Non-Semisimple Quantum Invariants and TQFTs from Small and Unrolled Quantum Goups
Marco De Renzi Nathan Geer Bertrand Patureau-Mirand






Non-Semisimple Quantum Invariants and TQFTs]Non-Semisimple Quantum Invariants and TQFTs from Small and Unrolled Quantum Groups


M. De Renzi]Marco De Renzi Department of Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1 Ōkubo, Shinjuku-ku, Tokyo, 169-8555, Japan m.derenzi@kurenai.waseda.jp


N. Geer]Nathan Geer
Mathematics & Statistics

  Utah State University 

  Logan, Utah 84322, USA nathan.geer@gmail.com


B. Patureau-Mirand]Bertrand Patureau-Mirand
Univ. Bretagne - Sud, UMR 6205, LMBA, F-56000 Vannes, France
bertrand.patureau@univ-ubs.fr



 We show that unrolled quantum groups at odd roots of unity give rise to relative modular categories. These are the main building blocks for the construction of 1+1+1-TQFTs extending CGP invariants, which are non-semisimple quantum invariants of closed 3-manifolds decorated with ribbon graphs and cohomology classes. When we consider the zero cohomology class, these quantum invariants are shown to coincide with the renormalized Hennings invariants coming from the corresponding small quantum groups.



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    December 30, 2023
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The goal of this paper is two-fold: first of all, we provide a new family of concrete examples of relative modular categories. These are ribbon categories which can be used as fundamental bricks for the construction of non-semisimple quantum invariants of closed manifolds in dimension 3 <cit.> and Extended Topological Quantum Field Theories (ETQFTs) in dimension 1+1+1 <cit.>. They are modeled on categories of finite-dimensional weight representations of the unrolled quantum group U_q^H at even roots of unity q, which were used in <cit.> to build TQFTs in dimension 2+1. They should be thought of as a non-semisimple analogue to standard modular categories, although differences with respect to their semisimple counterparts are many: first of all, a relative modular category  comes equipped with a structure group G that provides a grading on its objects; secondly, it enjoys finiteness properties only up to the action of a periodicity group Z of transparent objects; more importantly, it is only generically semisimple, with non-semisimple part confined to a critical set X ⊂ G whose complement is dense in G. When G = Z = { 0 } and X = ∅ the definition reduces to the standard one of <cit.>, see Section 1.7 of <cit.>. In Theorem <ref> we prove that categories ^H of finite-dimensional weight representations of unrolled quantum groups U_q^H associated with arbitrary simple complex Lie algebras  at odd roots of unity q are relative modular. These categories were already known to induce topological invariants _^H of certain decorated closed 3-manifolds (M,T,ω), where T ⊂ M is a ^H-colored ribbon graph, where ω∈ H^1(M ∖ T;G) is a cohomology class, and where the triple (M,T,ω) is subject to a crucial admissibility condition. These so-called Costantino-Geer-Patureau (CGP) quantum invariants are quite rich, but their definition involves certain technical aspects, such as the notion of computable surgery presentation introduced in <cit.>. The results of this paper imply these invariants can be extended to graded 1+1+1-TQFTs for all simple complex Lie algebras . In the case of , these invariants contain the Akutsu-Deguchi-Ohtsuki invariants of colored links and the abelian Reidemeister torsion of closed 3-manifolds, and they were already known to extend to graded 1+1+1-TQFTs.


The second main result of this paper is a Hennings-type formula for these CGP quantum invariants. More precisely, every simple complex Lie algebra  also determines a corresponding small quantum group U̅_q for every odd root of unity q <cit.>. These finite-dimensional factorizable quotients have been studied a lot in literature <cit.>, and they induce renormalized Hennings TQFTs in dimension 2+1, see <cit.>. In particular, their categories of finite-dimensional representations  yield quantum invariants '_ of certain admissible closed 3-manifolds (M,T), where T ⊂ M is a -colored bichrome graph, which is a very mild generalization of standard ribbon graphs obtained by specifying special components which correspond to surgery presentations. Then, we prove in Theorem <ref> that, for a fixed simple complex Lie algebra  at an odd root of unity q, the CGP invariant _^H of decorated closed 3-manifolds with zero cohomology classes coincides with the corresponding renormalized Hennings invariant '_. This result builds a bridge between the two theories, and, in this setting, it gives us a way of computing CGP quantum invariants which bypasses computable surgery presentations.



