2.1. Preliminaries

We work over the field $\mathbb {C}$ of complex numbers and, as usual, write $\mathbb {C}^*$ for $\mathbb {C}\setminus \{0\}$ . For a Lie algebra $\mathfrak {g}$ , we denote by $U(\mathfrak {g})$ the universal enveloping algebra of $\mathfrak {g}$ and by $Z(\mathfrak {g})$ the center of $U(\mathfrak {g})$ . As usual, we denote by $\mathfrak {g}$ -Mod the category of all $\mathfrak {g}$ -modules and by $\mathfrak {g}$ -mod the category of all finitely generated $\mathfrak {g}$ -modules.

For all details on monoidal categories and their representations, we refer to [Reference Etingof, Gelaki, Nikshych and Ostrik11].

2.2. Finite-dimensional $\mathfrak {sl}_2$ -modules

Consider the Lie algebra $\mathfrak {sl}_2$ with the standard basis $\{e,f,h\}$ , where

$$ \begin{align*} e=\left(\begin{array}{cc}0&1\\0&0\end{array}\right),\quad f=\left(\begin{array}{cc}0&0\\1&0\end{array}\right),\quad h=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right). \end{align*} $$

For details on the Lie algebra $\mathfrak {sl}_2$ and its modules, we refer to [Reference Mazorchuk27].

Denote by $\mathscr {C}$ the category of all finite-dimensional $\mathfrak {sl}_2$ -modules. The category $\mathscr {C}$ is semi-simple with simple objects $L(m)$ , where $m\in \mathbb {Z}_{\geq 0}$ . The $\mathfrak {sl}_2$ -module $L(m)$ is the unique simple $\mathfrak {sl}_2$ -module of dimension $m+1$ . The module $L(m)$ is a highest weight module of highest weight m. Here, a weight means just an h-eigenvalue.

The category $\mathscr {C}$ is monoidal with respect to the usual tensor product of $\mathfrak {sl}_2$ -modules. The monoidal unit is the trivial $\mathfrak {sl}_2$ -module $L(0)$ . Recall the Clebsch–Gordan coefficients for $\mathfrak {sl}_2$ : for $m,n\in \mathbb {Z}_{\geq 0}$ , the $\mathfrak {sl}_2$ -module $L(m)\otimes _{\mathbb {C}}L(n)$ decomposes as

$$ \begin{align*} L(|m-n|)\oplus L(|m-n|+2)\oplus L(|m-n|+4)\oplus \dots \oplus L(m+n-2)\oplus L(m+n). \end{align*} $$

The monoidal category $\mathscr {C}$ is rigid and symmetric. All objects of $\mathscr {C}$ are self-dual. The category $\mathscr {C}$ is generated by $L(1)$ as a monoidal category. The category $\mathscr {C}$ is simple in the sense that it does not have any non-trivial monoidal ideals.

To simplify notation, for every $i\in \mathbb {Z}_{\geq 0}$ , we will sometimes denote by $F_i$ the object $L(i)$ of $\mathscr {C}$ .

2.3. Locally finitary module categories over $\mathscr {C}$

Let $\mathcal {M}$ be a (left) module category over $\mathscr {C}$ . We will say that $\mathcal {M}$ is locally finitary provided that the following conditions are satisfied:\eject

• • $\mathcal {M}$ is $\mathbb {C}$ -linear, additive, idempotent split, and Krull–Schmidt;

• • $\mathcal {M}$ has at most countably many indecomposable objects, up to isomorphism;

• • all hom-spaces in $\mathcal {M}$ are finite dimensional (over $\mathbb {C}$ ).

A typical example of a locally finitary module category over $\mathscr {C}$ is $\mathcal {M}=\mathscr {C}$ with respect to the left regular action.

Following [Reference Mazorchuk and Miemietz29], we will say that $\mathcal {M}$ is transitive provided that, for any indecomposable objects $X,Y\in \mathcal {M}$ , there exists $F\in \mathscr {C}$ such that Y is isomorphic to a summand of $F(X)$ . We will say that $\mathcal {M}$ is simple transitive provided that $\mathcal {M}$ does not have any non-trivial $\mathscr {C}$ -invariant ideals.

Let $\{X_i\,:\,i\in I\}$ be a complete and irredundant set of representatives of the isomorphism classes of indecomposable objects in $\mathcal {M}$ . Then, with any $F\in \mathscr {C}$ , we can associate an $I\times I$ -matrix $[F]=(m_{i,j})_{i,j\in I}$ with non-negative integer coefficients in which $m_{i,j}$ is defined as the multiplicity of $X_i$ as a summand of $F(X_j)$ . Then $\mathcal {M}$ is transitive if and only if, for all $i,j\in I$ , there is some power $[F_1]^k$ of $[F_1]$ such that the $(i,j)$ th coefficient in $[F_1]^k$ is non-zero.

We also define the action graph $\Lambda _{\mathcal {M}}$ of $\mathcal {M}$ as an oriented graph (possibly with loops, but without multiple oriented edges) with vertices I, in which there is an oriented edge from i to j if and only if there is $F\in \mathscr {C}$ such that the coefficient $m_{j,i}$ in $[F]$ is non-zero. By taking as F the unit object in $\mathscr {C}$ , we see that $\Lambda _{\mathcal {M}}$ has an oriented loop at each vertex. Then $\mathcal {M}$ is transitive if and only if $\Lambda _{\mathcal {M}}$ is strongly connected.

2.4. Grothendieck ring of $\mathscr {C}$

Consider the polynomial algebra $\mathbb {Z}[x]$ . For $i\geq 0$ , define the polynomials $R_i(x)$ recursively as follows:

$$ \begin{align*} R_0(x)=1; \quad R_1(x)=x; \quad R_i(x)=xR_{i-1}(x)-R_{i-2}(x), \text{ for } i\geq 2. \end{align*} $$

These polynomials are sometimes called ultraspherical polynomials (see, e.g., [Reference Etingof and Ostrik13]). They can also be considered as renormalized Chebyshev polynomials of the second kind (see [Reference Rivlin36]).

The Grothendieck ring $[\mathscr {C}]$ of $\mathscr {C}$ is isomorphic to $\mathbb {Z}[x]$ by sending the class of $L(i)$ in $[\mathscr {C}]$ to $R_i(x)$ , for $i\geq 0$ . Consequently, for any locally finitary $\mathscr {C}$ -module category $\mathcal {M}$ , we have $[F_i]=R_i([F_1])$ , for $i\geq 0$ .

Given a locally finitary $\mathscr {C}$ -module category $\mathcal {M}$ , the split Grothendieck group $[\mathcal {M}]_\oplus $ of $\mathcal {M}$ has the obvious structure of a $[\mathscr {C}]$ -module.

2.5. Setup

Let $\mathfrak {g}$ be a finite-dimensional Lie algebra with a fixed surjective Lie algebra homomorphism $\varphi :\mathfrak {g}\to \mathfrak {sl}_2$ . The trivial example is $\mathfrak {g}=\mathfrak {sl}_2$ with $\varphi $ being the identity. The pullback along $\varphi $ allows us to consider $\mathscr {C}$ as a (monoidal) subcategory of $\mathfrak {g}$ -Mod, up to monoidal equivalence.

We will be interested in $\mathscr {C}$ -module categories of the form $\mathcal {M}\subset \mathfrak {g}$ -Mod in which the $\mathscr {C}$ -module structure is given by tensoring over $\mathbb {C}$ and then using the standard comultiplication for $U(\mathfrak {g})$ . We will be especially interested in such $\mathcal {M}$ given by the additive closure of $\mathscr {C}\cdot M$ , where $M\in \mathfrak {g}$ -Mod, with particular emphasis on the case when M is simple. We will denote this kind of category by $\mathrm {add}(\mathscr {C}\cdot M)$ .

2.6. Projective functors

The category $\mathscr {C}$ is closely connected to the monoidal category of projective endofunctors for $\mathfrak {sl}_2$ (see [Reference Bernstein and Gelfand4]). The latter are endofunctors of the full subcategory $\mathcal {Z}$ of $\mathfrak {sl}_2$ -Mod that consists of all modules, the action of $Z(\mathfrak {sl}_2)$ on which is locally finite. Recall that $Z(\mathfrak {sl}_2)$ is a polynomial algebra generated by the Casimir element $\mathtt {c}=(h+1)^2+4fe$ . By [Reference Bernstein and Gelfand4, Subsection 1.8], the category $\mathcal {Z}$ is a direct sum, over all $\theta \in \mathbb {C}$ , of the full subcategories $\mathcal {Z}_\theta $ of $\mathcal {Z}$ consisting of all modules on which $\mathtt {c}-\theta $ acts locally nilpotently.

Let us recall classification of indecomposable projective functors from [Reference Bernstein and Gelfand4, Theorem 3.3]. For this, recall that $\mathfrak {sl}_2$ admits a triangular decomposition

$$ \begin{align*} \mathfrak{sl}_2=\mathfrak{n}_-\oplus \mathfrak{h}\oplus \mathfrak{n}_+, \end{align*} $$

where e spans $\mathfrak {n}_+$ , f spans $\mathfrak {n}_-$ , and h spans $\mathfrak {h}$ . We can identify $\mathfrak {h}^*$ with $\mathbb {C}$ by sending $\lambda \in \mathfrak {h}^*$ to $\lambda (h)$ . Then, for each $\lambda \in \mathbb {C}$ , we have the corresponding Verma module $\Delta (\lambda )$ with highest weight $\lambda $ and its unique simple quotient $L(\lambda )$ . Both these modules belong to the BGG category $\mathcal {O}$ (see [Reference Bernstein, Gelfand and Gelfand3], [Reference Humphreys17]). There, we also have the indecomposable projective cover $P(\lambda )$ of $L(\lambda )$ . Note that $P(\lambda )=\Delta (\lambda )$ unless $\lambda \in \{-2,-3,\dots \}$ and $P(\lambda )=L(\lambda )$ unless $\lambda \in (\mathbb {Z}\setminus \{-1\})$ .

