First and second cohomology groups of the Lie algebras 𝒮𝒲(b)

Key Points

• This research aims to determine the second cohomology groups of a specific semi-direct product of Lie algebras and classify its derivations.
• Determined the second cohomology groups with trivial coefficients of the semi-direct product of the Witt algebra and loop Schrödinger algebra.
• Calculated three cocycles for the semi-direct product and two cocycles in the non-additive category of Leibniz algebras.
• Classified all derivations of the Lie algebra for calculating the first cohomology groups.
• Identified the second cohomology groups and universal central extension for the Lie algebra, yielding three and two cocycles respectively.
• Established a framework for understanding the derivations of the Lie algebra, contributing to the classification of the first cohomology groups.