Diagrammatic Construction of Representations of Small Quantum $\mathfrak{sl}_2$

We provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of $\mathfrak{sl}_2$ at a root of unity $q$ of odd order. Our approach is diagrammatic, and it relies on an extension of the Temperley-Lieb category specialized at $\delta = -q-q^{-1}$.


Introduction
The main goal of this paper is, very roughly speaking, to find a diagrammatic description of the categoryŪ -mod of finite-dimensional leftŪ -modules, wherē U =Ū q sl 2 is a finite-dimensional Hopf algebra called the small quantum group of sl 2 . First introduced by Lusztig in [Lu90],Ū arises from a quantum deformation of the universal enveloping algebra of sl 2 with deformation parameter q = e 2πi r , where 3 r ∈ Z is an odd integer called the level. The motivation behind this work is topological: indeed,Ū is a factorizable ribbon Hopf algebra, which means it can be used as an algebraic tool for building non-semisimple TQFTs [DGP17]. At present, constructions rely either on the structure of Hopf algebras, or on some general categorical machinery [DGGPR19]. Although this theory produces intreresting new phenomena, which differ strikingly from the semisimple case [DGGPR20], it exploits a rather elaborate technical setup, which can require some time to fully digest. However, in the case of the small quantum groupŪ , we can look for a more combinatorial approach by figuring out an explicit description of a large category ofŪ -modules in diagrammatic terms. This idea is pursued in [DM20], where a skein theoretic reformulation of the topological invariants of closed 3-manifold constructed fromŪ is obtained, based on results provided here.
The starting point for our work is a diagrammatic description of the category of representations of a different Hopf algebra U = U q sl 2 called the full quantum group, or the divided power quantum group of sl 2 . This version of quantum sl 2 is infinite-dimensional, and its definition is also due to Lusztig. In fact, the small quantum groupŪ was originally introduced as a finite-dimensional Hopf subalgebra of U . It was first shown in [MM92] that End U (X ⊗m ) is isomorphic to the mth Temperley-Lieb algebra TL(m) of [TL71] with parameter δ = −A 2 −A −2 specialized at A = q r+1 2 , where X is the fundamental simple U -module of dimension 2, see also [FK97,AST15]. Since the small quantum groupŪ is a Hopf subalgebra of U , we have a natural restriction functor from U -mod toŪ -mod. However, there exist indecomposable U -modules which become decomposable when considered as U -modules, and EndŪ (X ⊗m ) is not isomorphic to TL(m) in general.
The question of understanding EndŪ (X ⊗m ) has been studied in greater detail in the case of even roots of unity, when the deformation parameter is q = e 2πi r for some even integer 4 r ∈ Z. Indeed, in this case U contains a finite-dimensional Hopf subalgebraŪ which is called the restricted quantum group of sl 2 . In [GST12], it is shown that an extension of the mth Temperley-Lieb algebra TL(m), called the lattice W-algebra, is isomorphic to EndŪ (X ⊗m ) for every m 0 when r = 4, and it is conjectured that the isomorphism holds in general for every even r 4.
In [Mo17], generators and relations for a description of EndŪ (X ⊗m ) in terms of planar diagrams are provided for every m 0 and every even r 4, and it is conjectured these give a complete presentation. However, for even values of r, the Hopf algebraŪ is not ribbon, and although it admits a ribbon extension, the latter is not factorizable. In particular, the machinery of [DGP17] only produces quantum invariants of closed 3-manifolds, not TQFTs. This is why we turn our attention to the less discussed odd level case. In order to do this, we consider an extensioñ TL of the Temperley-Lieb category TL, which is obtained by introducing four additional generating morphisms, as well as several relations between them. These additional generators correspond to the non-trivial splitting, inŪ -mod, of a certain indecomposable U -module. The definition ofTL, which is given in Section 5, allows us to establish our main result. This allows us to describe in diagrammatic terms every morphism in the full monoidal subcategory ofŪ -mod generated by X.
