Lie algebra of the infinitesimal automorphisms on \(S^ 3\) and its central extension (Q1816605)

Similarly to the celebrated Virasoro algebra, which is a central extension of a Lie algebra of vector fields on a circle, the author attempts to find a central extension for a Lie algebra \(\text{Vect}(S^3)\) of vector fields on a 3-dimensional sphere \(S^3.\) Regarding \(S^3\) to be identified in a natural way with the unitary group \(\text{ SU}(2),\) the Lie algebra \(\text{Vect}(S^3)\) is, as a module, generated by complex invariant vector fields over the commutative associative algebra of harmonic polynomials. Since the regular representation of \(\text{ SU}(2)\) is well understood, commutation relations for this Lie algebra can be written explicitly. The 2-cocycle defining the central extension of \(\text{ Vect}(S^3)\) is given in the Radul-Kravchenko-Khesin form by \[ c(X,Y)=\int_{S^3}\text{ Res}\{ s(Y)\cdot [\ln (s(\Delta)),s(X)]\}, \] where \(s\) denotes the symbol of a differential operator, \(\Delta\) is the Laplace-Beltrami operator on \(S^3,\) and \(\text{ Res}\) is a slight modification of the noncommutative Wodzicki residue on the cotangent bundle \(T^*S^3.\) The problem is that non-triviality of this cocycle is not studied. Since it is known that the Lie algebra of all smooth vector fields on \(S^3\) has trivial second (continuous) cohomology, the paper presents a very complicated definition of a trivial cocycle.