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

For the parameter [Formula: see text], denote the semi-direct product of the Witt algebra and the loop Schrödinger algebra as [Formula: see text]. In this paper, we determine the second cohomology groups of [Formula: see text] with trivial coefficients and determine its universal central extension. This yields three [Formula: see text]-cocycles for [Formula: see text] and two [Formula: see text]-cocycles for [Formula: see text]. We then proceed to consider the universal central extension of the Lie algebra [Formula: see text] in the non-additive category of Leibniz algebras. Finally, we classify all derivations of [Formula: see text] to calculate the first cohomology groups [Formula: see text]