 §.§ Acknowledgments

We would like to thank the referee for their extremely careful review of our paper. Their deep understanding of our results and their detailed comments helped us improve the paper. 
 NG was partially supported by NSF grants 
DMS-1308196 and DMS-1452093. 



§ OVERVIEW OF NON-SEMISIMPLE CONSTRUCTIONS

 
 
In this section, we quickly review the two main constructions this paper deals with. References are provided by <cit.> for the CGP theory, and by <cit.> for the renormalized Hennings one. All the manifolds we consider are always assumed to be oriented.
 



 §.§ Ribbon categories and m-traces



First, let us fix our notation 

and conventions for categorical structures. Following <cit.>, a ribbon category is a braided rigid monoidal category  equipped with a natural transformation θ : 𝕀_⇒𝕀_, called the twist, satisfying

    θ_V ⊗ V' = (θ_V ⊗θ_V') ∘ c_V',V∘ c_V,V',     
     (θ_V)^* = θ_V^*

for all V,V' ∈, where c_V,V' : V ⊗ V' → V' ⊗ V denotes the braiding of . For every V ∈ we denote with

    _V    : V^* ⊗ V →,    _V    : → V ⊗ V^*, 
    _V    : V ⊗ V^* →,    _V    : → V^* ⊗ V

its left and right duality morphisms, 



and for every f ∈_(V) we denote with _(f) its categorical trace. As shown in <cit.>, every ribbon category  induces a ribbon functor F_ : _→ called the Reshetikhin-Turaev functor, where _ denotes the category of -colored ribbon graphs.



An ideal of  is a full subcategory of  whose class of objects is closed under retracts and absorbent under tensor products with arbitrary objects of .

We denote with () the ideal of projective objects of . We define the partial trace of an endomorphism f ∈_(V ⊗ V') to be the endomorphism (f) ∈_(V) given by

    (f) := (𝕀_V ⊗_V') ∘ (f ⊗𝕀_V'^*) ∘ (𝕀_V ⊗_V').

Then, following <cit.>, if  is a ribbon linear category over a field , an m-trace  on () is a family of linear maps
{_V : _(V) →| V ∈() } satisfying:

 
  * Cyclicity: _V(f' ∘ f) = _V'(f ∘ f') for all objects V,V' ∈() and for all morphisms f ∈_(V,V') and f' ∈_(V',V);
 
  * Partial trace: _V ⊗ V' (f) = _V((f)) for all objects V ∈() and V' ∈ and for every morphism f ∈_(V ⊗ V').

For every V ∈() we denote with (V) := _V(𝕀_V) its modified dimension. We say an m-trace  on () is non-degenerate if, for every V ∈() and V' ∈, the bilinear pairing _V( ·∘· ) : _(V',V) ×_(V,V') → is non-degenerate. 


Finally, let us fix some terminology which will be extensively used throughout the paper: every time we have a ribbon linear category  over a field , we have an associated notion of skein equivalence between formal linear combinations of -colored ribbon graphs. Indeed, if (ε,V) and (ε',V') are objects of _, if α_1,…,α_m,α'_1,…,α'_m' are scalar coefficients in , and if T_1,…,T_m,T'_1,…,T'_m' are morphisms of _ from (ε,V) and (ε',V'), then we say two formal linear combinations ∑_i=1^m α_i · T_i and ∑_i'=1^m'α'_i'· T'_i' are skein equivalent, and we write 

    ∑_i=1^m α_i · T_i ≐∑_i'=1^m'α'_i'· T'_i',

if we have the equality ∑_i=1^m α_i · F_(T_i) = ∑_i'=1^m'α'_i'· F_(T'_i') under the Reshetikhin-Turaev functor F_ : _→.




 §.§ 3-Manifold invariants from non-degenerate relative pre-modular categories



Let us start by recalling the definition of relative pre-modular categories, which, as we mentioned earlier, are ribbon linear categories carrying additional structures. First of all, if G is an abelian group, a compatible G-structure on a rigid

monoidal category  is an equivalence of linear categories
≅⊕_g ∈ G_g for a family
{_g | g ∈ G } of full subcategories of 
satisfying the following conditions: If V ∈_g, then
V^* ∈_- g; If V ∈_g and V' ∈_g',
then V ⊗ V' ∈_g + g'; If V ∈_g and
V' ∈_g' with g ≠ g', then _(V,V') =
0. Remark that, if  is ribbon, then _0 is also ribbon.



