The root-height completion for weightwise quasi-$R$ intertwiners

For the fixed library coproduct use the diagonal root-height completion $\widehat{C}={\prod }_{\nu \in Q_{+}}{U}^{+}[\nu ]\otimes U^{-}[-\nu ],$ with multiplication $(AB{)}_{\nu }={\sum }_{\beta +\gamma =\nu }{A}_{\beta }{B}_{\gamma }$; each component sum is finite because $I$ is finite and the root coefficients of $\beta ,\gamma \in Q_{+}$ are nonnegative integers. For intertwining with $\mathrm{\Delta }$, use the ambient completed tensor square obtained from the source completion by the exact presentation map of The Lusztig–Kashiwara form, parameter and coproduct dictionary ( $v=q^{-1}$, $K_h\mapsto K_{-h}$ ) followed by the tensor flip; equivalently, every generator identity is read coefficientwise in the root grading. The tensor orientation is positive first, negative second (Positive, negative and toral quantum subalgebras and their root gradings).

A family $A\in \widehat{C}$ acts on $m_{\mu }\otimes n_{\eta }$ by its finite sum whenever $U^{+}[\nu ]m_{\mu }\otimes U^{-}[-\nu ]n_{\eta }$ vanishes for all sufficiently high $\nu$. This holds if the first factor has finite-union bounded-above support, or if the second factor is a lowest module with bounded-below support. For both factors in the finite-weight-space category ${\mathcal{O}}_{\mathrm{i}\mathrm{n}\mathrm{t}}$ the relevant fixed total-weight subspace is finite dimensional, and every tail beyond a finite degree acts as $0$ there. An infinite series is not evaluated on a two-sided-unbounded pair without this explicit finiteness condition.

This completion and the normalization ${\mathrm{\Theta }}_0=1$ are field-level data of the based-module theory of The bar lattice and the divided-power integral lattice of a based module; no universal $R$-matrix, no braiding and no integral coefficient property is asserted here.

The distinction between the diagonal completion ${\prod }_{\nu }{U}^{+}[\nu ]\otimes U^{-}[-\nu ]$ and the ordinary tensor product is essential: an element of the completion may have infinitely many nonzero components, and its action is defined only under the stated support hypotheses. The tensor flip in the convention transport is the one that converts the source's opposite-coproduct intertwining into the library's $\mathrm{\Delta }$-intertwining of The Lusztig–Kashiwara form, parameter and coproduct dictionary, clause (ii).