2007/04/19 by Nir Ben David, David, Nir Ben, Yuval Ginosar +1
Mathematics · #20J06 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20J06
paper · pdf · doi:10.48550/arxiv.0704.2516
13 pages
arxiv created 2007/04/19 · openalex publication_date 2007/04/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A finite group G is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class [c]∈ H2(G,\C^*) (G acts trivially on \C^*). Groups of central type play a fundamental role in the classification of semisimple triangular complex Hopf algebras and can be determined by their representation theoretical properties. Suppose that a finite group Q acts on an abelian group A so that there exists a bijective 1-cocycle π∈ Z1(Q,\ach), where \ach=\rmHom(A,\C^*) is endowed with the diagonal Q-action. Under this assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate 2-cocycle in Z2(G,\C^*), where G:=A\rtimes Q. Hence, the semidirect product G is of central type. In this paper we present a more general correspondence between bijective and non-degenerate cohomology classes. In particular, given a bijective class [π]∈ H1(Q,\ach) as above, we construct non-degenerate classes [cπ]∈ H2(G,\C^*) for certain extensions 1→ A→ G→ Q→ 1 which are not necessarily split. We thus strictly extend the above family of central type groups.