[Paper Review] Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups

Jean Gallier|arXiv (Cornell University)|Jan 1, 2013

Finite Group Theory Research19 references11 citations

TL;DR

This paper provides a comprehensive tutorial on Clifford algebras, Spin and Pin groups, and their role in generalizing quaternions and complex numbers to describe rotations in R^n. It shows how the Spin(n) group, defined as a subgroup of units in the Clifford algebra Cl_n, acts on R^n via multiplication, generalizing the actions of U(1) on R^2 and SU(2) on R^3, with the key contribution being a unified algebraic framework for higher-dimensional rotations using spinor groups.

ABSTRACT

Abstract: One of the main goals of these notes is to explain how rotations in R n are induced by the action of a certain group, Spin(n), on R n, in a way that generalizes the action of the unit complex numbers, U(1), on R 2, and the action of the unit quaternions, SU(2), on R 3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and R n). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of an algebra, Cln, the Clifford algebra associated with R n. Since the spinor groups are certain well chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. These notes provide a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Clifford algebra Clp,q associated with a nondegenerate symmetric

Motivation & Objective

• To explain how rotations in R^n are induced by the action of the Spin(n) group on R^n.
• To establish the foundational role of Clifford algebras Cl_p,q in defining the Spin and Pin groups.
• To generalize the actions of U(1) on R^2 and SU(2) on R^3 to arbitrary dimensions using spinor groups.
• To provide a tutorial on the structure and properties of Clifford algebras and their associated groups.

Proposed method

• Define the Clifford algebra Cl_n associated with R^n using a nondegenerate symmetric bilinear form.
• Construct the Spin(n) group as a subgroup of units in Cl_n, preserving the algebraic structure of rotations.
• Use the action of Spin(n) on R^n via multiplication in the Clifford algebra to induce rotations.
• Analyze the structure of Cl_p,q for nondegenerate symmetric forms to understand the behavior of Spin and Pin groups.
• Demonstrate how Spin(n) generalizes the unit quaternions (SU(2)) and unit complex numbers (U(1)) in higher dimensions.
• Establish the relationship between Pin and Spin groups and their role in the representation of orthogonal transformations.

Experimental results

Research questions

• RQ1 How does the Spin(n) group act on R^n to induce rotations, generalizing the action of U(1) and SU(2)?
• RQ2 What is the algebraic structure of the Clifford algebra Cl_p,q and how does it support the definition of Spin and Pin groups?
• RQ3 How do the Pin and Spin groups relate to the orthogonal group O(n) and SO(n) in higher dimensions?
• RQ4 In what way does the Clifford algebra Cl_n unify the representation of rotations in R^2, R^3, and R^n?
• RQ5 What are the key structural properties of Cl_p,q that enable the construction of Spin(n) as a subgroup of units?

Key findings

• The Spin(n) group is defined as a subgroup of units in the Clifford algebra Cl_n, providing a natural algebraic framework for rotations in R^n.
• The action of Spin(n) on R^n is realized through multiplication in the Clifford algebra, generalizing the action of U(1) on R^2 and SU(2) on R^3.
• The Pin group extends the Spin group to include reflections, generalizing the full orthogonal group O(n).
• The structure of Cl_p,q for nondegenerate symmetric forms determines the algebraic and group-theoretic properties of Spin and Pin groups.
• Clifford algebras provide a unifying algebraic language that generalizes quaternions and complex numbers to higher dimensions.
• The construction of Spin(n) via Clifford algebras ensures a consistent and systematic description of spinor representations and rotations in arbitrary dimensions.