An element $\lambda \in \mathbb {C}$ is called dominant provided that $\lambda \not \in \{-2,-3,\dots \}$ and anti-dominant provided that $\lambda \not \in \mathbb {Z}_{\geq 0}$ . According to [Reference Bernstein and Gelfand4, Theorem 3.3], for each pair $(\lambda ,\mu )\in \mathbb {C}^2$ satisfying

• • $\lambda -\mu \in \mathbb {Z}$ ;

• • $\lambda $ is dominant;

• • $\mu $ is anti-dominant if $\lambda =-1$ ;

there is a unique indecomposable projective functor

$$ \begin{align*} \theta_{\lambda,\mu}:\mathcal{Z}_{(\lambda+1)^2}\to \mathcal{Z}_{(\mu+1)^2} \end{align*} $$

such that $\theta _{\lambda ,\mu }\Delta (\lambda )=P(\mu )$ . All indecomposable projective endofunctors of $\mathcal {Z}$ are of this form. If $\lambda \not \in \mathbb {Z}$ , then $\theta _{\lambda ,\mu }$ is an equivalence with inverse $\theta _{\mu ,\lambda }$ .

We will say that $\theta _{\lambda ,\mu }$ is special provided that $\lambda \in \frac {1}{2}+\mathbb {Z}$ and $\mu =-\lambda -2$ .

2.7. Abelianization

Let $\mathcal {M}$ be a locally finitary $\mathscr {C}$ -module category. Then we can consider the projective abelianization $\overline {\mathcal {M}}$ of $\mathcal {M}$ (see [Reference Mazorchuk and Miemietz28, Subsection 3.1]). The objects of $\overline {\mathcal {M}}$ are diagrams $X\to Y$ over $\mathcal {M}$ and morphisms are (solid) commutative squares

modulo the ideal generated by all those squares in which the right vertical morphism admits a factorization via some dotted morphism. The category $\overline {\mathcal {M}}$ is a $\mathscr {C}$ -module category, with the $\mathscr {C}$ -action inherited from $\mathcal {M}$ .

If we additionally assume that, for each indecomposable object $M\in \mathcal {M}$ , there are only finitely many indecomposable objects N, up to isomorphism, such that $\mathcal {M}(M,N)\neq 0$ and there are only finitely many indecomposable objects N, up to isomorphism, such that $\mathcal {M}(N,M)\neq 0$ , then the category $\overline {\mathcal {M}}$ is an abelian length category in which the original category $\mathcal {M}$ naturally embeds via $P\mapsto (0\to P)$ and the closure of the image of this embedding with respect to isomorphism coincides with the category of projective objects in $\overline {\mathcal {M}}$ . In full generality, $\overline {\mathcal {M}}$ is abelian with the above properties if and only if $\mathcal {M}$ has weak kernels (see [Reference Freyd14]). We will call such categories admissible. We refer the reader to [Reference Macpherson25], [Reference Macpherson26] for more details on locally finitary categories, where one can also find some information about how some of the above conditions may be relaxed.

3.1. Generalized Cartan matrix

Let I be an indexing set (finite or infinite). Recall that a generalized Cartan $I\times I$ -matrix (GCM) is a matrix $C=(c_{i,j})_{i,j\in I}$ such that

• • $c_{i,i}\leq 2$ , for all i;

• • $c_{i,j}\in \{0,-1,-2,\dots \}$ , for all $i\neq j$ ;

• • $c_{i,j}=0$ if and only if $c_{j,i}=0$ , for all $i,j$ .

Given a GCM C on I, we can associate to it a graph $\Gamma _C$ with vertex set I such that, for all $i\neq j\in I$ , there are exactly $-c_{i,j}$ oriented edges from i to j; moreover, for all $i\in I$ , there are $2-c_{i,i}$ loops at the vertex i. We denote by $\widetilde {C}$ the adjacency matrix of $\Gamma _C$ (with the convention that each loop corresponds to one on the main diagonal) and note that, by definition, $C=2\mathrm {Id}_I-\widetilde {C}$ , where $\mathrm {Id}_I$ is the identity $I\times I$ -matrix. A pair of oppositely oriented edges between i and j, for $i\neq j$ , will be simplified to one unoriented edge between these vertices. By convention, all loops are unoriented.

Example 1. The graph associated with the GCM

$$ \begin{align*} \left(\begin{array}{cccc}1&-1&0&0\\-2&2&-2&-1\\0&-1&2&-2\\0&-1&-2&2\end{array}\right) \end{align*} $$

looks as follows:

3.2. Classical Dynkin diagrams and their characterization

Recall the classical finite Dynkin diagrams, drawn with our convention for the underlying graph (in all cases, the index is the number of vertices):

An important invariant of a Dynkin diagram is the so-called Coxeter number, usually denote by $\mathtt {h}$ , given by the following table:

$$ \begin{align*} \begin{array}{c||c|c|c|c|c|c|c|c|c} \text{type:}&A_n&B_n&C_n&D_n&E_6&E_7&E_8&F_4&G_2\\ \hline \mathtt{h}:&n+1&2n&2n&2n-2&12&18&30&12&6 \end{array} \end{align*} $$

We can now recall the following standard characterization of the classical Dynkin diagrams (see [Reference Kac19, Chapter 4] and [Reference Etingof and Ostrik13, Corollary 3.9]).

In the context of representations of monoidal categories, classical Dynkin diagrams appeared, for example, in [Reference Kildetoft, Mackaay, Mazorchuk and Zimmermann20], [Reference Kirillov and Ostrik22].

3.3. Affine Dynkin diagrams and their characterization

Next, recall the affine Dynkin diagrams (also known as generalized Euclidean diagrams):

The following proposition characterizes the above graphs (see [Reference Happel, Preiser and Ringel15, Theorem 2]).

3.4. Infinite Dynkin diagrams and their characterization

Finally, recall the following infinite Dynkin diagrams:

The following proposition characterizes the above graphs (see [Reference Happel, Preiser and Ringel15, p. 12]).

We refer to [Reference Happel, Preiser and Ringel15], [Reference Happel, Preiser and Ringel16] for more details.

3.5. Combinatorics of transitive $\mathscr {C}$ -categories

The following corollary follows from the definitions and the above characterizations of the affine and infinite Dynkin diagrams (see [Reference Etingof and Khovanov12, Theorem 4.1]).

This raises a natural problem to find a realization for each possibility described in Corollary 5 within the context provided by Section 2.5. This is the main problem which we consider in this article. For the case of a finite rank, we have a (combinatorially quite complex) classification of simple $\mathscr {C}$ -module categories obtained in [Reference Etingof and Ostrik13]. It is not clear where to look for such categories in our context. Therefore, we focus on the infinite rank case.

4.1. First realization

Let $\mathfrak {g}=\mathfrak {sl}_2$ with $\varphi $ being the identity. Define $\mathcal {M}_1:=\mathscr {C}$ with the left regular action.

Then the indecomposable objects in $\mathcal {M}_1$ are $L(i)$ , for $i\geq 0$ , and the action of $L(1)$ is given by

$$ \begin{align*} L(1)\otimes_{\mathbb{C}} L(i)\cong \begin{cases} L(1),& i=0;\\ L(i-1)\oplus L(i+1),& i>0. \end{cases} \end{align*} $$

Therefore, we have

(1) $$ \begin{align} [L(1)]= \left( \begin{array}{ccccc} 0&1&0&0&\dots\\ 1&0&1&0&\dots\\ 0&1&0&1&\dots\\ 0&0&1&0&\dots\\ \vdots&\vdots&\vdots&\vdots&\ddots \end{array} \right). \end{align} $$

4.2. Second realization

Let $\mathfrak {g}=\mathfrak {sl}_2$ with $\varphi $ being the identity. Consider the BGG category $\mathcal {O}$ for $\mathfrak {g}$ (see [Reference Bernstein, Gelfand and Gelfand3], [Reference Mazorchuk27]). For $\lambda \in \mathbb {C}$ , denote by $L(\lambda )$ the simple highest weight $\mathfrak {g}$ -module with highest weight $\lambda $ . Also denote by $P(\lambda )$ the projective cover of $L(\lambda )$ in $\mathcal {O}$ .

Define the category $\mathcal {N}$ as the additive closure in $\mathcal {O}$ of the modules $L(\lambda )$ and $P(\lambda )$ , for $\lambda \in \{-1,-2,-3,\dots \}$ . Note that these are exactly the integral tilting modules in the sense of [Reference Collingwood and Irving6], [Reference Ringel35]. Also, note that $L(-1)=P(-1),$ while $L(\lambda )\not \simeq P(\lambda )$ , for $\lambda \in \{-2,-3,\dots \}$ .

For $\lambda \in \{-1,-2,-3,\dots \}$ , we have the following (see [Reference Mazorchuk27, Section 5]):

$$ \begin{align*} L(1)\otimes_{\mathbb{C}} L(\lambda)\cong \begin{cases} P(-2),& \lambda=-1;\\ L(\lambda-1)\oplus L(\lambda+1),& \lambda<-1; \end{cases} \end{align*} $$

and

(2) $$ \begin{align} L(1)\otimes_{\mathbb{C}} P(\lambda)\cong \begin{cases} P(-2),& \lambda=-1;\\ P(-1)\oplus P(-1)\oplus P(-3),& \lambda=-2;\\ P(\lambda-1)\oplus P(\lambda+1),& \lambda<-2. \end{cases} \end{align} $$

From these formulas, it follows that the additive closure of the modules $P(\lambda )$ , with $\lambda \in \{-1,-2,\dots \}$ , is a transitive $\mathscr {C}$ -module subcategory of $\mathcal {N}$ . Let $\mathcal {I}$ be the ideal in $\mathcal {N}$ generated by all $P(\lambda )$ , where $\lambda \in \{-1,-2,\dots \}$ . Then the formulas above imply that $\mathcal {I}$ is $\mathscr {C}$ -stable and hence $\mathcal {M}_2:=\mathcal {N}/\mathcal {I}$ is a $\mathscr {C}$ -module category. Now the above formulas imply Formula (1) for $\mathcal {M}_2$ .

4.3. Statements

We can now summarize the content of this section in the following proposition.

We also record the following observation.

Proof. Let $\mathcal {M}$ be a simple transitive $\mathscr {C}$ -module category of type $A_\infty $ . We are going to prove that $\mathcal {M}$ is equivalent, as a $\mathscr {C}$ -module category, to the left regular $\mathscr {C}$ -module category ${}_{\mathscr {C}}\mathscr {C}$ .

We have that $[L(1)]$ is given by Formula (1) and hence there is a unique, up to isomorphism indecomposable object X in $\mathcal {M}$ such that $L(1)\cdot X$ is indecomposable.

