Mathematics – Quantum Algebra
Scientific paper
2010-06-13
Mathematics
Quantum Algebra
38 pages. Added a lower boundedness condition on the orders of poles in Definition 3.5 and also added Remark 3.6 to explain th
Scientific paper
We introduce a cohomology theory of grading-restricted vertex algebras. To construct the "correct" cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to "rational functions valued in the algebraic completion of a module for the algebra," instead of linear maps from tensor powers of the algebra to a module for the algebra. One subtle complication arising from such "rational functions valued in the algebraic completion of a module" is that we have to carefully address the issue of convergence when we compose these linear maps with vertex operators. In particular, for each n in \N, we have an inverse system {H^{n}_{m}(V, W)}_{m\in Z_{+}} of n-th cohomologies and an additional n-th cohomology H_{\infty}^{n}(V, W) of a grading-restricted vertex algebra V with coefficients in a V-module $W$ such that H_{\infty}^{n}(V, W) is isomorphic to the inverse limit of the inverse system {H^{n}_{m}(V, W)\}_{m\in Z_{+}}. In the case of n=2, there is an additional second cohomology denoted by H^{2}_{1/2}(V, W) which will be shown in a sequel to the present paper to correspond to what we call square-zero extensions of V and to first order deformations of V when W=V.
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