Twisted group algebras and commutator envelopes

Proof of Proposition 8.1

The new cocycle is

The scalar coboundary factor is symmetric in \(a\) and \(b\), so it cancels from \(c'(a,b)c'(b,a)^{-1}\).

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Proof of Proposition 8.2

Associativity of the twisted product gives multiplicativity of scalar commutators in each variable. We have \(b_c(a,a)=1\) and \(b_c(b,a)=b_c(a,b)^{-1}\). The radical is the kernel of the homomorphism from \(A\) to its character group defined by \(a\mapsto b_c(a,\cdot)\). The induced pairing on the quotient is nondegenerate by construction.

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Proof of Theorem 8.3

Let

be central. Commuting with \(u_x\) gives

for every \(x\). Nondegeneracy forces \(z_a=0\) for every nonzero \(a\). Thus the center is \({\mathbb C}\). To prove semisimplicity directly, choose a linear projection \(P\) onto a submodule of a finite dimensional \({\mathbb C}^c[A]\)-module and form

Every summand maps into the submodule and is the identity there. Conjugation by any \(u_x\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \(\overline P\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \({\mathbb C}\) permits only one simple factor. A finite dimensional central simple algebra over \({\mathbb C}\) is a full matrix algebra. Its dimension is \({\left\lvert A\right\rvert}\), so the matrix size is \(\sqrt{{\left\lvert A\right\rvert}}\).

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Proof of Corollary 8.4

Every radical element commutes with all projective generators, so it contributes only central data. Quotienting by the radical leaves the nondegenerate commutator form. In the elementary abelian case, multiplicative values in the group of \(p\)th roots can be written as an alternating bilinear form over \({\mathbb F}_p\).

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