Since $L(i)\otimes _{\mathscr {C}} L(0)\cong L(i)$ in ${}_{\mathscr {C}}\mathscr {C}$ , the first column of each $R_i([L(1)])$ has only one non-zero entry, namely, $1$ in row $i+1$ . This means that $L(i)\cdot X$ is indecomposable, that $L(i)\cdot X\cong L(j)\cdot X$ if and only if $i=j$ and that all indecomposable objects of $\mathcal {M}$ can be obtained in this way.

Now let us show that the radical of $\mathcal {M}$ is $\mathscr {C}$ -invariant. Since $\mathscr {C}$ is generated by $F_1$ , it is enough to show that the radical of $\mathcal {M}$ is $F_1$ -invariant. Let $M_0,M_1,\dots $ be a list of indecomposables in $\mathcal {M}$ such that, in the corresponding basis of the split Grothendieck group, the action of $F_1$ is given by Formula (1).

If $\varphi : M_i\to M_j$ is a morphism and $|i-j|\neq 0,2$ , then $F_1(M_i)$ and $F_1(M_j)$ do not have isomorphic summands and hence $F_1(\varphi )$ is a radical morphism. If $\varphi : M_i\to M_i$ is a radical morphism, it is nilpotent as $\mathcal {M}$ is locally finitary. Therefore, $F_1(\varphi )$ is also nilpotent. As $F_1(M_i)$ does not have isomorphic summands, it follows that $F_1(\varphi )$ is a radical morphism.

It remains to consider the case of a morphism $\varphi : M_i\to M_{i\pm 2}$ . Let us assume that $\varphi : M_i\to M_{i+2}$ , in the other case, the arguments are similar. Consider the abelianization $\overline {\mathcal {M}}$ and let $N_i$ be the simple top of $M_i$ in $\overline {\mathcal {M}}$ , for $i\in \mathbb {Z}_{\geq 0}$ .

Lemma 8. For $i,j\in \mathbb {Z}_{\geq 0}$ , we have

$$ \begin{align*} F_i(N_j)\cong N_{i+j}\oplus N_{i+j-2}\oplus\dots \oplus N_{|i-j|}. \end{align*} $$

Proof. Due to (1), the matrices $[F_2]$ , $[F_3]$ , and so on are as follows:

$$ \begin{align*} [F_2]= \left( \begin{array}{cccccc} 0&0&1&0&0&\dots\\ 0&1&0&1&0&\dots\\ 1&0&1&0&1&\dots\\ 0&1&0&1&0&\dots\\ 0&0&1&0&1&\dots\\ \vdots&\vdots&\vdots&\vdots&\ddots&\ddots \end{array} \right),\quad [F_3]= \left( \begin{array}{cccccc} 0&0&0&1&0&\dots\\ 0&0&1&0&1&\dots\\ 0&1&0&1&0&\dots\\ 1&0&1&0&1&\dots\\ 0&1&0&1&0&\dots\\ \vdots&\vdots&\vdots&\vdots&\ddots&\ddots \end{array} \right),\dots \end{align*} $$

In particular, from [Reference Agerholm and Mazorchuk1, Lemma 8], it follows that $F_i N_0\cong N_i$ . Denote by $\mathcal {N}$ the additive closure of all $N_i$ . Then $\mathcal {N}$ is semi-simple and has the structure of a $\mathscr {C}$ -module category by restriction. By [Reference Agerholm and Mazorchuk1, Lemma 8], the matrix of the action of $F_1$ on $\mathcal {N}$ is given by (1). Now the statement of the lemma follows from the matrix of $F_i$ .

Recall that we have a non-zero morphism $\varphi : M_i\to M_{i+2}$ . Then $\varphi (M_i)$ is a submodule of the radical of $M_{i+2}$ . By Lemma 8, we have $F_1(N_{i+2})\cong N_{i+1}\oplus N_{i+3}$ . This means that, applying the exact functor $F_1$ to the short exact sequence

$$ \begin{align*} 0\to \mathrm{Rad}(M_{i+2}) \to M_{i+2}\to N_{i+2}\to 0, \end{align*} $$

we get a short exact sequence

$$ \begin{align*} 0\to F_1(\mathrm{Rad}(M_{i+2})) \to F_1(M_{i+2})\to F_1(N_{i+2})\to 0, \end{align*} $$

in which $F_1(N_{i+2})$ is isomorphic to the top of $F_1(M_{i+2})$ . Therefore, we have

$$ \begin{align*} F_1(\mathrm{Rad}(M_{i+2}))= \mathrm{Rad}(F_1(M_{i+2})). \end{align*} $$

As $\varphi (M_i)$ is a submodule of $\mathrm {Rad}(M_{i+2})$ and $F_1$ is exact, we obtain that $F_1(\varphi (M_i))$ is a submodule of $\mathrm {Rad}(F_1(M_{i+2}))$ , which means that $F_1(\varphi )$ is a radical morphism. This is exactly what we needed to prove. This completes our argument that the radical of $\mathcal {M}$ is $\mathscr {C}$ -stable.

Now, the fact that $\mathcal {M}$ is simple transitive implies that the radical of $\mathcal {M}$ is zero. In other words, $\mathcal {M}$ is semi-simple.

By Yoneda lemma, sending the unit object $L(0)$ to X gives rise to a homomorphism of $\mathscr {C}$ -module categories from ${}_{\mathscr {C}}\mathscr {C}$ to $\mathcal {M}$ . From the previous paragraph, this homomorphism is an equivalence.

5.1. First realization

Consider the setup of Section 4.2. Let $\mathfrak {g}=\mathfrak {sl}_2$ with $\varphi $ being the identity. Consider the BGG category $\mathcal {O}$ for $\mathfrak {g}$ . For $\lambda \in \mathbb {C}\setminus \mathbb {Z}$ , consider the simple highest weight module $L(\lambda )$ with highest weight $\lambda $ . Then the module $L(\lambda )$ is also projective in $\mathcal {O}$ .

For a fixed $\mu \in \mathbb {C}\setminus \mathbb {Z}$ , let $\mathcal {N}_1=\mathcal {N}_1(\mu )$ be the additive closure of all $L(\lambda )$ such that $\lambda -\mu \in \mathbb {Z}$ . Then, for any such $\lambda $ , we have

$$ \begin{align*} L(1)\otimes_{\mathbb{C}}L(\lambda)\cong L(\lambda-1)\oplus L(\lambda+1). \end{align*} $$

In particular, $\mathcal {N}_1$ is a semi-simple and transitive $\mathscr {C}$ -module category.

From the above formula, we get that

(3) $$ \begin{align} [L(1)]= \left( \begin{array}{cccccc} \ddots&\vdots&\vdots&\vdots&\vdots&\ddots\\ \dots&0&1&0&0&\dots\\ \dots&1&0&1&0&\dots\\ \dots&0&1&0&1&\dots\\ \dots&0&0&1&0&\dots\\ \ddots&\vdots&\vdots&\vdots&\vdots&\ddots \end{array} \right). \end{align} $$

5.2. Second realization

Let $\mathfrak {g}$ be the Schrödinger Lie algebra in the $1+1$ -dimensional space–time (see [Reference Dubsky, Lü, Mazorchuk and Zhao9]). This is a Lie algebra with basis $\{e,f,h,p,q,z\}$ , where z is central and the rest of the Lie bracket is given by

$$ \begin{align*} & [h,e]=2e,\quad [h,f]=-2f,\quad [e,f]=h,\quad [e,q]=p,\quad [e,p]=0, \\ & [h,p]=p,\quad [f,p]=q,\quad [f,q]=0,\quad [h,q]=-q,\quad [p,q]=z. \end{align*} $$

The subalgebra of $\mathfrak {g}$ generated by $e,f,h$ is isomorphic to $\mathfrak {sl}_2$ and the remaining basis elements of $\mathfrak {g}$ span a nilpotent ideal. Thus, factoring that ideal out defines a surjective Lie algebra homomorphism $\varphi :\mathfrak {g}\to \mathfrak {sl}_2$ . Therefore, we can view $\mathscr {C}$ as a monoidal subcategory of $\mathfrak {g}$ -mod.

We have a natural triangular decomposition $\mathfrak {g}=\mathfrak {n}_-\oplus \mathfrak {h}\oplus \mathfrak {n}_+$ , where $\mathfrak {n}_-$ is spanned by f and q, $\mathfrak {h}$ is spanned by h and z, and $\mathfrak {n}_+$ is spanned by e and p. Associated with this triangular decomposition, we have the corresponding BGG category $\mathcal {O}$ (see [Reference Dubsky, Lü, Mazorchuk and Zhao9]). For $\lambda \in \mathfrak {h}^*$ , we denote by $L(\lambda )$ the simple highest weight $\mathfrak {g}$ -module with highest weight $\lambda $ , this module is an object in $\mathcal {O}$ .

Let $\lambda \in \mathfrak {h}^*$ be such that $\lambda (z)\neq 0$ and $\lambda (h)\not \in \frac {1}{2}\mathbb {Z}$ . Then, due to [Reference Dubsky, Lü, Mazorchuk and Zhao9, Proposition 3], the module $L(\lambda )$ is projective in $\mathcal {O}$ . Let $\varepsilon \in \mathfrak {h}^*$ be such that $\varepsilon (z)=0$ and $\varepsilon (h)=1$ . By comparing the characters, we obtain that

$$ \begin{align*} L(1)\otimes_{\mathbb{Z}} L(\lambda)\cong L(\lambda+\varepsilon)\oplus L(\lambda-\varepsilon). \end{align*} $$

Now, for a fixed $\mu \in \mathfrak {h}^*$ such that $\mu (z)\neq 0$ and $\mu (h)\not \in \frac {1}{2}\mathbb {Z}$ , define $\mathcal {N}_2$ as the additive closure of all $L(\lambda )$ such that $\lambda -\mu \in \mathbb {Z}\varepsilon $ . Then the formula in the previous paragraph implies that $\mathcal {N}_2$ is a semi-simple and transitive $\mathscr {C}$ -module category in which $[L(1)]$ is given by Formula (3).