Structure of the paper. In Section 2, we recall the definition of the small quantum groupŪ and the main properties of its category of finite-dimensional represen-tationsŪ -mod. In particular, we collect definitions of simple and indecomposable projectiveŪ -modules, and we give explicit formulas forŪ -module morphisms between them. In Section 3, we recall the definition of the Temperley-Lieb category TL A of indeterminate A, and within it we highlight a special family of idempotent endomorphisms g m ∈ TL A (m) = TL A (m, m) for r m 2r − 2. Under the specialization A = q r+1 2 , these planar diagrams, which were first considered in [GW93], extend the standard family of (simple) Jones-Wenzl idempotents f m ∈ TL A (m) of [Jo83,We87], which are well-defined for 0 m r − 1. Since the endomorphisms g m can be interpreted in terms of indecomposable projectiveŪ -modules, we refer to them as non-semisimple Jones-Wenzl idempotents. Our explicit recursive formulas provide an odd level counterpart to the work of Ibanez [Ib15] and Moore [Mo18], who independently came up with similar ones for the restricted quantum group of sl 2 at even roots of unity. In Section 4, we recall the definition of a well-known monoidal linear functor F TL : TL →Ū -mod, where TL denotes the category obtained from TL A under the specialization A = q r+1 2 , and we explicitly compute the image of simple and non-semisimple Jones-Wenzl idempotents. In doing so, we show that the functor F TL is not full. In Section 5, we define the extended Temperley-Lieb categoryTL by introducing four additional generating morphisms to TL, and we extend F TL to a functor FT L :TL →Ū -mod. This allows us to prove our main result in Section 6. We do so in two steps: First, we show that the union of simple and non-semisimple Jones-Wenzl idempotents dominates the categoryTL; Then, we exhibit explicit morphisms which ensure fullness of FT L . In Section 7, we list additional relations satisfied by generating morphisms ofTL in the quotientTL =TL/ ker FT L , which will be useful in future works. It remains an open question to understand whether these relations provide a complete presentation ofTL, in the sense that they generate the kernel of FT L . However, it seems likely that further relations, generalizing equation (57) to the case r m 2r − 2, will be needed.

Small quantum group
In this section we recall the definition, due to Lusztig, of the small quantum group of sl 2 at odd roots of unity. To start, let us fix an odd integer 3 r ∈ Z and let us consider the primitive rth root of unity q = e 2πi r . For every integer k 0 we introduce the notation We denote withŪ the algebra over C with generators {E, F, K} and relations E r = F r = 0, K r = 1, This algebra was first introduced in [Lu90, Section 6.5] under the notationũ. We can makeŪ into a Hopf algebra, called the small quantum group of sl 2 , by setting Note however that all these references actually focus on the restricted quantum group of sl 2 , which corresponds to even values of r instead of odd ones. For every integer 0 m r − 1 we denote with X m the simpleŪ -module with basis {a m j | 0 j m} and action given, for all integers 0 j m, by , where a m −1 := a m m+1 := 0. As aŪ -module, X m is generated by the highest weight vector a m 0 . Next, for every integer r m 2r − 2 we denote with P m the indecomposable projectiveŪ -module with basis and action given, for all integers 0 j 2r − m − 2 and 0 k m − r, by where a m −1 := a m 2r−m−1 := x m −1 := y m m−r+1 := 0. As aŪ -module, P m is generated by the dominant vector b m 0 . Let us mention that P m is usually denoted P 2r−m−2 , because it is the projective cover of X 2r−m−2 for every integer r m 2r − 2. Our change of notation is motivated by later convenience. Next, we need to specify identifications for relevant submodules of iterated tensor products of the fundamental simpleŪ -module X := X 1 . Many details about decompositions of tensor products can be found in [Su94, Section 3], and an even more general treatment is given in [KS09, Section 3]. Again, these references discuss the restricted quantum group of sl 2 , which corresponds to even values of r rather than odd ones, but we will not actually need any of their results, as all the formulas we will list in here below can be easily checked by hand. For