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Indagationes Mathematicae


This paper is a sequel to J. Feldvoss and F. Wagemann: On Leibniz cohomology, (2021), where we mainly consider semisimple Leibniz algebras. It turns out that the analogue of the Hochschild-Serre spectral sequence for Leibniz cohomology cannot be applied to many ideals, and therefore this spectral sequence seems not to be applicable for computing the cohomology of non-semi-simple Leibniz algebras. The main idea of the present paper is to use similar tools as developed by Farnsteiner for Hochschild cohomology (see R. Farnsteiner: On the cohomology of associative algebras and Lie algebras (1987) and R. Farnsteiner: On the vanishing of homology and cohomology groups of associative algebras (1988)) to work around this. Unfortunately, it does not seem to be possible to relate the cohomology of a Leibniz algebra directly to Hochschild cohomology as is the case for Lie algebras, but all the desired results can be obtained in a similar way. In particular, this enables us to generalize the vanishing theorems of Dixmier and Barnes for nilpotent and (super)solvable Lie algebras to Leibniz algebras. Moreover, we compute the cohomology of the one-dimensional Lie algebra with values in an arbitrary Leibniz bimodule and show that it is periodic with period two. As a consequence, we prove the Leibniz analogue of a non-vanishing theorem of Dixmier. Although not needed in full for the aforementioned results, we prove a Fitting lemma for Leibniz bimodules that might be useful elsewhere.


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Mathematics and Statistics


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