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The affine vacuum character exists at every complex level

Statement

Let $g$ be finite-dimensional complex simple, let $\overline{L g}$ be the central extension of Untwisted affine central extension, write $K = c$ for its central basis vector, and let $k \in C$ . The subspace $p = g [t] \oplus C K$ is a Lie subalgebra, and the assignment $g [t] ∋ x \otimes t^{n} \mapsto 0$ , $K \mapsto k$ defines a one-dimensional representation $C_{k}$ of $p$ .

Facts & Assumptions

Given: The finite-dimensional complex simple Lie algebra, its normalized form, the loop central extension, and $k \in C$ .

[F1]

Untwisted affine central extension: $[x_{m}, y_{n}] = [x, y]_{m + n} + m δ_{m, - n} B (x, y) K$ , and $K$ is central.

Proof

1.1F1algebra

For $m, n \geq 0$ , the loop term $[x, y]_{m + n}$ belongs to $g [t]$ . The central term in [F1] can be nonzero only if $m + n = 0$ , which for nonnegative $m, n$ forces $m = n = 0$ and hence its coefficient $m$ is zero. Thus $g [t]$ is closed under brackets in the central extension, and $K$ commutes with it. Bilinearity extends this to all finite sums in $p$ .

2.1step 1.1F1algebra∎

The proposed scalar action sends every bracket in $g [t]$ to zero, agrees with the zero bracket between $K$ and $g [t]$ , and sends $[K, K] = 0$ to zero. Hence it is a Lie-algebra representation on the one-dimensional space $C_{k}$ ; the scalar $k$ is unrestricted.

Depends on

Untwisted affine central extension

Used by

Universal affine vacuum module Definition
Mode translation descends to the affine vacuum module Lemma
PBW basis of the universal affine vacuum module Lemma

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tomoyuki Arakawa and Anne Moreau, Arc spaces and vertex algebras, §3.1 (standard reference, not scraped)