The number of inversions of a permutation is the number of pairs (i < j) for which w_i > w_j. This important statistic that arises in many contexts, including as the minimum number of simple transpositions needed to express the permutation and, equivalently, as the rank function for weak Bruhat order on the symmetric group.…

Following Beilinson and Drinfeld, we describe vertex algebras as Lie algebras in a certain pseudo-tensor category. More precisely, starting from a vector superspace V with a linear operator on it, we introduce the operad of n-ary chiral operations on V. When V is equipped with a vertex algebra structure, this allows us to define the cohomology of V with…