Next, if Z is an abelian group, a free realization of  in a ribbon category  is a monoidal functor σ : Z →, where Z also denotes the discrete category over Z with tensor product given by the group operation +, satisfying ϑ_σ(k) = 𝕀_σ(k) for every k ∈ Z, and inducing a free action on isomorphism classes of simple objects of  by tensor product with σ(k). 

Next, we say a subset X of G is symmetric if X = -X, and we say it is small  if G ⊄⋃_i=1^m (g_i + X) for all m ∈ and all g_1, …, g_m ∈ G. 



 If G and Z are abelian groups, and if X ⊂ G is a small symmetric subset, then a pre-modular G-category relative to (Z,X) is a ribbon linear category  over a field  together with a compatible G-structure on , a free realization σ : →_0, and a non-zero m-trace  on (). These data are subject to the following conditions: 
 
  
  * Generic semisimplicity. For every g ∈ G ∖ X the homogeneous subcategory _g is semisimple and dominated by Θ(_g) ⊗σ(Z) for some finite set Θ(_g) = { V_i ∈_g | i ∈_g } of simple projective objects with epic evaluation;
  
  * Compatibility. There exists a bilinear map ψ : G × Z →^* such that c_σ(k),V∘ c_V,σ(k) = ψ(g,k) ·𝕀_V ⊗σ(k) for every g ∈ G, for every V ∈_g, and for every k ∈ Z.
 



The group G is called the structure group, the group Z is called the periodicity group, and the set X is called the critical set of .

Condition (1) implies that, for any g ∈ G ∖ X, every object of _g is a direct sum of simple objects in the set {V_i ⊗σ(k) ∈| i ∈_g, k ∈ Z }.

We point out that the bilinear map ψ is uniquely determined by the braiding c and by the free realization σ, and that, although the set of representatives Θ(_g) is not unique in general, its choice does not affect


the following construction.   

In particular, both ψ and Θ(_g) should not be considered relevant parts of the structure of .



Relative pre-modular G-categories are a slight generalization of the notion of relative G-modular category introduced for the first time in <cit.>, see Section 1.5 of <cit.> for a full discussion of the relation between the two definitions. The change in terminology is motivated by the semisimple theory, where quantum invariants are defined for any non-degenerate pre-modular category, and modularity is an additional condition ensuring the invariant extends to a TQFT.

If  is a pre-modular G-category relative to (Z,X) then the associated Kirby color of index g ∈ G ∖ X is the formal linear combination of objects

    Ω_g := ∑_i ∈_g(V_i) · V_i.

The name comes from Lemmas 5.9 and 5.10 of <cit.>. In particular, there exist constants Δ_-Ω,Δ_+Ω∈, called stabilization coefficients, which realize the skein equivalences of Figure <ref>, and which are independent of both V ∈_g and g ∈ G ∖ X. We say the relative pre-modular category  is non-­de­gen­er­ate if Δ_-ΩΔ_+Ω≠ 0.





In <cit.> it is shown that every non-degenerate relative pre-modular category  gives rise to a topological invariant _ of admissible triples (M,T,ω), where M is a closed 3-manifold, T ⊂ M is a -colored ribbon graph, and ω∈ H^1(M ∖ T;G) is a compatible cohomology class, meaning that every edge e ⊂ T is colored with an object of _⟨ω,m_e ⟩ for the homology class m_e of a positive meridian of e. The CGP invariant _ is defined only for admissible triples (M,T,ω), which are triples such that every component of M contains either a projective edge of T, that is an edge of T whose color is a projective object of , or a generic curve for ω, that is an embedded closed oriented curve whose homology class is sent to G ∖ X by ω. Its definition uses computable surgery presentations in S^3, which are surgery presentations L = L_1 ∪…∪ L_ℓ of M satisfying ⟨ω,m_j ⟩∈ G ∖ X for all integers 1 ⩽ j ⩽ℓ, where m_j denotes the homology class of a meridian of the component L_j. We interpret a surgery presentation of M which is computable with respect to some decoration (T,ω) as a -colored ribbon graph by arbitrarily choosing orientations, and by labeling every component with the corresponding Kirby color, with index prescribed by the evaluation of ω against the homology class of a positive meridian. This is a technical complication, because arbitrary surgery presentations are not computable in general. Computable surgery presentations do exist for admissible decorated closed 3-manifolds, but only up to replacing admissible decorations via certain operations called projective and generic stabilizations, see Section 3.1 of <cit.>. The idea is to build _ out of a renormalized invariant F'_ of admissible closed -colored ribbon graphs which combines the Reshetikhin-Turaev functor F_ on _ with the m-trace  on (). We say a closed -colored ribbon graph is admissible if one of his edges is projective. 
Example 3 of Section 1.5 in <cit.> (see also <cit.>) implies the formula


