ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Even-subalgebra types determine small graded Clifford modules

Example

For signatures $(1, 0)$ , $(0, 1)$ , $(2, 0)$ , $(1, 1)$ , and $(0, 2)$ , the even subalgebras are respectively $R$ , $R$ , $C$ , $R \oplus R$ , and $C$ . Hence their graded-simple module counts and coefficient types are respectively one real, one real, one complex, two real, and one complex.

Facts & Assumptions

Given: The positive-square-first convention and the five real Clifford algebras in the Example.

[F1]

The even subalgebra of ${C l}_{p, q}$ is ${C l}_{q, p - 1}$ for $p > 0$ and ${C l}_{0, q - 1}$ for $p = 0, q > 0$ (Even subalgebras of real and complex Clifford algebras).

[F2]

Graded-simple Clifford modules correspond to simple modules over the even subalgebra; matrix algebra factors determine their coefficient types (Irreducible graded Clifford modules and real, complex, and quaternionic types).

[F3]

The Clifford algebra of the zero vector space over $R$ is $R$ (The zero-dimensional Clifford algebra is the base field).

[F4]

${C l}_{0, 1} ≅ C$ and ${C l}_{1, 0} ≅ R \oplus R$ (Low-dimensional real Clifford algebra models).

Verification

1.1givenF1F3F4algebra

For ${C l}_{1, 0}$ and ${C l}_{2, 0}$ , [F1] gives even algebras ${C l}_{0, 0}$ and ${C l}_{0, 1}$ ; [F3]–[F4] identify these with $R$ and $C$ . For ${C l}_{0, 1}$ and ${C l}_{0, 2}$ it gives ${C l}_{0, 0} = R$ and ${C l}_{0, 1} = C$ . For ${C l}_{1, 1}$ it gives ${C l}_{1, 0} = R \oplus R$ by [F4].

2.1F2step 1.1algebra∎

The fields $R$ and $C$ each have one simple module of their own coefficient type, while $R \oplus R$ has two simple modules, one supported on each factor. Applying [F2] therefore gives the five counts and types in the Example. In every positive-dimensional case an invertible odd generator identifies the two parity components, so each resulting graded module has equal dimensions in degrees $0$ and $1$ .

Depends on

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