every integer 0 m r − 1, there is a unique isomorphic copy of X m among the submodules of X ⊗m , which we still denote X m by abuse of notation. We construct it recursively. If m = 0, a standard basis of X 0 ⊂ X ⊗0 is obtained by setting a 0 0 := 1. If 0 < m r − 1, a standard basis of X m ⊂ X ⊗m is obtained by setting Similarly, for every integer r m 2r − 2, there is an isomorphic copy of P m of multiplicity one among the submodules of X ⊗m , which we still denote P m by abuse of notation. However, this submodule is no longer unique (for instance, although X r−1 ⊗X 1 and X 1 ⊗X r−1 are isomorphic, they are not the same submodule of X ⊗r ). Therefore, we fix a choice recursively. If m = r, a standard basis of P r ⊂ X ⊗r is obtained by setting ⊗ a 1 1 , y r 0 := a r−1 r−1 ⊗ a 1 1 . (2) If r < m 2r − 2, a standard basis of P m ⊂ X ⊗m is obtained by setting For m = 2r + 1, we denote with X + r−1 and X − r−1 the two submodules of X ⊗2r−1 isomorphic to X r−1 with standard bases obtained by setting The direct sum X 2r−1 := X + r−1 ⊕X − r−1 can be lifted to a simple representation of the divided power quantum group, and it is this non-trivial splitting which will motivate the introduction of four additional generating morphisms in the Temperley-Lieb category. It will also be convenient to fix isomorphisms for supplements of these submodules. If m = 2, we denote with X 0 the subspace of X 1 ⊗ X 1 isomorphic to X 0 with standard basis obtained by setting If 2 < m r − 1, we denote with X m−2 the subspace of X m−1 ⊗ X 1 isomorphic to X m−2 with standard basis obtained by setting a m−2 j If m = r + 1, we denote with X + r−1 and X − r−1 the submodules of X r−1 ⊗ X 1 ⊗ X 1 isomorphic to X r−1 with standard bases obtained by setting a r−1,+ j If r m 2r − 1, we denote with P m−2 the subspace of P m−1 ⊗ X 1 isomorphic to P m−2 with standard basis obtained by setting a m−2 j Finally, let us fix our notation forŪ -module morphisms. Dimensions of non-zero vector spaces of morphisms between simple and indecomposable projective objects ofŪ -mod are given by dim C HomŪ (P m , X 2r−m−1 ) = dim C HomŪ (X 2r−m−1 , P m ) = 1 r m 2r − 2, dim C HomŪ (P m , P 3r−m−2 ) = 2 r m 2r − 2, dim C HomŪ (X r−1 , X 2r−1 ) = dim C HomŪ (X 2r−1 , X r−1 ) = 2.

Temperley-Lieb category
In this section we discuss a generalization of Jones-Wenzl idempotents at odd roots of unity. Non-semisimple Temperley-Lieb algebras at roots of unity and evaluable idempotents were first studied by Goodman and Wenzl in [GW93]. Recursive formulas for these idempotents already appeared in the work of Ibanez [Ib15] and Moore [Mo18], who independently treated the case of even roots of unity, which is complementary to ours.
For an indeterminate A and for every integer k 0 we introduce the notation The Temperley-Lieb category TL A is the C(A)-linear category with set of objects given by N, and with vector space of morphisms from m ∈ TL A to m ∈ TL A denoted TL A (m, m ) and linearly generated by planar (m, m )-tangles modulo the subspace linearly generated by planar (m, m )-tangles of the form where t is a planar (m, m )-tangle, and where u is the unknot. Composition in TL A is defined by vertical gluing, and denoted like multiplication, while tensor product is defined by horizontal juxtaposition. Equipped with this monoidal structure, it is easy to see that TL A is a rigid category: for every m ∈ TL A the dual m * ∈ TL A is given by m itself, while for every t ∈ TL A (m, m ) the dual t * ∈ TL A (m , m) is obtained by applying a rotation of angle π to t. For every m ∈ TL A , the mth Let us recall now the definition of simple Jones-Wenzl idempotents. These endomorphisms of TL A were first discovered by Jones [Jo83], although the recursive definition given here is due to Wenzl [We87], see also Lickorish [Li97]. For every integer m 0 the mth simple Jones-Wenzl idempotent f m ∈ TL A (m) is recursively defined as These endomorphisms satisfy for all integers 1 j m − 1. This implies for all integers 0 n m. It can be shown that f * m = f m for every integer m 0. Furthermore, for all integers 0 k m we have Now we can define a new family of endomorphisms g m , h m ∈ TL A (m) for every integer r m 2r − 2 by setting where These endomorphisms satisfy for all integers r m 2r − 2 and 1 j m − 1, provided j is not equal to r − 1.
