Weak $\mathfrak{sl}_2$-categorifications

R. Virk

Department of Mathematics
University of Wisconsin
Madison, WI 53706

virk@math.wisc.edu

Put $e=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$, $f=\begin{pmatrix}0&0\\ 1&0\end{pmatrix}$ and $h=ef-fe=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$. Then $e,f,h$ give a basis for the Lie algebra $\mathfrak{sl}_2$.

Let $\mathcal{C}$ be an artinian abelian category with Grothendieck group $K(\mathcal{C})$. Following [CR08], a weak $\mathfrak{sl}_2$-categorification is the data of an adjoint pair $(E,F)$ of exact endofunctors of $\mathcal{C}$ such that

• the action of $e=[E]$ and $f=[F]$ on $\mathbb{Q}\otimes K(\mathcal{C})$ gives a locally finite $\mathfrak{sl}_2$-representation,
• the classes of the simple objects of $\mathcal{C}$ are weight vectors,
• $F$ is isomorphic to a left adjoint of $E$.

We put $E_+=E$, $E_-=F$ and $e_{\pm}=[E_{\pm}]$. By the weight space of an object of $\mathcal{C}$, we will mean the weight space of its class (whenever this is meaningful).

Let $V$ be a locally finite representation of $\mathfrak{sl}_2$. Given $\lambda \in \mathbb{Z}$, we denote by $V_{\lambda}$ the weight space of $V$ for the weight $\lambda$. For $v\in V$ let

$$h_{\pm}(v)=\max\{n\geq 0\,\vert\, e_{\pm}^n\neq 0\}.$$

Proposition 0.0.1   [CR08, Proposition 5.5] Fix a weak $\mathfrak{sl}_2$-categorification of $\mathcal{C}$ and let $V=\mathbb{Q}\otimes K(\mathcal{C})$. Let $\mathcal{C}_{\lambda}$ be the full subcategory of objects of whose class is in $V_{\lambda}$. Then, $\mathcal{C}=\bigoplus_{\lambda}\mathcal{C}_{\lambda}$. In particular, the class of an indecomposable object of $\mathcal{C}$ is a weight vector.

Proof. Let $L_1$ and $L_2$ be simple objects in different weight spaces. Then, there is $\varepsilon\in\{\pm\}$ and $\{i,j\}=\{1,2\}$ such that $h_{\varepsilon}(L_i) > h_{\varepsilon}(L_j)$. Set $r=h_{\varepsilon}(L_i)$. Suppose $M$ is an extension of $L_1$ by $L_2$. Then, $E_{\varepsilon}^rM \cong E_{\varepsilon}^rL_i\neq 0$. So, all the composition factors of $E_{-\varepsilon}^rE_{\varepsilon}^rM$ are in the same weight space as $L_i$. Now,

$$\mathrm{Hom}(E_{-\varepsilon}^rE_{\varepsilon}^rM, M)\cong\mathrm... ...M,E_{\varepsilon}^rM)\cong\mathrm{Hom}(M, E_{-\varepsilon}^rE_{\varepsilon}^rM)$$

and these spaces are not zero. So $M$ has a non-zero simple quotient and a non-zero simple submodule in the same weight space as $L_i$. Hence, $L_i$ is both a submodule and quotient of $M$. Consequently $M=L_1\oplus L_2$.

Thus, $\mathrm{Ext}^1(L_1,L_2)=0$ whenever $L_1$ and $L_2$ are simple objects in different weight spaces. $\qedsymbol$