For the double complex structure of grading-restricted vertex algebra cohomology defined in [6], [7], we introduce a multiplication of elements of double complex spaces. We show that the orthogonality and bi-grading conditions applied on double complex spaces, provide in relation among mappings and actions of co-boundary operators. Thus, we endow the double complex spaces with structure of bi-graded differential algebra. We then introduce the simplest cohomology classes for a grading-restricted vertex algebra, and show their independence on the choice of mappings from double complex spaces. We prove that its cohomology class does not depend on mappings representing of the double complex spaces. Finally, we show that the orthogonality relations together with the bi-grading condition bring about generators and commutation relations for a continual Lie algebra.

对于[6]、[7]中定义的分级限制顶点代数上同调的双复结构，我们引入了双复空间元素的乘法。我们展示了应用于双复空间的正交性和双分级条件，提供了共边界算子的映射和动作之间的关系。因此，我们赋予双复空间以二阶微分代数的结构。然后，我们介绍了分级限制顶点代数的最简单的上同调类，并展示了它们在选择双复空间映射时的独立性。我们证明了它的上同调类不依赖于表示双复空间的映射。最后，