[Paper Review] On the Structure Theorem of Clifford Algebras

Rafał Abłamowicz|arXiv (Cornell University)|Oct 7, 2016

Algebraic and Geometric Analysis5 references3 citations

TL;DR

This paper reviews the structure theorem of real Clifford algebras $\mathrm{Cl}_{p,q}$ using the framework of semisimple rings and the Wedderburn-Artin theorem, establishing that these algebras decompose into direct sums of minimal left ideals (spinor modules) isomorphic to matrix algebras over $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, depending on $p-q \mod 8$. The key contribution is the systematic connection between Clifford algebra representation theory and the general theory of semisimple rings, with explicit constructions of spinor representations via primitive idempotents.

ABSTRACT

In this paper, theory and construction of spinor representations of real Clifford algebras $\cl_{p,q}$ in minimal left ideals are reviewed. Connection with a general theory of semisimple rings is shown. The actual computations can be found in, for example, [2].

Motivation & Objective

• To reframe the structure of real Clifford algebras $\mathrm{Cl}_{p,q}$ within the general theory of semisimple rings and modules.
• To clarify how spinor representations arise from minimal left ideals in $\mathrm{Cl}_{p,q}$, particularly through primitive idempotents.
• To establish the isomorphism type of the division ring $f\mathrm{Cl}_{p,q}f$ for a primitive idempotent $f$, as required by the Wedderburn-Artin theorem.
• To demonstrate that $\mathrm{Cl}_{p,q}$ decomposes as a direct sum of minimal left ideals, each carrying an irreducible representation.
• To explain the distinction between faithful and non-faithful irreducible representations in semisimple versus simple Clifford algebras.

Proposed method

• Utilizes the Wedderburn-Artin theorem to classify $\mathrm{Cl}_{p,q}$ as a semisimple ring, decomposing it into a finite direct sum of simple two-sided ideals.
• Applies the theory of left $R$-modules to show that minimal left ideals in $\mathrm{Cl}_{p,q}$ are simple modules, corresponding to irreducible spinor representations.
• Identifies primitive idempotents $f$ such that $\mathrm{Cl}_{p,q}f$ is a minimal left ideal, with $f\mathrm{Cl}_{p,q}f$ isomorphic to a division ring $\mathbb{K}$ ($\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$).
• Uses the orthogonal decomposition $1 = c_1 + c_2$ with $c_1 = \frac{1}{2}(1 + \mathbf{e}_{12\ldots n})$, $c_2 = \frac{1}{2}(1 - \mathbf{e}_{12\ldots n})$ to split $\mathrm{Cl}_{p,q}$ into two simple subalgebras when $n = p+q$ is odd.
• Constructs a complete set of $2^k$ primitive mutually annihilating idempotents via independent sign choices in products of $k$ commuting basis monomials.
• Relies on the CLIFFORD Maple package for explicit matrix realizations of generators $\mathbf{e}_1, \dots, \mathbf{e}_n$ in the spinor representation, with algebra maps $\gamma$ preserving multiplication.

Experimental results

Research questions

• RQ1 How does the Wedderburn-Artin theorem apply to the classification of real Clifford algebras $\mathrm{Cl}_{p,q}$?
• RQ2 What is the role of primitive idempotents in constructing irreducible spinor representations of $\mathrm{Cl}_{p,q}$?
• RQ3 Why do semisimple Clifford algebras require a double spinor space $S \oplus \hat{S}$ to realize faithful representations, while simple ones do so in a single minimal left ideal?
• RQ4 How does the center of $\mathrm{Cl}_{p,q}$, particularly the unit pseudoscalar $\mathbf{e}_{12\ldots n}$, influence the decomposition into simple subalgebras?
• RQ5 What determines the isomorphism type of the division ring $f\mathrm{Cl}_{p,q}f$ for a primitive idempotent $f$, and how does it depend on $p-q \mod 8$?

Key findings

• The Clifford algebra $\mathrm{Cl}_{p,q}$ decomposes as a direct sum of $2^k$ minimal left ideals $\mathrm{Cl}_{p,q}f_i$, each isomorphic to a simple left $\mathrm{Cl}_{p,q}$-module, where $k = \lfloor (p+q)/2 \rfloor$.
• When $n = p+q$ is odd, the unit pseudoscalar $\mathbf{e}_{12\ldots n}$ commutes with all generators, enabling the orthogonal decomposition $\mathrm{Cl}_{p,q} = \mathrm{Cl}_{p,q}c_1 \oplus \mathrm{Cl}_{p,q}c_2$ into two simple subalgebras.
• Each subalgebra $\mathrm{Cl}_{p,q}c_i$ is isomorphic to $\mathrm{Mat}(2^{k-1}, \mathbb{K})$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{H}$ depending on $p-q \mod 8$.
• The division ring $f\mathrm{Cl}_{p,q}f$ for a primitive idempotent $f$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, with the type determined by $p-q \mod 8$.
• In semisimple Clifford algebras, irreducible representations on single spinor spaces $S$ or $\hat{S}$ are not faithful; faithfulness requires the direct sum $S \oplus \hat{S}$, realized over $\mathbb{K} \oplus \hat{\mathbb{K}}$.
• The matrix representation of any element $u \in \mathrm{Cl}_{p,q}$ can be computed via the algebra map $\gamma$, using the CLIFFORD package to store and compute generator matrices for a chosen primitive idempotent.