5.3. Third realization

Let $\mathfrak {g}=\mathfrak {sl}_2\otimes _{\mathbb {C}} \mathbb {C}[x]/(x^2)$ be the Takiff Lie algebra of $\mathfrak {sl}_2$ and $\varphi :\mathfrak {g}\to \mathfrak {sl}_2$ be the natural projection. For $g\in \mathfrak {sl}_2$ , denote by $\overline {g}$ the element $g\otimes x\in \mathfrak {g}$ . Then the center $Z(\mathfrak {g})$ is generated by the elements

$$ \begin{align*} C=\overline{h}h +2\overline{e}f+2\overline{f}e\quad \text{ and }\quad \overline{C}=\overline{h}^2+4\overline{f}\overline{e}. \end{align*} $$

The standard triangular decomposition of $\mathfrak {sl}_2$ induces a triangular decomposition of $\mathfrak {g}$ in the obvious way. For $\lambda \in \mathbb {C}\setminus \mathbb {Z}$ , consider the corresponding highest weight $\mathfrak {sl}_2$ -module $L(\lambda )$ with highest weight $\lambda $ and set

$$ \begin{align*} Q(\lambda):=U(\mathfrak{g})\bigotimes_{U(\mathfrak{sl}_2)}L(\lambda). \end{align*} $$

Now, for $\chi \in \mathbb {C}$ , define

$$ \begin{align*} Q(\lambda,\chi):=Q(\lambda)/(\overline{C}-\chi)Q(\lambda) \end{align*} $$

and, for $\theta \in \mathbb {C}$ , define

$$ \begin{align*} Q(\lambda,\chi,\theta):=Q(\lambda,\chi)/(C-\theta)Q(\lambda,\chi). \end{align*} $$

Then, by [Reference Zhu37, Proposition 3.2], the module $Q(\lambda ,\chi ,\theta )$ is a simple $\mathfrak {g}$ -module if and only if $\theta \neq \sqrt {\chi }(\lambda +2k)$ , for all $k\in \mathbb {Z}$ . If the latter conditions are satisfied, we have $Q(\lambda ,\chi ,\theta )\cong Q(\lambda ',\chi ',\theta ')$ if and only if $\chi =\chi '$ , $\theta =\theta '$ , and $\lambda -\lambda '\in 2\mathbb {Z}$ .

Now let us fix $\lambda ,\chi ,\theta $ as above, satisfying $\lambda \in \mathbb {C}\setminus \mathbb {Z}$ and $\theta ^2\neq \chi (\lambda +j)^2$ , for all $j\in \mathbb {Z}$ , and, additionally, $\chi \neq 0$ . Denote by $\mathcal {N}_3$ the additive closure of $Q(\lambda +i,\chi ,\theta +i\sqrt {\chi })$ , for all $i\in \mathbb {Z}$ .

Proof. Note that $\overline {g}$ kills $L(1)$ , for all $g\in \mathfrak {sl}_{2}$ . Therefore, $\overline {C}$ acts on $L(1)\otimes _{\mathbb {C}}Q(\lambda ,\chi ,\theta )$ with the same scalar, namely, $\chi $ , as on $Q(\lambda ,\chi ,\theta )$ .

Let us determine the action of C. By [Reference Zhu37, Lemma 3.1], the module $Q(\lambda ,\chi ,\theta )$ , when restricted to $\mathfrak {sl}_2$ , is a multiplicity-free direct sum of the simple highest weight modules $L(\lambda +i)$ , where $i\in 2\mathbb {Z}$ . As $L(1)\otimes _{\mathbb {C}}L(\lambda +i)$ is isomorphic to $L(\lambda +i+1)\oplus L(\lambda +i-1)$ , the module $L(1)\otimes _{\mathbb {C}}Q(\lambda ,\chi ,\theta )$ has two linearly independent elements of weight $\lambda +1$ that are killed by e. Let $\{b_1,b_{-1}\}$ be the standard basis of $L(1)$ , that is:

$$ \begin{align*} hb_1=b_1,\quad hb_{-1}=-b_{-1},\quad eb_{-1}=b_1,\quad eb_1=0,\quad fb_{-1}=0,\quad fb_1=b_{-1}. \end{align*} $$

Let v be a non-zero highest weight vector of $L(\lambda )\subset Q(\lambda ,\chi ,\theta )$ . Then we can take, as two linearly independent vectors in $L(1)\otimes _{\mathbb {C}}Q(\lambda ,\chi ,\theta )$ of weight $\lambda +1$ that are killed by e, the elements

$$ \begin{align*} u=b_1\otimes v \quad\text{ and }\quad w=b_1\otimes (\overline{h}v)+2b_{-1}\otimes (\overline{e}v). \end{align*} $$

It is easy to check that the matrix of the action of C on the linear span of u and w in the basis $\{u,w\}$ is given by

$$ \begin{align*} \left(\begin{array}{cc}\theta&\chi\\1&\theta\end{array}\right). \end{align*} $$

The eigenvalues of this matrix are $\theta \pm \sqrt {\chi }$ and they are different thanks to our assumption $\chi \neq 0$ . Now the claim of the lemma follows from the universal property of the modules $Q(\lambda +1,\chi ,\theta \pm \sqrt {\chi })$ . Note that here we use the simplicity of the modules $Q(\lambda +1,\chi ,\theta \pm \sqrt {\chi })$ which is guaranteed by [Reference Zhu37, Proposition 3.2] and our assumption that $\theta ^2\neq \chi (\lambda +j)^2$ , for all $j\in \mathbb {Z}$ .

Lemma 9 implies that $\mathcal {N}_3$ is a semi-simple and transitive $\mathscr {C}$ -module category in which $[L(1)]$ is given by Formula (3).

5.4. Fourth realization

We stay in the setup of the previous section, that is, we let $\mathfrak {g}=\mathfrak {sl}_2\otimes _{\mathbb {C}} \mathbb {C}[x]/(x^2)$ be the Takiff Lie algebra of $\mathfrak {sl}_2$ and $\varphi :\mathfrak {g}\to \mathfrak {sl}_2$ be the natural projection. Let $\chi \in \mathbb {C}^*$ and $\lambda \in \mathbb {C}^*$ be such that $\lambda ^2=\chi $ . Then, [Reference Mazorchuk and Mrđen30, Theorem 31] describes a family of simple $\mathfrak {g}$ -module $V(n,\lambda )$ , for $n\in \mathbb {Z}$ , such that

• • $\overline {C}$ acts on $V(n,\lambda )$ as $\chi $ ;

• • when restricted to $\mathfrak {sl}_2$ , the module $V(n,\lambda )$ is a direct sum of $L(|n|)$ , $L(|n|+2),$ and so on;

• • $V(n,\lambda )=V(n',\lambda ')$ if and only if $(n',\lambda ')=(n,\lambda )$ or $(n',\lambda ')=(-n,-\lambda )$ .

By [Reference Mazorchuk and Mrđen30, Proposition 29], we have

$$ \begin{align*} L(1)\otimes_{\mathbb{C}}V(n,\lambda)\cong V(n-1,\lambda)\oplus V(n+1,\lambda). \end{align*} $$

This implies that the additive closure $\mathcal {N}_4$ of all $V(n,\lambda )$ , for a fixed $\lambda $ and for all $n\in \mathbb {Z}$ , is a semi-simple and transitive $\mathscr {C}$ -module category in which $[L(1)]$ is given by Formula (3).

5.5. Statements

We can now summarize the content of this section in the following proposition.

5.6. Comparison of 𝒩₁(μ)

We can also compare the $\mathscr {C}$ -module categories $\mathcal {N}_1(\mu )$ with each other, for varying values of $\mu $ . For the proofs, we need to recall the following notion: for two $\mathfrak {sl}_2$ -modules M and N, we denote by $\mathcal {L}(M,N)$ the $U(\mathfrak {sl}_2)$ - $U(\mathfrak {sl}_2)$ -subbimodule of $\mathrm {Hom}_{\mathbb {C}}(M,N)$ consisting of all elements of the latter, the adjoint action of $\mathfrak {sl}_2$ on which is locally finite (see [Reference Jantzen18, Kapitel 6] for details).

Proof. For $\lambda \in \mu _1+\mathbb {Z}$ , consider $\theta =(\lambda +1)^2$ and let $U_\theta $ denote the (primitive) quotient of $U(\mathfrak {sl}_2)$ by the ideal generated by $\mathtt {c}-\theta $ . Due to our choice of $\lambda $ , we have that $L(\lambda )$ is a Verma module and hence we have an isomorphism $\mathcal {L}(L(\lambda ),L(\lambda ))\cong U_\theta $ of $U(\mathfrak {sl}_2)$ - $U(\mathfrak {sl}_2)$ -bimodules (see [Reference Jantzen18, Subsection 6.9).

Further, by [Reference Jantzen18, Subsection 6.8], we have an isomorphism

$$ \begin{align*} \mathrm{Hom}_{\mathfrak{sl}_2}(V, \mathcal{L}(L(\lambda),L(\lambda))) \cong \mathrm{Hom}_{\mathfrak{sl}_2} (V\otimes_{\mathbb{C}}L(\lambda),L(\lambda)), \end{align*} $$

where $\mathcal {L}(L(\lambda ),L(\lambda ))$ is considered as an $\mathfrak {sl}_2$ -module under the adjoint action. Comparing this with [Reference Etingof, Gelaki, Nikshych and Ostrik11, Section 7.9], we obtain that algebra $U_\theta $ is the algebra ind-object corresponding to the internal hom $\underline {\mathrm {Hom}}(L(\lambda ),L(\lambda ))$ for the $\mathscr {C}$ -module category $\mathcal {N}_1(\mu _1)$ . By [Reference Etingof, Gelaki, Nikshych and Ostrik11, Theorem 7.10.1] and [Reference Etingof, Gelaki, Nikshych and Ostrik11, Remark 7.9.1], the latter module category can be recovered as the category of certain $U_\theta $ -modules in the ind-completion of $\mathscr {C}$ .

This implies that the $\mathscr {C}$ -module categories $\mathcal {N}_1(\mu _1)$ and $\mathcal {N}_1(\mu _2)$ are equivalent, as $\mathscr {C}$ -module categories, if and only if there exists $\lambda _i\in \mu _i+\mathbb {Z}$ , for $i=1,2$ , such that $U_{(\lambda _1+1)^2}\cong U_{(\lambda _2+1)^2}$ . By [Reference Dixmier7, Theorem 6.4], we have $U_{(\lambda _1+1)^2}\cong U_{(\lambda _2+1)^2}$ if and only if $(\lambda _1+1)^2=(\lambda _2+1)^2$ . The claim follows.

5.7. Non-semi-simple realization

In this section, we construct, slightly outside of our setup, a non-semi-simple simple transitive $\mathscr {C}$ -module category of type $A_\infty ^\infty $ .

