Example. a central group element inside a Group Algebra [fgap-000X]
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Let \(G\) be a group and let \(z_G\in G\) satisfy \[ z_G^2=1_G,\qquad z_Gg=gz_G\quad \text {for every }g\in G. \] In the Group Algebra \(R[G]\), write \([z_G]\) for the basis element indexed by \(z_G\). Then \[ [z_G]^2=[1_G]=1_{R[G]}, \] and \([z_G]\) commutes with every basis element \([g]\). By linearity it is a central involution in the algebra. The Group Algebra and the passage from group representations to its modules are developed in [sengupta2010representations, secs. 3.1--3.3, pp. 39--42]; compare [webb2007finite, pp. 1--4].

The brackets matter. Assume \(R\) is nontrivial. Then \([z_G]\neq -1_{R[G]}\): their supports differ when \(z_G\neq 1_G\); when \(z_G=1_G\), this is \(1_{R[G]}\neq -1_{R[G]}\), since 2 is invertible.

Example. a central group element inside a Group Algebra [fgap-000X]
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Reference. Representations of Algebras and Finite Groups: An Introduction [sengupta2010representations]

Reference. Finite Group Representations for the Pure Mathematician [webb2007finite]

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]
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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]

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

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