Remark. mathematical summands and physical readings [fgap-0013]
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A central involution canonically separates the algebra and its modules into two mathematical summands. This alone does not identify either summand with particles, interactions, chirality, or any other physical sector. Such an interpretation requires additional data: a representation, selected observables or forms, dynamics, and a map from the mathematics to measurable quantities.

The split is useful precisely because it can be calculated before those choices are made. Later notes may test proposed physical correspondences against it, but the algebra decomposition remains valid independently of whether any such proposal succeeds.

Remark. mathematical summands and physical readings [fgap-0013]
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Central involutions and complementary summands [fgap-000V]

Definition. central involution in an algebra [fgap-000W]

Example. a central group element inside a Group Algebra [fgap-000X]

Lemma. the two idempotents of a central involution [fgap-000Y]

Proof.

Example. the central split of the real Group Algebra of 2T [fgap-0015]

Theorem. the eigenspace decomposition of a module [fgap-000Z]

Proof.

Theorem. the central-idempotent algebra product [fgap-0010]

Proof.

Example. splitting the Group Algebra of C₂ [fgap-0011]

Remark. three levels of splitting [fgap-0012]

Remark. mathematical summands and physical readings [fgap-0013]
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