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

For the elements \(e_+\) and \(e_-\) associated with a central involution, \[ e_+^2=e_+,\qquad e_-^2=e_-,\qquad e_+e_-=e_-e_+=0,\qquad e_++e_-=1_A. \] Both idempotents are central.

Proof.

First, \[ e_++e_-=h(1+z)+h(1-z)=2h=1_A, \] so \(e_-=1_A-e_+\). Since \(2h=1_R\) and \(z^2=1_A\), \[ e_+^2 =h^2(1+2z+z^2) =2h^2(1+z) =e_+. \] The remaining identities now follow from this one idempotence calculation: \[ \begin {aligned} e_-^2&=(1_A-e_+)^2=1_A-e_+=e_-,\\ e_+e_-&=e_+(1_A-e_+)=0,\\ e_-e_+&=(1_A-e_+)e_+=0. \end {aligned} \] Finally, each \(e_\pm \) is a scalar linear combination of the central elements \(1_A\) and \(z\), so it is central.

These equations are the elementary two-idempotent instance of the projection calculus discussed in [sengupta2010representations, sec. 4.5, pp. 63--68].

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

Proof.

Reference. Representations of Algebras and Finite Groups: An Introduction [sengupta2010representations]

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]
✍️source

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]

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

Proof.

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