Example. the real Group Algebra of a cyclic group of order three [fgap-000N]
Example. the real Group Algebra of a cyclic group of order three [fgap-000N]
Let \(C_3=\langle g\mid g^3=1\rangle \). Sending \(g\) to \(x\) gives \[ \mathbb {R}[C_3]\cong \mathbb {R}[x]/(x^3-1). \] Over the real numbers, \[ x^3-1=(x-1)(x^2+x+1), \] and the two factors are coprime. The Chinese remainder theorem and the identification \(\mathbb {R}[x]/(x^2+x+1)\cong \mathbb {C}\) therefore give \[ \mathbb {R}[C_3]\cong \mathbb {R}\times \mathbb {C}. \]
More concretely, if \(\zeta \) is a primitive complex cube root of unity, the isomorphism is \[ a+bg+cg^2\longmapsto \bigl (a+b+c,\ a+b\zeta +c\zeta ^2\bigr ). \] The two factors can be read in three compatible ways: \[ \begin {array}{c|c|c} \text {factor of }x^3-1& \text {real algebra block}& \text {real representation}\\ \hline x-1&\mathbb {R}&\text {trivial line}\\ x^2+x+1&\mathbb {C}&\text {rotation plane}. \end {array} \] The rotation plane is the linearization of the nontrivial part of the regular Group Action. More explicitly, the generator cyclically permutes the basis \(1,g,g^2\). The averaging element \[ e_0=\frac {1+g+g^2}{3} \] projects onto the fixed line \(\mathbb {R}(1+g+g^2)\), while its complementary kernel is the augmentation plane \[ \{a+bg+cg^2:a+b+c=0\}. \] Orbits, freeness, and transitivity belong to the action; the projector and invariant subspaces appear only after linearization.
Since the spectrum of a product is the disjoint union of the spectra, \[ \operatorname {Spec}\mathbb {R}[C_3] \cong \operatorname {Spec}\mathbb {R} \sqcup \operatorname {Spec}\mathbb {C}. \] This is a useful commutative picture. It is not yet a reason to introduce a topos, and it does not extend without choices to noncommutative Group Algebras.