identify the quaternionic units i, j, k with, respectively, e 1 , e 2 , e 12 (= e 1 e 2 ) in C 0,2 where the standard anti-commuting orthonormal basis elements e 1 , e 2 satisfy (e 1 ) 2 = (e 2 ) 2 = (e 1 e 2 ) 2 = −1 and e 1 e 2 = −e 2 e 1 , see [13].…”

Unit dual split quaternions and arcs of dual hyperbolic spherical triangles

“…Dual hyperbolic spherical geometry which is studied by means of dual time-like unit vectors is analogous to real hyperbolic spherical geometry which is studied by means of real time-like unit vectors. Quaternions and split quaternions have many applications in mathematics (see [1], [2], [3]). Some of the recent works include [4], [5].…”