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

Let \(C_2=\{1,s\}\) with \(s^2=1\), and suppose 2 is invertible in the nonzero commutative ring \(R\). The basis element \([s]\) is central, and \[ e_+=\frac {[1]+[s]}2,\qquad e_-=\frac {[1]-[s]}2. \] The algebra-product theorem becomes \[ R[C_2]\cong R\times R. \]

The isomorphism and its inverse are explicit: \[ \begin {aligned} a[1]+b[s]&\longmapsto (a+b,a-b),\\ (x,y)&\longmapsto \frac {x+y}{2}[1]+\frac {x-y}{2}[s]. \end {aligned} \] Under this map, \(e_+\) goes to \((1,0)\) and \(e_-\) goes to \((0,1)\). Thus the two idempotents are the coordinate projections, not merely a dimension count.

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

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]
✍️source

Remark. three levels of splitting [fgap-0012]

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

Theorem. Projectors of an involution on a module [fdip-003O]

Proof.