For a Lie comodule M of a Lie coalgebra C and the Lie algebra g(C, M) (respectively, the group G(C, M)) of diagonal coderivations (respectively, Lie coalgebra automorphisms) of the semidirect sum C x M, we show that there is a representation of g(C, M) (respectively, G(C, M)) on the second cohomology group H-c(2)(M, C), from which we construct obstruction classes of extensibility of pairs of coderivations (respectively Lie coalgebra automorphisms) and exact sequences of Wells type for abelian extensions of Lie coalgebras.
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