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Kissin, Edward and Shulman, Victor (2023) Cohomology and extensions of representations of groups with normal Engel subgroups. European Journal of Mathematics, 9 (84). ISSN 2199-6768
Let λ, U be representations of a group G with a normal subgroup N. The paper studies the first cohomology group H1 (G, λ, U) under various spectral type conditions imposed on the restrictions of λ, U to N. We assume often that N is an Engel group and examine various decompositions of the extension e(λ, U, ξ) of λ by U associated with non-trivial (λ, U)-cocycles ξ.
Aiming at applications to double extensions and the theory of J-unitary group representations on indefinite metric spaces, we describe (λ, U)-cocycles when G = D n N is the semidirect product, D is an Engel group and the restrictions of λ, U to N are χ1 for some character χ on N (such pairs of representations form in a sense a base class in the variety of all pairs of representations). Our description is complete if G = D n Rn, where D is the group of all diagonal matrices with positive entries, or G = D n T is the group of all upper triangular matrices d = (dij ) with dii > 0 and T is its subgroup of all matrices with dii = 1, or G = SO(2) n R2.
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