An Explicit Construction of Commuting Quantum KdV Hamiltonians in the Universal Virasoro Vertex Algebra

We give an explicit formula for elements H₂ₖ of conformal weight $2k$, k≥1, in the universal Virasoro vertex operator algebra V at every central charge, such that H₂ₖ(0)H₂ₗ∈ TV for all k,l≥1. Modulo translation, the kernel of H₄(0) is generated by [H₂ₖ] in each positive even weight and is zero in positive odd weights. The formula uses a realization on differential polynomials by possibly infinite-order differential operators in a completion of the Weyl algebra, together with the inverse of the identity plus a nilpotent operator on each weight space. A Koszul-type exactness argument for the classical KdV flows proves conservation; uniqueness and the Jacobi identity give commutativity. The classical limit recovers the normalized Gelfand--Dickey densities. Explicit low-weight elements and exact symbolic verifications illustrate the construction.