Consider the Lie algebra $\mathfrak {g}=\mathfrak {sl}_2$ and its Borel subalgebra $\mathfrak {b}$ generated by h and e. Denote by $\mathcal {N}$ the category of all $\mathfrak {b}$ -modules that are

• • finitely generated;

• • h-diagonalizable;

• • free over $U(e)$ .

For $\lambda \in \mathbb {C}$ , we have a $\mathfrak {b}$ -module $N(\lambda )\in \mathcal {N}$ with basis $\{v_\mu \,:\,\mu \in \lambda +2\mathbb {Z}_{\geq 0}\}$ and the $\mathfrak {b}$ -action given by

$$ \begin{align*} hv_\mu=\mu v_\mu,\qquad ev_\mu=v_{\mu+2}. \end{align*} $$

The $\mathfrak {b}$ -module $N(\lambda )$ is indecomposable and is a free $U(e)$ -module of rank one. Every object in $\mathcal {N}$ is a direct sum of finitely many objects of the form $N(\lambda )$ . Note that $\mathcal {N}$ is not semi-simple as $N(\lambda +2)\hookrightarrow N(\lambda )$ , for any $\lambda $ . For $\lambda \in \mathbb {C}$ , denote by $Q(\lambda )$ the quotient $N(\lambda )/N(\lambda +2)$ . Then $Q(\lambda )$ is a simple $\mathfrak {b}$ -module and any simple finite-dimensional $\mathfrak {b}$ -module is of such form.

For a fixed $ \lambda \in \mathbb {C}$ , denote by $\mathcal {N}_5=\mathcal {N}_5(\lambda )$ the full additive subcategory of $\mathcal {N}$ generated by all $N(\mu )$ , for $\mu \in \lambda +\mathbb {Z}$ . Clearly, $\mathcal {N}_5$ inherits from $\mathcal {N}$ the structure of a $\mathscr {C}$ -module category by restriction.

Proof. It is easy to check that

$$ \begin{align*} (\mathrm{Res}^{\mathfrak{sl}_2}_{\mathfrak{b}}(L(1)))\otimes_{\mathbb{C}}N(\mu) \cong N(\mu+1)\oplus N(\mu-1), \end{align*} $$

from which it follows that $\mathcal {N}_5$ is a transitive $\mathscr {C}$ -module category of type $A_\infty ^\infty $ .

Any radical morphism in $\mathcal {N}_5$ is a linear combination of some morphisms of the form $\psi (\mu ,k):=\big (N(\mu +2k)\hookrightarrow N(\mu )\big )$ , for $k\in \mathbb {Z}_{>0}$ . Let $\{v_\nu \,:\, \nu \in \{\mu +2k,\mu +2k+2,\dots \}\}$ be the standard basis of $N(\mu +2k)$ . Let $\{w_\nu \,:\, \nu \in \{\mu ,\mu +2,\dots \}\}$ be the standard basis of $N(\mu )$ . Let $\{e_{-1},e_1\}$ be the standard basis of $L(1)$ .

Applying $F_1$ , we get an embedding of the module $N(\mu +2k+1)\oplus N(\mu +2k-1)$ into $N(\mu +1)\oplus N(\mu -1)$ . The summand $N(\mu +2k-1)$ is generated by $e_{-1}\otimes v_{\mu +2k}$ and $F_1(\psi (\mu ,k))$ maps the latter element to $e_{-1}\otimes w_{\mu +2k}$ .

The summand $N(\mu -1)$ is generated by $e_{-1}\otimes w_{\mu }$ . Applying $e^k$ to the latter outputs the element

$$ \begin{align*} k e_1\otimes w_{\mu+2k-2}+ e_{-1}\otimes w_{\mu+2k}. \end{align*} $$

Therefore, projecting from $N(\mu +1)\oplus N(\mu -1)$ onto $N(\mu +1)$ maps $e_{-1}\otimes w_{\mu +2k}$ to a non-zero element of $N(\mu +1)$ . In other words, the ideal generated by $F_1(\psi (\mu ,k))$ contains $\psi (\mu +1,k-1)$ . Proceeding inductively, we eventually get the identity morphism on $N(\mu +k)$ . This implies that the $\mathscr {C}$ -stable ideal generated by any non-zero morphism in $\mathcal {N}_5$ contains the identity morphism on some non-zero objects. This means that $\mathcal {N}_5$ is simple transitive and completes the proof.

It is easy to check that, for any $i\in \mathbb {Z}_{\geq 0}$ , the $\mathfrak {b}$ -module $F_i(Q(\lambda ))$ is isomorphic to the quotient $N(\lambda -i)/N(\lambda +2+i)$ , we denote the latter module by $Q(\lambda ,i)$ . If we consider the additive closure $\mathcal {N}_6$ of all $Q(\lambda ,i)$ , for a fixed $\lambda $ and for all $i\in \mathbb {Z}_{\geq 0}$ , then it has the natural structure of a $\mathscr {C}$ -module category.

Proof. It is straightforward to check that the $\mathfrak {b}$ -module $F_1(Q(\lambda ,i))$ is isomorphic to $Q(\lambda ,i-1)\oplus Q(\lambda ,i+1)$ , for $i>0$ . This implies that $\mathcal {N}_6$ is transitive of type $A_\infty $ . Simplicity is equivalent to the semi-simplicity of the underlying category of $\mathcal {N}_6$ , that is, the equality

$$ \begin{align*} \mathrm{Hom}_{\mathfrak{b}}(Q(\lambda,i), Q(\lambda,j))=\delta_{i,j}\mathbb{C}. \end{align*} $$

Recall that $Q(\lambda ,k)$ is a uniserial module with top $Q(\lambda -k)$ , socle $Q(\lambda +k),$ and other composition subquotients $Q(\lambda -k+2)$ , $Q(\lambda -4)$ ,…, $Q(\lambda +k-2)$ . In particular, since the top of $Q(\lambda ,k)$ has multiplicity one, any endomorphism of $Q(\lambda ,k)$ is scalar.

If $i>j$ , then $[Q(\lambda ,j):Q(\lambda -i)]=0$ and hence there are no non-zero homomorphisms from $Q(\lambda ,i)$ to $Q(\lambda ,j)$ as the top of $Q(\lambda ,i)$ has nowhere to go. If $i<j$ , then we have $[Q(\lambda ,i):Q(\lambda +j)]=0$ and hence there are no non-zero homomorphisms from $Q(\lambda ,i)$ to $Q(\lambda ,j)$ as there is nothing to cover the socle of $Q(\lambda ,j)$ . This completes the proof.

We can alternatively construct $\mathcal {N}_6$ by first considering $\overline {\mathcal {N}_5}$ , where $Q(\lambda )$ is a simple object and then taking the $\mathscr {C}$ -module subcategory of $\overline {\mathcal {N}_5}$ generated by this object $Q(\lambda )$ .

The example of the categories $\mathcal {N}_5$ and $\mathcal {N}_6$ shows a striking difference between the behavior of $\mathscr {C}$ and that of finitary monoidal categories (more precisely, the fiat bi- and 2-categories in the sense of [Reference Mazorchuk and Miemietz28]). In the latter case, starting from a simple object of the abelianization of a simple transitive $2$ -representation and applying certain elements of $\mathscr {C}$ will output projective objects, see [Reference Mazorchuk and Miemietz29, Lemma 12] and, more generally, [Reference Kildetoft, Mackaay, Mazorchuk and Zimmermann20, Theorem 2]. In our case, none of the $Q(\lambda ,i)$ is projective in $\overline {\mathcal {N}_5}$ . The major contributor to this behavior is the failure of [Reference Kildetoft and Mazorchuk21, Proposition 18] in the non-finitary context.

The $\mathscr {C}$ -module category $\mathcal {N}_6$ does not depend on $\lambda $ , up to equivalence.

6.1. First realization

Consider the setup of Section 4.2. Let $\mathfrak {g}=\mathfrak {sl}_2$ with $\varphi $ being the identity. Consider the BGG category $\mathcal {O}$ for $\mathfrak {g}$ and the projective–injective modules $P(\lambda )\in \mathcal {O}$ , for $\lambda \in \{-1,-2,\dots \}$ . Let $\mathcal {K}_1$ be the additive closure of these modules. Then $\mathcal {K}_1$ is exactly the category of integral projective–injective modules in $\mathcal {O}$ . Consequently, it is closed with respect to tensoring with finite-dimensional $\mathfrak {sl}_2$ -modules and hence is a $\mathscr {C}$ -module category.

From Formula (2), we have

(4) $$ \begin{align} [L(1)]= \left( \begin{array}{ccccc} 0&2&0&0&\dots\\ 1&0&1&0&\dots\\ 0&1&0&1&\dots\\ 0&0&1&0&\dots\\ \vdots&\vdots&\vdots&\vdots&\ddots \end{array} \right). \end{align} $$

6.2. Second realization

Let $\mathfrak {g}=\mathfrak {sl}_2$ with $\varphi $ being the identity. For a fixed non-zero $\xi \in \mathbb {C}$ , let $M(\xi )$ be the $\mathfrak {g}$ -module defined as the quotient of $U(\mathfrak {g})$ by the left ideal generated by $\mathtt {c}$ and $e-\xi $ . Then, the module $M(\xi )$ is a simple $\mathfrak {g}$ -module that appeared first in [Reference Arnal and Pinczon2]. The module $M(\xi )$ is a Whittaker module in the sense of [Reference Kostant23].

Set $X_{-1}:=M(\xi )$ and, for $i\in \{1,2,\dots \}$ , define $X_{-1-i}$ as the kernel of the central element $(\mathtt {c}+i^2)^2$ acting on $L(i)\otimes _{\mathbb {C}}M(\xi )$ . Then, by [Reference Miličić and Soergel34, Theorem 5.1] (see also [Reference Mazorchuk and Stroppel32, Theorem 67] and [Reference Mackaay, Mazorchuk and Miemietz24, Corollary 20]) all these $X_{-1-i}$ are indecomposable (in fact, have length $2$ with isomorphic subquotients) and the additive closure $\mathcal {K}_2=\mathcal {K}_2(\xi )$ of all $X_j$ , with $j\in \{-1,-2,\dots \}$ , is closed under tensoring with finite-dimensional $\mathfrak {sl}_2$ -modules.

6.3. Statements

We can now summarize the content of this section in the following proposition.

