Null root, central coroot, and affine level

Fix the finite Cartan $\mathfrak{h}$ supplied by Finite semisimple Cartan, root and string structure. In Degree derivation and full untwisted affine algebra put $\widehat{\mathfrak{h}}=\mathfrak{h}\oplus \mathbb{C}c\oplus \mathbb{C}d\mathrm{.}$ It is abelian, and its centralizer is itself: commuting with $d$ forces all nonzero modes to vanish, and commuting with $\mathfrak{h}$ forces the degree-zero finite part into $\mathfrak{h}$, by the cited self-centralization result. Extend each finite root $\alpha$ to $\widehat{\mathfrak{h}}$ by $\alpha (c)=\alpha (d)=0$.

The null root is the unique functional $\delta \in {\widehat{\mathfrak{h}}}^{\ast }$ with $\delta {\mathrm{\mid }}_{\mathfrak{h}\oplus \mathbb{C}c}=0$ and $\delta (d)=1$. The chosen central generator $c$ is called the affine central coroot in the normalized loop convention. Its identification with the primitive central combination of the affine simple coroots belongs to the presentation comparison.

The level of a weight $\mathrm{\Lambda }\in {\widehat{\mathfrak{h}}}^{\ast }$ is $\mathrm{\Lambda }(c)$. A module has level $k$ when $c$ acts as $k\mathrm{id}$ on the whole module. In particular a module generated by a weight vector $v$ of weight $\mathrm{\Lambda }$ has level $\mathrm{\Lambda }(c)$: centrality gives $c(x_1\cdots x_{m}v)=x_1\cdots x_{m}cv=\mathrm{\Lambda }(c)x_1\cdots x_{m}v$ on every finite word, and these words span. An arbitrary weight module may have several levels, so no single level is asserted for it. The zero module satisfies every scalar-action identity; it does not determine a unique scalar level.