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Cohomology of some nilpotent Lie algebras. (Russian) Zbl 0682.17008
Let $$\mathfrak g$$ be a complex finite-dimensional simple Lie algebra and $$\mathfrak n$$ be the nilradical of a parabolic subgroup of $$\mathfrak g$$. In the paper the cohomology spaces $$H^ 1(\mathfrak n;\mathfrak n)$$ and $$H^ 2(\mathfrak n;\mathfrak n)$$ are described. The description of $$H^ 2(\mathfrak n;\mathfrak n)$$ is reduced to the study of the space $$H^ 1(\mathfrak n;\mathfrak g/\mathfrak n)$$. In the proofs B. Kostant ’s results are used [Ann. Math. (2) 74, 329–387 (1961; Zbl 0134.03503)]. As an application, the manifold of Lie structures of a finite-dimensional vector space is studied.

##### MSC:
 17B56 Cohomology of Lie (super)algebras 17B30 Solvable, nilpotent (super)algebras