Twisted cocycles of Lie algebras and corresponding invariant functions
The aim of this paper is to study a natural generalization of the cohomology cocycles of a (complex) Lie algebra with values in a representation \(V\). To this extent, the notion of \(k\)-twisted cocycles is introduced, where \(k\) corresponds to a complex symmetric matrix. As a special case corresponding to dimension one, the \((\alpha,\beta,\gamma)\)-derivations of the Lie algebra are recovered [\textit{P. Novotný} and \textit{J. Hrivnák}, J. Geom. Phys. 58, No. 2, 208--217 (2008; Zbl 1162.17014)]. In dimension 2, 16 cases for the adjoint representation are found, described by four essential parameters. New invariants of the Lie algebra are derived starting from these twisted cocycles. As an illustration of the method, it is shown that the classification of complex Lie algebras in dimensions three and four can be obtained by means of the invariant functions obtained. Further, the method is applied to the contraction problem and the existence of special types of contractions. The contractions of three dimensional algebras is also considered from this perspective. Finally, these invariants are applied to nilpotent parametric continua of Lie algebras resulting from graded contractions of the Pauli graded \(\text{sl}(3, \mathbb C)\).
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