The simplest cohomological invariants for vertex algebras

For the double complex structure of grading-restricted vertex algebra cohomology defined in \cite{Huang}, we introduce a multiplication elements spaces. We show that orthogonality and bi-grading conditions applied on spaces, provide relation among mappings actions co-boundary operators. Thus, endow spaces with bi-graded differential algebra. then simples classes for algebra, their independence choice from prove its class does not depend representing Finally, relations together condition bring about generators commutation continual Lie