Proof. The statement is proved by direct computation. For m = r, equation (37) follows from the recurrence formula (20) defining f r . For m = r + 1, equation (38) follows from a double application of equation (20). For r + 1 < m 2r − 2, let us look at the right-hand side of equation (39). For what concerns the first term, equation (24) yields For what concerns the second term, equations (29) and (31) yield For what concerns the third term, equations (31) and (32) yield This means that, using the equality we can rewrite the right-hand side of equation (39) as where the first equality follows from equation (25), the second equality follows from equation (23), and the third equality follows from equation (20). This means that the right-hand side of equation (39) can be further rewritten as Now the statement follows from equation (24).
Proof. We follow the proof of equation (39) in Lemma 3.2, but this time we suppose m = 2r − 1. Then, the same argument implies that the right-hand side of equation (40) can be rewritten as where the first equality follows from equation (25), the second equality follows from equation (23), and the third equality follows from equation (20). Now the statement follows from equation (20).
From now on, we will focus on the specialization We observe that f m can be specialized to TL only for 0 m r − 1, because the formula defining f r has a pole at A = q r+1 2 . On the other hand, thanks to Lemma 3.2, g m can be specialized to TL for all r m 2r − 2, and similarly, thanks to Lemma 3.3, f 2r−1 can be specialized to TL. We call g m the mth nonsemisimple Jones-Wenzl idempotent.
Remark 3.4. It can be proven by induction that in TL the idempotent f r−1 satisfies In particular, in TL we have h * m = h m for every integer r m 2r − 2, because a rotation of angle π fixes h m . Remark however that this is true only once we specialize A to q r+1 2 , so the same property does not hold for g m .

Monoidal linear functor
In this section, we study a monoidal linear functor F TL : TL →Ū -mod, compare with [FK97]. By abuse of notation, we still denote by TL the idempotent completion of TL. This means we promote idempotent endomorphisms p m ∈ TL(m) to objects of TL, and we set with the morphism p m being the identity of the object p m . As a consequence, we will occasionally confuse direct summands of objects ofŪ -mod with the corresponding idempotent endomorphisms. We also start adopting a graphical notation featuring labels given by integers m 0 placed next to endpoints of edges, standing for the number of parallel strands contained in the plane.
Let us consider the monoidal linear functor F TL : TL →Ū -mod sending the object 1 ∈ TL to X ∈Ū -mod, and sending the morphism ∪ ∈ TL(0, 2) to c ∈ HomŪ (C, X ⊗ X) defined by In order to study F TL , let us define projection morphisms p m ∈ TL(g m , f 2r−m−2 ) and injection morphisms i m ∈ TL(f 2r−m−2 , g m ) as for every integer r m 2r − 2, which immediately gives The fact that p m g m = p m and that g m i m = i m follows from a direct computation.
Lemma 4.1. For every integer 0 m r − 1 the object f m satisfies for r m 2r − 2 the object g m and the morphisms h m , p m , and i m satisfy and for m = 2r − 1 the object f 2r−1 satisfies where the morphisms ε m , π m , and ι m are defined by equations (9) to (11).
The proof of Lemma 4.1 is a lengthy computation which will occupy the remainder of this section.
