Reading the figure
A projective commutator space passes to its quotient by the radical. Radical directions commute with every generator; the remaining quotient carries the nondegenerate alternating form.
Mathematical labels
\(T_aT_b=c(a,b)T_{a+b}\)
\(b_c(a,b)\)
\(A/\mathrm{rad}(b_c)\)
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
Projective action TaTb = c(a, b)Ta + b Scalar commutator bc(a, b) Symplectic quotient A/rad(bc) Radical directions commute with every projective generator.
Original figure caption
A cocycle contains choices; its alternating commutator forgets scalar rescaling.
For \(T_aT_b=c(a,b)T_{a+b}\) define \(b_c(a,b)=c(a,b)c(b,a)^{-1}\). Replacing \(T_a\) by \(\lambda_aT_a\) multiplies \(c\) by the symmetric coboundary \(\lambda_a\lambda_b\lambda_{a+b}^{-1}\), so \(b_c\) is unchanged.
The radical consists of \(a\) commuting projectively with every \(b\). Its quotient carries a nondegenerate alternating pairing. In the nondegenerate case the twisted group algebra has scalar center and is a full matrix algebra:
Scope
This constructs the finite envelope for a supplied projective action. It does not produce a canonical action for every arbitrary wild inertia atom.