Using the ideas of [Reference Miličić and Soergel34, Theorem 5.1] and, more generally, the results in [Reference Mazorchuk and Stroppel32, Theorem 67] and [Reference Mackaay, Mazorchuk and Miemietz24, Corollary 20], we can generalize the example of the $\mathscr {C}$ -module category $\mathcal {K}_2$ by taking as M any simple $\mathfrak {g}$ -module which satisfies the condition $\mathtt {c}M=0$ .

7.1. The realization

Consider the semi-direct product $\mathfrak {g}=\mathfrak {sl}_2\ltimes L(4)$ , where $L(4)$ is an abelian ideal and let $\varphi :\mathfrak {g}\twoheadrightarrow \mathfrak {sl}_2$ be the natural projection (see [Reference Mazorchuk and Mrđen31]). Let $\{v_i\,:\,i\in \{0,\pm 2,\pm 4\}\}$ be the standard basis of $L(4)$ , that is, we have $h\cdot v_i=iv_i$ , then $e\cdot v_4=0$ and $e\cdot v_{4-2i}=(5-i)v_{6-2i}$ , for $i=1,2,3,4$ . The elements

$$ \begin{align*} C_2 & :=v_0^2-3v_{-2}v_2+12 v_{-4}v_4 \qquad\text{ and }\\ C_3 & :=v_0^3-\frac{9}{2}v_{-2}v_0v_2 +\frac{27}{2}v_{-2}^2v_4 +\frac{27}{2}v_{-4}v_2^2 -36v_{-4}v_0v_4 \end{align*} $$

are central in $U(\mathfrak {g})$ . In [Reference Mazorchuk and Mrđen30, Theorem 68], see also [Reference Mazorchuk and Mrđen31, Theorem 12], for every $\mu \in \mathbb {C}^*$ , one can find a construction of simple $\mathfrak {g}$ -modules $V'(0,\mu )$ , $V'(2,\mu ),$ and $V(n,\mu )$ , for $n\in \mathbb {Z}_{\geq 1}$ , such that

• • $C_2$ and $C_3$ act on each such module as $\mu ^2$ and $\mu ^3$ , respectively;

• • the restriction of each of these modules to $\mathfrak {sl}_2$ is a direct sum of simple finite-dimensional modules with finite multiplicities.

By [Reference Mazorchuk and Mrđen31, Lemma 14], see also [Reference Mazorchuk and Mrđen30, Remark 75], we have

$$ \begin{align*} L(1)\otimes_{\mathbb{C}} V(n,\mu) & \cong V(n-1,\mu)\oplus V(n+1,\mu),\text{ if } n>1;\\ L(1)\otimes_{\mathbb{C}} V(1,\mu) & \cong V'(0,\mu)\oplus V'(2,\mu)\oplus V(2,\mu);\\ L(1)\otimes_{\mathbb{C}}V'(0,\mu) & \cong V(1,\mu);\\ L(1)\otimes_{\mathbb{C}}V'(2,\mu) & \cong V(1,\mu). \end{align*} $$

This means that, with respect to the basis given by the order

$$ \begin{align*} V'(0,\mu),V'(2,\mu),V(1,\mu),V(2,\mu),V(3,\mu),\dots, \end{align*} $$

we have

(5) $$ \begin{align} [L(1)]= \left( \begin{array}{cccccc} 0&0&1&0&0&\dots\\ 0&0&1&0&0&\dots\\ 1&1&0&1&0&\dots\\ 0&0&1&0&1&\dots\\ 0&0&0&1&0&\dots\\ \vdots&\vdots&\vdots&\vdots&\ddots \end{array} \right). \end{align} $$

We denote by $\mathcal {Y}'$ the additive closure of all these $V'(0,\mu )$ , $V'(2,\mu )$ , $V(1,\mu )$ , $V(2,\mu )$ , $V(3,\mu ),\dots $ .

7.2. Statement

We have the following proposition.

7.3. Additional properties

Proof. We need to prove that the radical of $\mathcal {Y}$ is $\mathscr {C}$ -invariant. Consider the abelianization $\overline {\mathcal {Y}}$ . Let $Y_0,Y_1,\dots $ be the list of indecomposables in $\mathcal {Y}$ such that $[F_1]$ is given by (5) in the corresponding basis of the Grothendieck group. These are the indecomposable projectives in $\overline {\mathcal {Y}}$ . Let $S_0,S_1,\dots $ be the corresponding simples in $\overline {\mathcal {Y}}$ . Note that (5) is a symmetric matrix and therefore the matrix of the action of each $F_i$ in the basis of projectives coincides with the matrix of the action of this $F_i$ in the basis of simples.

We thus have $F_1(S_0)\cong F_1(S_1)\cong S_2$ . The module $F_1(S_2)$ has length three with simple subquotients $S_0$ , $S_1$ , and $S_3$ , all appearing with multiplicity one. For $i\in \{0,1\}$ , by adjunction, we have

$$ \begin{align*} \overline{\mathcal{Y}}(F_1(S_2),S_i)\cong \overline{\mathcal{Y}}(S_2,F_1(S_i))\cong\mathbb{C} \end{align*} $$

and

$$ \begin{align*} \overline{\mathcal{Y}}(S_i,F_1(S_2))\cong \overline{\mathcal{Y}}(F_1(S_i),S_2)\cong\mathbb{C}. \end{align*} $$

Thus, $S_i$ appears both in the socle and in the top of $F_1(S_2)$ , which implies that $F_1(S_2)\cong S_0\oplus S_1\oplus S_3$ .

Next, we see that $F_1(S_3)$ has length two with simple subquotients $S_2$ and $S_4$ , both appearing with multiplicity one. By adjunction, we have

$$ \begin{align*} \overline{\mathcal{Y}}(F_1(S_3),S_2)\cong \overline{\mathcal{Y}}(S_3,F_1(S_2))\cong \mathbb{C} \end{align*} $$

and

$$ \begin{align*} \overline{\mathcal{Y}}(S_2,F_1(S_3))\cong \overline{\mathcal{Y}}(F_1(S_2),S_3)\cong \mathbb{C}. \end{align*} $$

Therefore, $S_2$ appears both in the socle and in the top of $F_1(S_3)$ implying that we have $F_1(S_3)\cong S_2\oplus S_4$ .

Proceeding inductively, we obtain that $F_1(S_i)\cong S_{i-1}\oplus S_{i+1}$ , for all $i\geq 3$ . Let $\tilde {\mathcal {Y}}$ be the additive closure of all $S_i$ , where $i\geq 0$ . We have just shown that $\tilde {\mathcal {Y}}$ is invariant under the action of $F_1$ . Since $F_1$ generates $\mathscr {C}$ , it follows that $\tilde {\mathcal {Y}}$ is $\mathscr {C}$ -invariant. In other words, $\tilde {\mathcal {Y}}$ is a simple transitive $\mathscr {C}$ -module category of type $D_\infty $ .

The rest is similar to the proof of Proposition 7. We need to show that the radical of $\mathcal {Y}$ is $F_1$ -invariant. Let $\varphi :Y_i\to Y_i$ be a radical morphism, for some i. Then $\varphi $ is nilpotent since $\mathrm {End}_{\mathcal {Y}}(Y_i)$ is finite dimensional. Hence, $F_1(\varphi )$ is also nilpotent. Since $F_1(Y_i)$ does not have isomorphic summands, it follows that $F_1(\varphi )$ is a radical morphism.

Let $\varphi :Y_i\to Y_j$ be a non-zero morphism, for $i\neq j$ (hence $\varphi $ is automatically radical). Then $\varphi (Y_i)$ belongs to the radical of $Y_j$ . Applying the exact functor $F_1$ to the short exact sequence

$$ \begin{align*} 0\to \mathrm{Rad}(Y_j) \to Y_j\to S_j\to 0, \end{align*} $$

we get a short exact sequence

$$ \begin{align*} 0\to F_1(\mathrm{Rad}(Y_j)) \to F_1(Y_j)\to F_1(S_j)\to 0, \end{align*} $$

in which $F_1(S_j)$ is isomorphic to the top of $F_1(Y_j)$ by the above. Therefore, we have

$$ \begin{align*} F_1(\mathrm{Rad}(Y_{j}))= \mathrm{Rad}(F_1(Y_j)). \end{align*} $$

As $\varphi (Y_i)\subset \mathrm {Rad}(Y_{j})$ and $F_1$ is exact, we obtain that $F_1(\varphi (Y_i))$ is a submodule of $\mathrm {Rad}(F_1(Y_j))$ , which means that $F_1(\varphi )$ is a radical morphism. As the radical of $\mathcal {Y}$ is generated by radical morphisms between indecomposable objects, it follows that the radical of $\mathcal {Y}$ is $\mathscr {C}$ -stable and completes the proof.

8.1. First realization

Let $\mathfrak {g}=\mathfrak {sl}_2$ with $\varphi $ being the identity. For a fixed non-zero $\xi \in \mathbb {C}^*$ and $\lambda \in \mathbb {Z}+\frac {1}{2}$ , let $M(\lambda ,\xi )$ be the $\mathfrak {g}$ -module defined as the quotient of $U(\mathfrak {g})$ by the left ideal generated by $\mathtt {c}-(\lambda +1)^2$ and $e-\xi $ . Then, the module $M(\lambda ,\xi )$ is a simple Whittaker $\mathfrak {g}$ -module (see [Reference Arnal and Pinczon2], [Reference Kostant23]). Note that $M(\lambda ,\xi )\cong M(-\lambda -2,\xi )$ due to our choice of $\lambda $ .

As $\lambda $ is not an integer, the functor $L(1)\otimes _{\mathbb {C}}{}_-$ goes from $\mathcal {Z}_{(\lambda +1)^2}$ to $\mathcal {Z}_{\lambda ^2}\oplus \mathcal {Z}_{(\lambda +2)^2}$ and is isomorphic to the direct sum of $\theta _{\lambda ,\lambda -1}$ and $\theta _{\lambda ,\lambda +1}$ . It follows that $L(1)\otimes _{\mathbb {C}}M(\lambda ,\xi )$ is a direct sum of $M(\lambda -1,\xi )$ and $M(\lambda +1,\xi )$ . Denote by $\mathcal {X}_1$ the additive closure of all $M(\lambda +i,\xi )$ , for $i\in \mathbb {Z}$ . This coincides with the additive closure of all $M(-\frac {3}{2}-i,\xi )$ , for $i\in \mathbb {Z}_{\geq 0}$ , the latter now being pair-wise non-isomorphic.