Proof. In order to discuss the proof, we first need to fix some notation. Let us denote with Y m the unique submodule of X ⊗m satisfying for every integer 0 m r − 1, and let us denote with ϕ m ∈ EndŪ (X ⊗m ) the idempotent endomorphism with image X m and kernel Y m . Next, let us denote with Q m the unique submodule of X ⊗m satisfying for every integer r m 2r − 2, and let us denote with ψ m ∈ EndŪ (X ⊗m ) the idempotent endomorphism with image P m and kernel Q m . Similarly, let us denote with Y 2r−1 the unique submodule of X ⊗2r−1 satisfying and for every ε ∈ {+, −} let us denote with ϕ ε 2r−1 ∈ EndŪ (X ⊗2r−1 ) the idempotent endomorphism with image X ε 2r−1 and kernel X ε r−1 ⊕ Y 2r−1 , where ε ∈ {+, −} denotes the opposite of ε.
The strategy will be to prove by induction on 0 m 2r−1 that F TL (f m ) = ϕ m , if 0 m r − 1, that F TL (g m ) = ψ m , if r m 2r − 2, and finally that F TL (f 2r−1 ) = ϕ + 2r−1 + ϕ − 2r−1 . This is precisely what equations (44), (45), and (49) stand for. This will allow us to establish equations (47) and (48), which immediately imply equation (46), by restriction to the submodule P m of X ⊗m . An important observation is that, since the functor F TL sends all morphisms if TL to intertwiners, it is sufficient to check these equalities only for highest-weight vectors, in the case of simpleŪ -modules, and dominant vectors, in the case of indecomposable projectivē U -modules. gives We know F TL (f m−1 ⊗ f 1 ) restricts to the identity on X m−1 ⊗ X 1 and to zero on Y m−1 ⊗ X 1 thanks to the induction hypothesis. Then, using equation (1), we have . This means that, using the definition of F TL and equation (53), we get On the other hand, using equation (6), we have ⊗ a 1 0 ⊗ a 1 0 , and, using equations (1) and (6) gives g r = f r−1 ⊗ f 1 . We know F TL (f r−1 ⊗ f 1 ) restricts to the identity on X r−1 ⊗ X 1 and to zero on Y r−1 ⊗X 1 thanks to equation (44), and so F TL (g r ) = ψ r . This proves equation (45). gives We know F TL (f r−1 ⊗ f 1 ⊗ f 1 ) restricts to the identity on X r−1 ⊗ X 1 ⊗ X 1 and to zero on Y r−1 ⊗ X 1 ⊗ X 1 . Then, using equation (3), we have and, using equation (6) This means that, using the definition of F TL and equation (54), we get On the other hand, using equation (7), we have . This means that, using the definition of F TL and equation (54), we get gives We know F TL (g m−1 ⊗ f 1 ) restricts to the identity on P m−1 ⊗ X 1 and to zero on Q m−1 ⊗ X 1 thanks to the induction hypothesis. Then, using equation (3), we have This means that, using the definition of F TL and equation (55), we get On the other hand, using equation (8) ⊗ a 1 0 ⊗ a 1 0 , and, using equations (3) and (8), we have This means that, using the definition of F TL and equation (55), we get where We know F TL (g 2r−2 ⊗ f 1 ) restricts to the identity on P 2r−2 ⊗ X 1 and to zero on Q 2r−2 ⊗ X 1 thanks to equation (45). Then, using equation (4), we have This means that, using the definition of F TL and equation (56), we get On the other hand, using equation (8) ⊗ a 1 0 ⊗ a 1 0 , and, using equations (3) and (8), we have This means that, using the definition of F TL and equation (56), we get Therefore, F TL (f 2r−1 ) = ϕ + 2r−1 + ϕ − 2r−1 . This proves equation (49). The functor F TL : TL →Ū -mod is not full. Indeed, we have dim C TL(g m , g 3r−m−2 ) = 0 for every integer r m 2r − 2, as follows directly from a parity argument, 3r −2m−2 being odd. This does not match the dimension of HomŪ (P m , P 3r−m−2 ), as recalled in Section 2.

