Casimir KZ connection on ordered configurations

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Killing form $B$, let $n\ge 2$, and let $V_1,\dots ,V_n$ be finite-dimensional complex vector spaces. A representation here is a complex-linear Lie algebra homomorphism ${\rho }_i:\mathfrak{g}\to {\mathrm{End}}_{\mathbb{C}}(V_i)$, so $[{\rho }_i(x),{\rho }_i(y)]={\rho }_i([x,y])$. Let $F_n(\mathbb{C})=\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:z_i\mathrm{\ne }{z}_j\text{ for }i\mathrm{\ne }j\}$ be the ordered configuration space and put $V:=V_1\otimes \cdots \otimes V_n$ in the indicated factor order. For any pair of $B$-dual bases $(x_a),(x^a)$ define ${\mathrm{\Omega }}_{ij}:={\sum }_a{\rho }_i(x_a{)}^{(i)}{\rho }_j(x^a{)}^{(j)}\in {\mathrm{End}}_{\mathbb{C}}(V)$, where each superscript means that the operator acts on that tensor factor and as the identity on all other factors. For $i<j$ put ${\eta }_{ij}:=(d{z}_i-d{z}_j)\mathrm{/}(z_i-z_j)$, a globally defined complex-valued one-form on $F_n(\mathbb{C})$. For $\kappa \in \mathbb{C}\setminus \{0\}$, the Casimir Knizhnik–Zamolodchikov connection on the trivial vector bundle $F_n(\mathbb{C})\times V\to F_n(\mathbb{C})$ is

${\mathrm{\nabla }}_{\kappa }^{\mathrm{K}\mathrm{Z}}:=d-{\kappa }^{-1}{\sum }_{i<j}{\mathrm{\Omega }}_{ij}{\eta }_{ij}\mathrm{.}$

Here a connection on this trivial bundle is a complex-linear operator on smooth $V$-valued sections satisfying $\mathrm{\nabla }(fs)=df\otimes s+f\mathrm{\nabla }s$ for every smooth scalar function $f$ and section $s$. The associated quadratic Casimir is $C_B:={\sum }_a{x}_a{x}^a$ in the universal enveloping algebra; with the Killing-form convention above it is central, and ${\mathrm{\Omega }}_{ij}$ is its tensor-level Casimir counterpart.

Ordered configuration spaces $F_n(X)$

The Killing form of a semisimple Lie algebra

The quadratic Casimir element is independent of the choice of dual bases

The quadratic Casimir element

The quadratic Casimir element is central