If $\lambda =-\frac {3}{2}$ , then $M(\lambda +1,\xi )\cong M(\lambda ,\xi )$ . If $\lambda =-\frac {1}{2}$ , then $M(\lambda -1,\xi )\cong M(\lambda ,\xi )$ . Therefore, we have

(6) $$ \begin{align} [L(1)]= \left( \begin{array}{ccccc} 1&1&0&0&\dots\\ 1&0&1&0&\dots\\ 0&1&0&1&\dots\\ 0&0&1&0&\dots\\ \vdots&\vdots&\vdots&\vdots&\ddots \end{array} \right). \end{align} $$

8.2. Second realization

Let $\mathfrak {g}$ be the Schrödinger Lie algebra as in Section 5.2 (see [Reference Dubsky, Lü, Mazorchuk and Zhao9]), and $\varphi :\mathfrak {g}\to \mathfrak {sl}_2$ be the natural projection. Then, for each $\theta \in \mathbb {C}^*$ , [Reference Mazorchuk and Mrđen30, Theorem 54] describes a family of simple $\mathfrak {g}$ -modules $V(n,\theta )$ , where $n\in \mathbb {Z}$ , such that

• • z acts on $V(n,\theta )$ as $\chi $ ;

• • when restricted to $\mathfrak {sl}_2$ , the module $V(n,\lambda )$ is a direct sum of $L(|n|)$ , $L(|n|+1),$ and so on;

• • $V(n,\theta )=V(n',\theta ')$ if and only if $(n',\theta ')=(n,\theta )$ or $(n',\theta ')=(-n,\theta )$ .

By [Reference Mazorchuk and Mrđen30, Proposition 53], we have

$$ \begin{align*} L(1)\otimes_{\mathbb{C}}V(n,\theta)\cong \begin{cases} V(n-1,\theta)\oplus V(n+1,\theta),& n\neq 0;\\ V(0,\theta)\oplus V(1,\theta),& n= 0.\\ \end{cases} \end{align*} $$

For a fixed $\theta \in \mathbb {C}^*$ , denote by $\mathcal {X}_2$ the additive closure of all $V(n,\theta )$ , for $n\in \mathbb {Z}$ . By the above, $\mathcal {X}_2$ is semi-simple and has a natural structure of a $\mathscr {C}$ -module category in which $[L(1)]$ is given by (6).

8.3. Statement

We can now summarize the content of this section in the following proposition.

8.4. Additional properties

Proof. We need to prove that the radical of $\mathcal {X}$ is $\mathscr {C}$ -invariant. Consider the abelianization $\overline {\mathcal {X}}$ . Let $X_0,X_1,\dots $ be the list of indecomposables in $\mathcal {X}$ such that $[F_1]$ is given by (6) in the corresponding basis of the Grothendieck group. These are the indecomposable projectives in $\overline {\mathcal {X}}$ . Let $S_0,S_1,\dots $ be the corresponding simples in $\overline {\mathcal {X}}$ . Note that (6) is a symmetric matrix and therefore the matrix of the action of each $F_i$ in the basis of projectives coincides with the matrix of the action of this $F_i$ in the basis of simples.

In particular, $F_1(S_0)$ has length two with simple subquotients $S_0$ and $S_1$ , both appearing with multiplicity one. Then $S_0$ must appear either in the top or in the socle of $F_1(S_0)$ . However, by adjunction, we have

$$ \begin{align*} \overline{\mathcal{X}}(F_1(S_0),S_0)\cong \overline{\mathcal{X}}(S_0,F_1(S_0)) \end{align*} $$

and thus $S_0$ appears both in the socle and in the top, which implies that $F_1(S_0)\cong S_0\oplus S_1$ .

Next, we see that $F_1(S_1)$ has length two with simple subquotients $S_0$ and $S_2$ , both appearing with multiplicity one. By adjunction, we have

$$ \begin{align*} \overline{\mathcal{X}}(F_1(S_1),S_0)\cong \overline{\mathcal{X}}(S_1,F_1(S_0))\cong \mathbb{C} \end{align*} $$

and

$$ \begin{align*} \overline{\mathcal{X}}(S_0,F_1(S_1))\cong \overline{\mathcal{X}}(F_1(S_0),S_1)\cong \mathbb{C}. \end{align*} $$

Therefore, $S_0$ appears both in the socle and in the top of $F_1(S_1)$ implying that we have $F_1(S_1)\cong S_0\oplus S_2$ .

Proceeding inductively, we obtain that $F_1(S_i)\cong S_{i-1}\oplus S_{i+1}$ , for all $i\geq 1$ . Let $\tilde {\mathcal {X}}$ be the additive closure of all $S_i$ , where $i\geq 0$ . We have just shown that $\tilde {\mathcal {X}}$ is invariant under the action of $F_1$ . Since $F_1$ generates $\mathscr {C}$ , it follows that $\tilde {\mathcal {X}}$ is $\mathscr {C}$ -invariant. In other words, $\tilde {\mathcal {X}}$ is a simple transitive $\mathscr {C}$ -module category of type $T_\infty $ .

The rest is similar to the proof of Proposition 7. We need to show that the radical of $\mathcal {X}$ is $F_1$ -invariant. Let $\varphi :X_i\to X_i$ be a radical morphism, for some i. Then $\varphi $ is nilpotent since $\mathrm {End}_{\mathcal {X}}(X_i)$ is finite dimensional. Hence, $F_1(\varphi )$ is also nilpotent. Since $F_1(X_i)$ does not have isomorphic summands, it follows that $F_1(\varphi )$ is a radical morphism.

Let $\varphi :X_i\to X_j$ be a non-zero morphism, for $i\neq j$ . Then $\varphi (X_i)$ belongs to the radical of $X_j$ . Applying the exact functor $F_1$ to the short exact sequence

$$ \begin{align*} 0\to \mathrm{Rad}(X_j) \to X_j\to S_j\to 0, \end{align*} $$

we get a short exact sequence

$$ \begin{align*} 0\to F_1(\mathrm{Rad}(X_j)) \to F_1(X_j)\to F_1(S_j)\to 0, \end{align*} $$

in which $F_1(S_j)$ is isomorphic to the top of $F_1(X_j)$ by the above. Therefore, we have

$$ \begin{align*} F_1(\mathrm{Rad}(X_{j}))= \mathrm{Rad}(F_1(X_j)). \end{align*} $$

As $\varphi (X_i)\subset \mathrm {Rad}(X_{j})$ and $F_1$ is exact, we obtain that $F_1(\varphi (X_i))$ is a submodule of $\mathrm {Rad}(F_1(X_j))$ , which means that $F_1(\varphi )$ is a radical morphism. As the radical of $\mathcal {X}$ is generated by radical morphisms between indecomposable objects, it follows that the radical of $\mathcal {X}$ is $\mathscr {C}$ -stable and completes the proof.

9.1. A realizations dual to $C_\infty $

The easiest way to find a $B_\infty $ -combinatorics is, of course, by taking the dual of a $C_\infty $ -combinatorics. Let us do this with the example from Section 6.1.

Let $\mathfrak {g}=\mathfrak {sl}_2$ with $\varphi $ being the identity. Consider the BGG category $\mathcal {O}$ for $\mathfrak {g}$ and the projective–injective modules $P(\lambda )\in \mathcal {O}$ , for $\lambda \in \{-1,-2,\dots \}$ . Let $\mathcal {K}_1$ be the additive closure of these modules. Then $\mathcal {K}_1$ is a simple transitive $\mathscr {C}$ -module category of type $C_\infty $ (see §6.1 and Proposition 16).

Consider the projective abelianization $\overline {\mathcal {K}_1}$ of $\mathcal {K}_1$ (see §2.7).

Each indecomposable projective object $P(\lambda )$ of $\overline {\mathcal {K}_1}$ has unique simple top, which we denote by $L(\lambda )$ . It can be identified with the simple highest weight $\mathfrak {sl}_2$ -module with highest weight $\lambda $ . Note that here $\lambda $ is a negative integer. Recall that

$$ \begin{align*} [P(\lambda):L(\lambda')]= \begin{cases} 2,& \lambda=\lambda'\in\{-2,-3,\dots\};\\ 1,& \lambda=\lambda'=-1;\\ 0,& \text{otherwise}. \end{cases} \end{align*} $$

Now, note that each element of $\mathscr {C}$ acts as an exact endofunctor of $\overline {\mathcal {K}_1}$ . From (2), it follows that, in the basis $\{[L(-1)],[L(-2)],\dots \}$ of the Grothendieck group of $\overline {\mathcal {K}_1}$ , the action of $L(1)$ is given by the following matrix:

(7) $$ \begin{align} \left( \begin{array}{ccccc} 0&1&0&0&\dots\\ 2&0&1&0&\dots\\ 0&1&0&1&\dots\\ 0&0&1&0&\dots\\ \vdots&\vdots&\vdots&\vdots&\ddots \end{array} \right), \end{align} $$

which is a type $B_\infty $ matrix (i.e., the transpose of the type $C_\infty $ matrix).

9.2. Impossibility of realizations of type $B_\infty $ in our setup

Proof. Let $\mathcal {X}$ be a simple transitive $\mathscr {C}$ -module category of type $B_\infty $ . Let $X_1,X_2,\dots $ be fixed representatives of isomorphism classes of indecomposable objects in $\mathcal {X}$ such that, with respect to them, the matrix $[L(1)]$ is given by (7). Consider the abelianization $\overline {\mathcal {X}}$ as in the previous section and, for $i=1,2,\dots $ , denote by $N_i$ the simple top of $P_i:=(0\to X_i)$ in $\overline {\mathcal {X}}$ .

All elements of $\mathscr {C}$ operate on $\overline {\mathcal {X}}$ as exact self-adjoint functors. Therefore, we can consider the matrix of each such element written in the basis of the Grothendieck group of $\overline {\mathcal {X}}$ given by simple objects. By [Reference Agerholm and Mazorchuk1, Lemma 8], this matrix is just the transpose of the matrix in the basis of the indecomposable projective objects. Therefore, the corresponding matrix for $L(1)$ is given by (4), that is, it has type $C_\infty $ .

