Residue two cocycle on a loop algebra

Use Loop algebra of a simple Lie algebra. Fix the positive-real rescaling $B$ of the Killing form whose induced form on the real finite-root span makes every long root have square $2$. The Killing form is nondegenerate, invariant and symmetric by Engel, the trace criterion, and Killing nondegeneracy, and its restriction gives a positive definite real root form by Finite semisimple Cartan, root and string structure, so this rescaling exists. In particular $B([x,y],z)=B(x,[y,z])$.

For $f={\sum }_m{a}_m{t}^m$ put $f^{\mathrm{\prime }}={\sum }_{m}m{a}_m{t}^{m-1}$ and $\mathrm{Res}(fdt)=a_{-1}$. Define the residue bilinear form by $\omega (x\otimes f,y\otimes q)=B(x,y)\mathrm{Res}(f^{\mathrm{\prime }}qdt)\mathrm{.}$ The formula is balanced and complex bilinear, so extends uniquely to the tensor product. In particular, $\omega (x_m,y_n)=m{\delta }_{m,-n}B(x,y)\mathrm{.}$ The Kronecker symbol is one when $m=-n$ and zero otherwise. This form is in fact an alternating Lie-algebra two-cocycle. For Laurent polynomials $f,q$, the residue of $(fq{)}^{\mathrm{\prime }}$ is zero, so $\mathrm{Res}(f^{\mathrm{\prime }}qdt)=-\mathrm{Res}(f{q}^{\mathrm{\prime }}dt)$; symmetry of $B$ gives skew-symmetry, and in characteristic zero also $\omega (a,a)=0$. For pure tensors $x\otimes f,y\otimes q,z\otimes h$, invariance and symmetry of $B$ make the three factors $B([x,y],z)$, $B([y,z],x)$, and $B([z,x],y)$ equal. The cyclic cocycle sum is therefore that common factor times $\mathrm{Res}(((fq{)}^{\mathrm{\prime }}h+(qh{)}^{\mathrm{\prime }}f+(hf{)}^{\mathrm{\prime }}q)dt)=2\mathrm{Res}((fqh{)}^{\mathrm{\prime }}dt)=0.$ Trilinearity extends the identity to all loop-algebra elements. Thus the terminology “two-cocycle” records a proved property of the defined form, not merely an intended later use.

For any long root $\alpha$, opposite root vectors normalized by $[e_{\alpha },f_{\alpha }]={\alpha }^{\vee }$ satisfy $B(e_{\alpha },f_{\alpha })=\frac{2}{(\alpha ,\alpha )}=1.$ The equality follows by pairing $[e_{\alpha },f_{\alpha }]$ with the Cartan and using invariance. A later result identifies the highest root and proves that it is long; no highest-root existence claim is used here.