proposition. the real Group Algebra of the quaternion group [fgap-001B]
AGENTDRAFTED
Let \(Q_8=\{\pm 1,\pm i,\pm j,\pm k\}\) be the concrete quaternion group
from the quaternion group inside Hamilton's Quaternions, and put
\[
\mathcal {E}=\{-1,1\}\subset \mathbb {R}^{\times },
\qquad
\mathcal {X}=\mathcal {E}\times \mathcal {E}.
\]
For \((\epsilon ,\delta )\in \mathcal {X}\), there is a real character
\[
\chi _{\epsilon ,\delta }:Q_8\longrightarrow \mathbb {R}^{\times }
\]
determined by
\[
\chi _{\epsilon ,\delta }(-1)=1,\qquad
\chi _{\epsilon ,\delta }(i)=\epsilon ,\qquad
\chi _{\epsilon ,\delta }(j)=\delta ,\qquad
\chi _{\epsilon ,\delta }(k)=\epsilon \delta .
\]
These are the four characters pulled back from
\(Q_8/\{\pm 1\}\cong C_2\times C_2\); compare
[etingof2024introduction, sec. 4.3, pp. 63--64 and ex. 4.8.1, p. 73].
Write \(\mathbb {R}^{\mathcal {X}}\) for the algebra of functions
\(\mathcal {X}\to \mathbb {R}\), with pointwise operations. Extending the four
characters and the inclusion
\(Q_8\subset \mathbb {H}^{\times }\) linearly gives an algebra homomorphism
\[
\Phi :\mathbb {R}[Q_8]\longrightarrow
\mathbb {R}^{\mathcal {X}}\times \mathbb {H}.
\]
Then \(\Phi \) is an isomorphism of real algebras. The displayed map retains
the chosen character index and quaternionic realization as part of its
data.
The quaternion coordinate alone is surjective, because its basis values
include \(1,i,j,k\). It vanishes exactly when
\(d_1=d_i=d_j=d_k=0\), or equivalently when \(a_q=a_{-q}\) for
\(q\in \{1,i,j,k\}\). Its kernel is therefore the 4-dimensional span of
\[
u_1+u_{-1},\quad
u_i+u_{-i},\quad
u_j+u_{-j},\quad
u_k+u_{-k}.
\]
It therefore induces
\[
\mathbb {R}[Q_8]/\ker (\Phi _{\mathbb {H}})
\cong _{\mathbb {R}\text {-alg}}\mathbb {H}.
\]
This quotient depends on the chosen quaternionic realization. The combined
map \(\Phi \) is what exhibits the selected quotient as the quaternion factor
of the displayed product; a quotient map by itself does not supply that
direct-factor statement.
The quaternion model is grounded in
[voight2021quaternion, sec. 11.2, p. 166]. The extension of group
representations to Group-Algebra maps follows
[sengupta2010representations, secs. 3.1--3.2, pp. 39--41]. The
coefficient recovery above is direct and does not assume a general
classification theorem. No physical meaning is assigned to a factor merely
from its dimension or familiar algebra.