Extended Temperley-Lieb category
In this section we introduce an extended version of the category TL over which we define a monoidal linear functor with targetŪ -mod.
Definition 5.1. The extended Temperley-Lieb categoryTL is the smallest linear category containing TL as a linear subcategory, as well as morphisms The proof of Lemma 5.3 is a lengthy computation which will occupy the remainder of this section.
Proof. In order to discuss the proof, let us first choose names for the morphisms appearing in equations (57) and (58), which we rewrite as t ε,ε r−1,m = δ ε,ε · t r−1,m , u ε,ε r−1,m = δ ε,ε · u r−1,m , respectively. We also give names to the images of these morphisms under the functor FT L by setting Therefore, what we need to show is When m = 0, the relations reduce to equations (18) we have we have and we have Similarly, let us choose names for the morphisms appearing in equation (59), which we rewrite as c ε r = −c ε r , d ε r = −d ε r . Equations , and so equation (80) gives Furthermore, equations (74) and (75) give , y r 0 = a r−1 r−1 ⊗ a 1 1 = a 1 1 ⊗ a r−1 r−1 , and so equations (81) and (82) give 6. Proof of Theorem 1.1 In this section we establish our main result. We do this in two main steps. First of all, for convenience of notation, let us set for every j ∈ J m−1,n . Therefore we can set J m,n := The second step in the proof of Theorem 1.1 consists in exhibiting morphisms ofTL whose image under FT L recovers the missing morphisms between indecomposable projectiveŪ -modules. In order to do this, let us define morphisms c ε m ∈TL(g m , g 3r−m−2 ) and d ε m ∈TL(g 3r−m−2 , g m ) as where γ + m and γ − m are defined by equations (12) and (13).
Proof. The statement is proved by induction on m. For m = r it follows from computations in the proof of Lemma 5.3 establishing the image under FT L of equation (59). For r < m 2r − 2, equation (3) gives and equation (63) gives This proves equations (64) and (65). Furthermore, equations (3) and (8) give  (63) gives We can now prove Theorem 1.1.
Proof of Theorem 1.1. For convenience of notation, let us set for all integers 0 n, n 2r − 2 and for all j ∈ J m,n and j ∈ J m ,n . Now, thanks to Lemmas 4.1 and 6.2, the linear map FT L :TL(t n , t n ) → HomŪ (T n , T n ) is surjective for all integers 0 n, n 2r − 2, and therefore FT L (t n u n ,j m ) • f • FT L (v m n,j t n ) ∈ HomŪ (T n , T n ) belongs to its image. This means that f belongs to the image of the linear map FT L :TL(m, m ) → HomŪ (X ⊗m , X ⊗m ).

Additional relations
In this section we list additional relations satisfied by images of generating mor- In order to do this, we define the small Temperley-Lieb categoryTL as the quotient linear categorȳ TL :=TL/ ker FT L .
In other words, objects ofTL are the same as objects ofTL, while vector spaces of morphisms ofTL are quotients of vector spaces of morphisms ofTL with respect to kernels of linear maps determined by the linear functor FT L .

Appendix A. Computations
In this appendix, we provide computations to be used in Sections 5 and 7. We start with the identity Lemma A.1. For all integers 0 j r − 1 we have Proof. The statement follows from a direct computation.
Next, let us fix some notation. For all integers 0 j m, let us set V m j := span C {a 1 j1 ⊗ . . . ⊗ a 1 jm ∈ X ⊗m | j 1 + . . . + j m = j}, and let us give special names to highest and lowest weight vectors for every integer r m 2r − 2 we have and for m = 2r − 1 we have Proof. The statement can be proved by induction on m.
Lemma A.3. For all integers 0 m r − 1 and 0 j m we have for all integers r m 2r − 2, 0 k m − r, and 0 2r − m − 2 we have and for every integer 0 j r − 1 we have Proof. The statement follows from equations (72) to (76).