On groups of central type,non-degenerate and bijective cohomology classes |
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Authors: | Nir Ben David |
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Institution: | (1) Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel |
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Abstract: | A finite group G is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class c] ∈ H
2(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 ∈ Z
1(Q,Ǎ), where Ǎ = Hom(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 Z
2(G, ℂ*), where G:= A × 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] ∈ H
1(Q,Ǎ) as above, we construct non-degenerate classes cπ] ∈ H
2(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. |
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Keywords: | |
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