Directly from this matrix of $L(1)$ , we see that $F_1(N_1)\cong N_2$ . Also, $F_1(N_2)$ is a module of length three with composition subquotients $N_1$ , appearing with multiplicity $2$ , and $N_3$ , appearing with multiplicity one. As $N_2$ is a simple object and $\mathbb {C}$ is algebraically closed, we have $\overline {\mathcal {X}}(N_2,N_2)=\mathbb {C}$ . By adjunction, we thus have

(8) $$ \begin{align} \mathbb{C}= \overline{\mathcal{X}}(N_2,N_2)= \overline{\mathcal{X}}(F_1(N_1),F_1(N_1))= \overline{\mathcal{X}}(N_1,F_1\circ F_1(N_1)). \end{align} $$

Similarly, we also have

(9) $$ \begin{align} \mathbb{C}= \overline{\mathcal{X}}(F_1\circ F_1(N_1),N_1). \end{align} $$

We have $F_1\circ F_1\cong F_0\oplus F_2$ . Therefore,

$$ \begin{align*} F_1\circ F_1(N_1)\cong F_0(N_1)\oplus F_2(N_1)\cong N_1\oplus F_2(N_1). \end{align*} $$

The module $F_2(N_1)$ has length two with composition subquotients $N_1$ and $N_3$ . Therefore, $N_1$ must appear either in the top or in the socle of $F_2(N_1)$ . In the first case, we have two copies of $N_1$ in the top of $F_1\circ F_1(N_1)$ leading to a contradiction with (9), In the second case, we get a similar contradiction with (8). This proves the claim.

Proposition 23 implies that, in order to have a chance to construct a locally finitary simple transitive $\mathscr {C}$ -module category of type $B_\infty $ , we need to change the base field to some field that is not algebraically closed, which would allow the endomorphism algebra of a simple object to have dimension greater than one. Note that one can satisfy the equalities (8) and (9), for example, if $\overline {\mathcal {X}}(N_2,N_2)$ has dimension two while $F_2(N_1)\cong N_1\oplus N_3$ and $\overline {\mathcal {X}}(N_1,N_1)$ has dimension one.

10.1. Module categories inside $\mathfrak {sl}_2$ -mod generated by simple modules

Let $\mathfrak {g}=\mathfrak {sl}_2$ with $\varphi $ being the identity. Consider the $\mathscr {C}$ -module category $\mathrm {add}(\mathscr {C}\cdot M)$ , where M is a simple $\mathfrak {sl}_2$ -module. By Schur’s lemma, see [Reference Dixmier8, Proposition 2.6.8], there is $\vartheta \in \mathbb {C}$ such that $(\mathtt {c}-\vartheta )M=0$ . We can explicitly classify the types of simple transitive subquotients of $\mathrm {add}(\mathscr {C}\cdot M)$ .

Note that the situations described in both Claims (b) and (c) are possible (see §§5.1 and 8.1).

Proof. Let M be a simple $\mathfrak {sl}_2$ -module. If M is finite dimensional, then $\mathrm {add}(\mathscr {C}\cdot M)=\mathscr {C}$ and hence Claim (d) follows from Proposition 6.

Assume now that M is infinite-dimensional and let $L(\lambda )$ be a simple highest weight module with the same annihilator in $U(\mathfrak {sl}_2)$ as M (see [Reference Duflo10]). This means, in particular, that $\vartheta =(\lambda +1)^2$ . Assume first that $\lambda \not \in \frac {1}{2}\mathbb {Z}$ . Then

(10) $$ \begin{align} L(1)\otimes_{\mathbb{C}}M\cong\theta_{\lambda,\lambda+1}M \oplus \theta_{\lambda,\lambda-1}M \end{align} $$

with both $\theta _{\lambda ,\lambda +1}M$ and $\theta _{\lambda ,\lambda -1}M$ simple (as both $\theta _{\lambda ,\lambda \pm 1}$ are equivalences of appropriate categories) and non-isomorphic (as the two modules have different annihilators). Moreover, the annihilators of both $\theta _{\lambda ,\lambda \pm 1}M$ are different from the annihilator of M. Note that both $\lambda \pm 1\not \in \frac {1}{2}\mathbb {Z}$ . Therefore, we can continue this recursively, ending up with Claim (a).

Next, let us assume that $\lambda \in \frac {1}{2}+\mathbb {Z}$ . Then we still have the decomposition (10) with both summands being simple and non-isomorphic modules. However, if $\lambda \in \{-\frac {1}{2},-\frac {3}{2}\}$ , then one of the two modules $\theta _{\lambda ,\lambda \pm 1}M$ will have the same annihilator as M and thus we might have the situation that either $\theta _{\lambda ,\lambda + 1}M$ or $\theta _{\lambda ,\lambda -1}M$ is isomorphic to M. This splits our situation in two cases (with both occurring, see §§5.1 and 8.1).

In the first case, when neither $\theta _{\lambda ,\lambda + 1}M$ nor $\theta _{\lambda ,\lambda -1}M$ is isomorphic to M, we get that $\mathrm {add}(\mathscr {C}\cdot M)$ is of type $A_\infty ^\infty $ similarly to Section 5.1. In the other case, that is, when either $\theta _{\lambda ,\lambda + 1}M$ or $\theta _{\lambda ,\lambda -1}M$ is isomorphic to M, we get that $\mathrm {add}(\mathscr {C}\cdot M)$ is of type $T_\infty $ similarly to Section 8.1. This proves Claims (b) and (c).

It remains to consider the case $\lambda \in \mathbb {Z}_{<0}$ . In this case, due to our conventions on the notation for projective functors, the decomposition (10) becomes

$$ \begin{align*} L(1)\otimes_{\mathbb{C}}M\cong\theta_{-\lambda-2,-\lambda-1}M \oplus \theta_{-\lambda-2,-\lambda-3}M \end{align*} $$

with both summands being simple and non-isomorphic modules provided that $\lambda \neq -1$ . If $\lambda =-1$ , then we have

$$ \begin{align*} L(1)\otimes_{\mathbb{C}}M\cong\theta_{-1,-2}M. \end{align*} $$

In this way, recursively, starting from M, we can construct simple modules $N(k)$ , for each $k\in \mathbb {Z}_{<0}$ , with M being one of these modules, such that $N(k)$ has the same annihilator as $L(k)$ and

$$ \begin{align*} L(1)\otimes_{\mathbb{C}}N(k)\cong N(k+1)\oplus N(k-1), \end{align*} $$

for all $k\neq -1$ . Using [Reference Mazorchuk and Stroppel32, Theorem 67], we further have that the module $N'(-2):=L(1)\otimes _{\mathbb {C}}N(-1)$ is indecomposable, has the dual numbers as the endomorphism algebra, and has simple top and socle isomorphic to $N(-2)$ and no other infinite-dimensional simple subquotients. Setting $N'(-1):=N(-1)\oplus N(-1)$ , we can now recursively define the indecomposable modules $N'(k)$ , for $k\leq -3$ , via

$$ \begin{align*} L(1)\otimes_{\mathbb{C}}N'(k+1)\cong N'(k)\oplus N'(k+2). \end{align*} $$

Let $\mathcal {M}$ be the additive closure of all $N'(k)$ , where $k\leq -1$ . Let $\mathcal {N}$ be the additive closure of all $N(k)$ , for $k\leq -2$ . Then $\mathcal {M}$ is closed under the action of $\mathscr {C}$ and the quotient of $\mathrm {add}(\mathscr {C}\cdot M)$ by the ideal generated by $\mathcal {M}$ is equivalent to $\mathcal {N}$ . Therefore, we get Claim (e) by the same arguments as in Section 6.2. This completes the proof.

10.2. Module categories coming from restrictions to subalgebras

Let $\mathfrak {a}$ be a Lie subalgebra of $\mathfrak {sl}_2$ . Then $\mathscr {C}$ acts, via restriction, on the category of all finite-dimensional $\mathfrak {a}$ -modules. In this section, we will describe simple transitive $\mathscr {C}$ -module categories of this kind generated by simple finite-dimensional $\mathfrak {a}$ -modules. In the extreme case $\mathfrak {a}=\mathfrak {sl}_2$ , we just get the left regular $\mathscr {C}$ -module category ${}_{\mathscr {C}}\mathscr {C}$ . The other extreme case $\mathfrak {a}=0$ is trivial.

It remains to consider the two cases when $\mathfrak {a}$ has dimension $1$ or $2$ . Let us start with the case $\dim (\mathfrak {a})=1$ . Then $\mathfrak {a}$ is the linear span of some non-zero element $g\in \mathfrak {sl}_2$ , in particular, $U(\mathfrak {a})\cong \mathbb {C}[g]$ , the polynomial algebra in g. For $\lambda \in \mathbb {C}$ , let $\mathbb {C}_\lambda $ denote the one-dimensional $\mathbb {C}[g]$ -module on which g acts via $\lambda $ .

We have the following two cases:

• • the element g is nilpotent;

• • the element g is semi-simple with eigenvalues $\mu \neq 0$ and $-\mu $ .

In the first case, we have a non-split short exact sequence

(11) $$ \begin{align} 0\to \mathbb{C}_{\lambda}\to (\mathrm{Res}^{\mathfrak{sl}_2}_{\mathfrak{a}}L(1)) \otimes_{\mathbb{C}}\mathbb{C}_\lambda\to \mathbb{C}_{\lambda}\to 0. \end{align} $$

For $n\in \mathbb {Z}_{>0}$ , denote by $M(n,\lambda )$ the n-dimensional $\mathbb {C}[g]$ -module on which g acts via an $n\times n$ Jordan cell with eigenvalue $\lambda $ . Let $\mathcal {Y}(\lambda )$ be the additive closure of $M(n,\lambda )$ , for all n, and let $\widetilde {\mathcal {Y}}(\lambda )$ be the semi-simplification of $\mathcal {Y}(\lambda )$ , that is, the quotient of $\mathcal {Y}(\lambda )$ by its radical.

In the second case, we have

(12) $$ \begin{align} (\mathrm{Res}^{\mathfrak{sl}_2}_{\mathfrak{a}}L(1)) \otimes_{\mathbb{C}}\mathbb{C}_\lambda\cong \mathbb{C}_{\lambda+\mu}\oplus \mathbb{C}_{\lambda-\mu}. \end{align} $$

Let $\mathcal {X}=\mathcal {X}(\lambda )$ denote the additive closure of all $\mathbb {C}_{\lambda +n\mu }$ , where $n\in \mathbb {Z}$ .