The specified rank-one divided-power tilting category

Fix an integer $\mathrm{\ell }\ge 3$ and set $q=e^{\pi i\mathrm{/}\mathrm{\ell }}$, so the multiplicative order of $q^2$ is $\mathrm{\ell }$. Use the divided-power integral form of The Drinfeld-Jimbo quantized enveloping algebra by generators and relations from The Lusztig divided-power integral form $U_A$, specialized by $v\mapsto q$ to $\mathbb{C}$. In type $A_1$, write $K=K_h$ and $E,F$ for the rank-one generators. The convention is

$KE{K}^{-1}=q^{2}E,KF{K}^{-1}=q^{-2}F,[E,F]=\frac{K-K^{-1}}{q-q^{-1}},$

$\mathrm{\Delta }(K)=K\otimes K,\mathrm{\Delta }(E)=E\otimes K+1\otimes E,\mathrm{\Delta }(F)=F\otimes 1+K^{-1}\otimes F\mathrm{.}$

This is the Sawin coproduct, rather than the inverse-$K$ coproduct in The Drinfeld–Jimbo formulas define a Hopf algebra. Its counit is $\epsilon (K)=1$, $\epsilon (E)=\epsilon (F)=0$ and its antipode is $S(E)=-E{K}^{-1}$, $S(F)=-KF$, $S(K)=K^{-1}$. The exact integral transport is from the library algebra with opposite coproduct, by $E_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}=K{E}_{\mathrm{l}\mathrm{i}\mathrm{b}}$, $F_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}=F_{\mathrm{l}\mathrm{i}\mathrm{b}}{K}^{-1}$ and $K_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}=K_{\mathrm{l}\mathrm{i}\mathrm{b}}$. Substituting these generators gives the displayed coproduct and unchanged mixed relation. On divided powers the formulas are $E_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}^{(r)}=v^{-r(r-1)}{K}^r{E}_{\mathrm{l}\mathrm{i}\mathrm{b}}^{(r)}$ and $F_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}^{(r)}=v^{r(r-1)}{F}_{\mathrm{l}\mathrm{i}\mathrm{b}}^{(r)}{K}^{-r}$; the inverse formulas also have Laurent-monomial coefficients, so the integral lattice and toral generators are preserved.

The Lusztig specialization includes all divided powers $E^{(r)},F^{(r)}$ and the toral quantum binomials; this is the divided-power form, not the small quantum group. A finite-dimensional module is type 1 when it carries its integral-weight grading $M={⨁}_{m\in \mathbb{Z}}{M}_m$ with $K{\mathrm{\mid }}_{M_m}=q^m$, the full toral character $T_{c,t}{\mathrm{\mid }}_{M_m}={\left[\begin{array}{c}m+c\\ t\end{array}\right]}_{q}I$ for every $c\in \mathbb{Z}$ and $t\ge 0$, and $E^{(r)}{M}_m\subseteq M_{m+2r}$, $F^{(r)}{M}_m\subseteq M_{m-2r}$. The binomial is its Laurent-polynomial specialization, including negative upper indices. The grading is part of the data and is not collapsed when distinct weights have the same $K$ eigenvalue. Every module intertwiner preserves it: for $m<n$, $T_{-m,n-m}$ has eigenvalues $0$ and $1$ on weights $m$ and $n$.

For $a\ge 0$, the Weyl module $\mathrm{\Delta }(a)$ has basis $v_0,\dots ,v_a$ and actions

$K{v}_j=q^{a-2j}{v}_j,E^{(r)}{v}_j={\left[\begin{array}{c}a-j+r\\ r\end{array}\right]}_q{v}_{j-r},F^{(r)}{v}_j={\left[\begin{array}{c}j+r\\ r\end{array}\right]}_q{v}_{j+r},$

where the Gaussian binomials are the Laurent-polynomial specializations of Quantum integers, factorials, Gaussian binomials and divided powers at $q_i$, and $v_j=0$ outside $0\le j\le a$. Its dual is the dual Weyl module $\mathrm{\nabla }(a)=\mathrm{\Delta }(a{)}^{\vee }$. A Weyl filtration (respectively dual-Weyl filtration) is a finite submodule filtration whose successive quotients are modules $\mathrm{\Delta }(a)$ (respectively $\mathrm{\nabla }(a)$). A tilting module is a module with both filtrations.

Take $s=e^{\pi i\mathrm{/}(2\mathrm{\ell })}$, so $s^2=q$. The ribbon category is the full weight-lattice extension ${\mathcal{U}}_s^{†}({\mathfrak{s}\mathfrak{l}}_2)$ of this Lusztig form, not its root-lattice restriction. Its toral factor is the completed function algebra on the weight lattice $P=\mathbb{Z}\omega$, acting on each weight space by its weight character. Before specialization, the rank-one universal $R$-matrix is the Sawin weight-lattice expression

$R=D{\sum }_{r\ge 0}{q}^{r(r+1)\mathrm{/}2}(1-q^{-2}{)}^r[r{]}_q!E^{(r)}\otimes F^{(r)},D{\mathrm{\mid }}_{M_m\otimes N_n}=s^{mn}\mathrm{.}$

interpreted in the completed tensor product; at the root it means the image of the integral completed form, not termwise division by a vanishing $[r{]}_q!$. In ordinary powers the coefficient is $q^{r(r-1)\mathrm{/}2}(q-q^{-1}{)}^r\mathrm{/}[r{]}_q!$; changing to two divided powers multiplies that coefficient by $([r{]}_q!{)}^2$. Thus the displayed divided-power coefficient is Laurent integral and vanishes for $r\ge \mathrm{\ell }$ after specialization. The toral factor is the weight pairing, fixed explicitly by $D$ above. The pivotal map is $j_M(x)(\phi )=\phi (Kx)$ and the twist is $\theta =u^{-1}j$ for the Drinfeld map $u$. The object class of this page is the full subcategory of finite-dimensional integrable type-1 tilting modules for this weight-lattice ribbon extension. No assertion about semisimplicity or the trace radical is part of this definition.

The parameter dictionary is fixed: $q$ here is the standard rank-one parameter in $KE{K}^{-1}=q^{2}E$ and in $\delta =q+q^{-1}$. In Sawin's notation, type $A_1$ has $D=1$, source root degree $l=2\mathrm{\ell }$, $l^{\mathrm{\prime }}=\mathrm{\ell }$, and $q=s^2$; this is the weight-lattice/ribbon branch.

Their full tensor, duality and ribbon identities on the tilting category are proved in The corrected finite rank-one R series gives the full tilting ribbon structure. This is a forward exposition pointer, not a premise of the definition.