Remark. the qualified Clifford bridge [fgap-001Z]
Remark. the qualified Clifford bridge [fgap-001Z]
A generalized Clifford algebra becomes a twisted group algebra only after its parameters are fixed. Let \(k\) contain a primitive \(n\)-th root of unity \(\omega \), assume \(\operatorname {char}k\nmid n\), and choose \(q_1,\ldots ,q_m\in k^\times \). The algebra generated by \(e_1,\ldots ,e_m\) with \[ e_i^n=q_i, \qquad e_i e_j=\omega e_j e_i\quad (j<i) \] is isomorphic to a twisted group algebra of \((\mathbb Z/n\mathbb Z)^m\) for an explicit 2-cocycle. The monomials in the \(e_i\) correspond to its twisted basis. See [cheng2019new, sec. 2.3 and prop. 2.1, pp. 3--4].
This is a qualified bridge, not an identification of every Clifford algebra with an ordinary Group Algebra. The field, root of unity, grading, parameters, and twisting are part of the theorem. It is also different from Clifford theory in the single orbit in Clifford restriction and Clifford homogeneity and the semidirect route, which concerns restriction of representations to normal subgroups.