[Paper Review] Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions

Jean Gallier|arXiv (Cornell University)|May 2, 2008

Algebraic and Geometric Analysis16 citations

TL;DR

This paper provides a comprehensive tutorial on Clifford algebras, Clifford groups, and their associated spin and pin groups, generalizing quaternions to higher dimensions. It establishes that Spin(n) acts on R^n via algebraic multiplication in the Clifford algebra Cl_n, proving Spin(n) is a double cover of SO(n) and deriving the 8-periodicity theorem of Cartan and Bott for Cl_{p,q} algebras.

ABSTRACT

One of the main goals of these notes is to explain how rotations in reals^n are induced by the action of a certain group, Spin(n), on reals^n, in a way that generalizes the action of the unit complex numbers, U(1), on reals^2, and the action of the unit quaternions, SU(2), on reals^3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and reals^n). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of an algebra, Cl_n, the Clifford algebra associated with reals^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 Cl_{p, q} associated with a nondegenerate symmetric bilinear form of signature (p, q) and culminating in the beautiful "8-periodicity theorem" of Elie Cartan and Raoul Bott (with proofs).

Motivation & Objective

• To provide a self-contained tutorial on Clifford algebras and their associated groups, particularly Spin(n) and Pin(n), for researchers in mathematics and theoretical physics.
• To generalize the role of complex numbers and quaternions in representing rotations in R^2 and R^3 to higher-dimensional rotations using Clifford algebras.
• To establish the topological and algebraic structure of the groups Spin(p,q) and Pin(p,q) as double covers of SO(p,q) and O(p,q), respectively.
• To prove and explain the 8-periodicity theorem of Cartan and Bott for the classification of real Clifford algebras Cl_{p,q}.
• To clarify the relationship between the Lie groups SO(p,q), Spin(p,q), and their topological properties via the polar decomposition theorem for pseudo-algebraic groups.

Proposed method

• Define the Clifford algebra Cl_{p,q} as the associative algebra generated by a vector space R^{p+q} equipped with a symmetric bilinear form of signature (p,q), subject to the relation v^2 = η(v)v for v in R^{p+q}.
• Construct the Clifford group as the subgroup of invertible elements in Cl_{p,q} that stabilize the vector space under twisted conjugation.
• Define the Pin(p,q) and Spin(p,q) groups as subgroups of the Clifford group that preserve the vector space under the adjoint action and satisfy specific norm conditions.
• Use the polar decomposition theorem for pseudo-algebraic groups to show that O(p,q) and SO(p,q) are homeomorphic to products of orthogonal groups and Euclidean spaces, respectively.
• Derive the Lie algebra of O(p,q) and SO(p,q) by differentiating the defining condition A^T J_{p,q} A = J_{p,q}, yielding matrices with skew-symmetric blocks.
• Prove the 8-periodicity theorem by analyzing the isomorphism types of Cl_{p,q} over R, showing that Cl_{p+8,q} ≅ Cl_{p,q} ⊗ R(16) and similarly for other shifts.

Experimental results

Research questions

• RQ1 How can rotations in R^n be systematically generalized beyond quaternions using algebraic structures in higher-dimensional Clifford algebras?
• RQ2 What is the precise algebraic and topological relationship between the groups Spin(p,q) and SO(p,q), and under what conditions is Spin(p,q) a double cover of SO(p,q)?
• RQ3 How does the 8-periodicity theorem of Cartan and Bott classify the structure of real Clifford algebras Cl_{p,q}?
• RQ4 What is the role of pseudo-algebraic groups in understanding the topology of orthogonal groups O(p,q) and SO(p,q)?
• RQ5 How can the polar decomposition theorem be applied to decompose the topological structure of O(p,q) and SO(p,q) into simpler components?

Key findings

• The group Spin(n) acts on R^n via the adjoint action in the Clifford algebra Cl_n, generalizing the action of U(1) on R^2 and SU(2) on R^3.
• For n ≥ 3, Spin(n) is simply connected and is a double cover of SO(n), making it a universal cover of SO(n).
• The Lie algebra of O(p,q) consists of matrices of the form [[X1, X2], [X2^T, X3]] with X1 and X3 skew-symmetric, and X2 arbitrary.
• The group O(p,q) is homeomorphic to O(p) × O(q) × R^{pq}, and SO(p,q) is homeomorphic to S(O(p) × O(q)) × R^{pq}, where S(O(p) × O(q)) is the subgroup with determinant 1.
• The 8-periodicity theorem states that Cl_{p+8,q} ≅ Cl_{p,q} ⊗ R(16), and Cl_{p,q+8} ≅ Cl_{p,q} ⊗ R(16), with periodicity in both p and q.
• The connected component SO₀(p,q) of SO(p,q) containing the identity is homeomorphic to SO(p) × SO(q) × R^{pq}, confirming its topological structure.