)^2} + \frac{c_+}{c_++c_-} \frac{\bar \partial \mathcal{A}^{a \bar a}(w)}{\bar z-\bar w} \right)+ ... \cr
%
j^a_{L,z}(z) j^{\bar a}_{R,\bar z}(w) &= 
-\frac{c^2_+}{c_++c_-} \left(\mathcal{A}^{a \bar a}(w) 2\pi \delta^{(2)}(z-w)
- \frac{c_-}{c_++c_-} \frac{\partial \mathcal{A}^{a \bar a}(w)}{\bar z - \bar w} + \frac{c_-}{c_++c_-} \frac{\bar \partial \mathcal{A}^{a \bar a}(w)}{z - w}\right) + ...\cr
%
j^a_{L,\bar z}(z) j^{\bar a}_{R, z}(w) &= 
-\frac{c^2_-}{c_++c_-} \left(\mathcal{A}^{a \bar a}(w) 2\pi \delta^{(2)}(z-w)
+ \frac{c_+}{c_++c_-} \frac{\partial \mathcal{A}^{a \bar a}(w)}{\bar z - \bar w} - \frac{c_+}{c_++c_-} \frac{\bar \partial \mathcal{A}^{a \bar a}(w)}{z - w}\right)+ ...\end{aligned}$$ The first two OPEs can be written in the alternative form: j\^a\_[L,z]{}(z) j\^[|a]{}\_[R,z]{}(w) = + ... j\^a\_[L,|z]{}(z) j\^[|a]{}\_[R,|z]{}(w) = + ...It is straightforward to show that the OPEs are compatible with current conservation and the Maurer-Cartan equation. These OPEs are also compatible with the fact that the stress-tensor can be written either in terms of the left-current or in terms of the right currents. As an example of these consistency checks, it is shown in appendix \[TRJL\] that when we express the energy-momentum tensor in terms of right currents, it satisfies the expected OPE with the left current: T(z) j\^a\_[L,z]{}(w) = \_[|c |b]{}:j\^[|b]{}\_[R,z]{}j\^[|c]{}\_[R,z]{}:(z) j\^a\_[L,z]{}(w) = + + ((z-w)\^0)

When the theory is defined on a cylinder we can Fourier expand the currents along the angular coordinate, at a given time. It was shown in [@Ashok:2009xx] that the modes of the time components of the left (or the right) currents generate an affine Lie algebra at level $k$. The full commutator algebra computed in appendix \[commutators\] shows that these two affine Lie algebras commute.

Summary {#summary .unnumbered}
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In this section we have determined the pole order parts of the left and right current operator product expansions. The algebra closes on the current components and the 