Tilting objects and negligible morphisms in a root-of-unity category

Fix a finite-type symmetrizable Cartan datum (Symmetrizable Cartan data for quantum groups), a root parameter $\zeta$, and the Lusztig algebra De Concini–Kac, Lusztig, and small quantum-group forms at roots of unity. Put $P^{+}=\{\lambda \in P:⟨\lambda ,h_i⟩\ge 0\text{ for every }i\}$. A finite-dimensional $U_{\zeta }^L(\mathfrak{g})$-module $M$ is type 1 here if it is equipped with a $P$-grading $M={⨁}_{\mu \in P}{M}_{\mu }$ such that $K_{h}m={\zeta }^{⟨\mu ,h⟩}m$ and $T_{i,c,t}m={\left[\begin{array}{c}⟨\mu ,h_i⟩+c\\ t\end{array}\right]}_{{\zeta }_i}m$ for every $m\in M_{\mu }$, $c\in \mathbb{Z}$, and $t\ge 0$, and $E_i^{(r)}{M}_{\mu }\subseteq M_{\mu +r{\alpha }_i}$, $F_i^{(r)}{M}_{\mu }\subseteq M_{\mu -r{\alpha }_i}$. The binomial is evaluated as its Laurent polynomial, including negative upper indices, rather than by dividing specialized factorials. The grading and the full toral-binomial characters are part of the type-1 structure, so coincident ordinary torus eigenvalues at a root do not identify different labels.

The root-representation input for the following definitions is a full $\mathbb{C}$-linear additive braided rigid monoidal subcategory ${\mathcal{C}}_{\zeta }$ of the category of finite-dimensional left $U_{\zeta }^L(\mathfrak{g})$-modules (Unital left and right modules over a ring; unqualified module means left module), together with a pivotal structure $j:\mathrm{id}\to (-{)}^{\vee \vee }$ and the braiding $c_{X,Y}$ (k-linear categories and k-linear functors, Additive category, Monoidal category, Rigid object and rigid monoidal category, Braided monoidal category, Pivotal structure), and two specified finite-dimensional module families $({\mathrm{\Delta }}_{\zeta }(\lambda ){)}_{\lambda \in P^{+}}$ and $({\mathrm{\nabla }}_{\zeta }(\lambda ){)}_{\lambda \in P^{+}}$ whose members are objects of ${\mathcal{C}}_{\zeta }$. In the intended quantum-group application these are intended to be the Weyl and dual Weyl families. Their construction, the tensor action on modules, and the compatible braiding and rigidity with the required root-order hypotheses are separate proof obligations; this definition treats them as supplied data.

A $\mathrm{\Delta }$-filtration of $M$ is a finite chain of submodules in the ambient module category, $0=M_0\subseteq M_1\subseteq \cdots \subseteq M_r=M$, whose successive quotients are isomorphic to specified modules ${\mathrm{\Delta }}_{\zeta }(\lambda )$. A $\mathrm{\nabla }$-filtration is defined in the same way using the specified ${\mathrm{\nabla }}_{\zeta }(\lambda )$ family. A tilting object is an object with both filtrations; write ${\mathcal{T}}_{\zeta }$ for the full subcategory of such objects.

For $M,N\in {\mathcal{T}}_{\zeta }$, a morphism $f:M\to N$ is negligible if ${\mathrm{Tr}}_L(j_M\circ g\circ f)=0\text{for every }g:N\to M,$ where ${\mathrm{Tr}}_L$ is the categorical trace The categorical trace of a morphism into the double dual. This defines a subspace ${\mathcal{N}}_{\zeta }(M,N)\subseteq \mathrm{Hom}(M,N)$. Linearity of composition and of the categorical trace makes this an intersection of kernels, hence a vector subspace. The vector spaces $\mathrm{Hom}(M,N)\mathrm{/}{\mathcal{N}}_{\zeta }(M,N)$ are candidate quotient morphism spaces. This definition does not assert that the ${\mathcal{N}}_{\zeta }(M,N)$ form a composition or tensor ideal, or that the resulting quotient inherits the braiding, is semisimple, ribbon, or modular.