Definition. factor sets and twisted Group Algebras [fgap-001S]
Definition. factor sets and twisted Group Algebras [fgap-001S]
Let \(k\) be a field, \(G\) a group, and \(\alpha :G\times G\to k^\times \) a normalized 2-cocycle: \[ \alpha (1,g)=\alpha (g,1)=1, \qquad \alpha (g,h)\alpha (gh,\ell )= \alpha (h,\ell )\alpha (g,h\ell ). \] The twisted Group Algebra \(k_\alpha [G]\) has basis \(u_g\) and multiplication \[ u_gu_h=\alpha (g,h)u_{gh}. \] The cocycle equation is exactly the associativity condition.
A projective representation with factor set \(\alpha \), namely operators \(T_g\) satisfying \(T_gT_h=\alpha (g,h)T_{gh}\), extends uniquely to an algebra map \(k_\alpha [G]\to \operatorname {End}_k(V)\); conversely an algebra map gives such a projective representation on its basis operators. Cohomologous cocycles rescale the basis and give isomorphic twisted algebras. See [conlon1964twisted, the introduction and sec. 1, pp. 152--155].