    F'_(T) := _V(F_(T_V))
 
defines a topological invariant of the admissible closed -colored ribbon graph T, where V ∈ is projective, and where T_V is a cutting presentation of T, i.e. an endomorphism of (+,V) in _ whose trace is T. Then, for a fixed choice of a square root _Ω∈ of Δ_- ΩΔ_+ Ω, the formula

    _(M,T,ω) := _Ω^-1-ℓδ_Ω^-σ(L) F'_ (L ∪T̃)

defines a topological invariant of the admissible triple (M,T,ω) thanks to Proposition 3.1 of <cit.>, where L ⊂ S^3 is an ℓ-component surgery presentation of M of signature σ(L) which is computable with respect to an admissible decoration (T̃,ω̃) obtained from (T,ω) by performing projective or generic stabilization, and where δ_Ω = _Ω / Δ_-Ω.




 §.§ 2+1-TQFTs from relative modular categories



A stronger non-de­gen­er­a­cy condition is required in order to extend CGP invariants to graded TQFTs.



 A pre-modular G-cat­e­go­ry  relative to (Z,X) is relative modular if there exists a relative modularity parameter ζ_Ω∈^* realizing the skein equivalence of Figure <ref> for all g,h ∈ G ∖ X and for all i,j ∈_g.






This condition automatically implies non-degeneracy, because the relative modularity parameter satisfies ζ_Ω = Δ_-ΩΔ_+Ω, see <cit.>. When  is relative modular then, as explained in Section 6.2 of <cit.>, _ extends to a Z-graded 2+1-TQFT _^ : _^G →_^Z via a Z-graded refinement of the universal construction of <cit.>, where _^G is the category of admissible cobordisms of dimension 2+1, and where _^Z is the category of Z-graded vector spaces. More precisely, an object of _^G is a 5-tuple = (,P,ϑ,B,), where  is a closed surface, where P ⊂ is a -colored ribbon set, where ϑ∈ H^1(∖ P;G) is a compatible cohomology class, where B ⊂∖ P is a finite set composed of exactly one base point in every connected component of , and where ⊂ H_1(;) is a Lagrangian subspace. A morphism of _^G from (,P,ϑ,B,) to (',P',ϑ',B',') is an equivalence class of admissible 4-tuples = (M,T,ω,n), where M is a 3-dimensional cobordism from  to ', where T ⊂ M is a -colored ribbon graph from P to P', where ω∈ H^1(M ∖ T,B ∪ B';G) is a compatible relative cohomology class restricting to ϑ and ϑ' on the incoming and outgoing boundary of M respectively, and where n ∈ is a signature defect. A 4-tuple (M,T,ω,n) is admissible if every component of M which is disjoint from the incoming boundary ∂_- M contains either a projective edge of T, or a generic curve for ω, and two 4-tuples (M,T,ω,n) and (M',T',ω',n') are equivalent if n = n', and if there exists a positive diffeomorphism f : M → M' which preserves boundary identifications and satisfies f(T) = T' and f^*(ω') = ω. Then, _ can be extended to an invariant of closed morphisms of _^G by setting

    _(M,T,ω,n) := δ_Ω^n _(M,T,ω) .

Remark that the category _^G thus obtained is not rigid, as objects (,P,ϑ,) such that P does not contain any projective point and such that ϑ does not admit any generic curve are not dualizable. 


State spaces associated with objects of _^G by the Z-graded TQFT _^ can be described in skein theoretical terms in all degrees. We have two relevant notions of skein equivalence between morphisms of _^G from  to ', one which is local, the other which is not. Indeed, we say a formal linear combination of morphisms of _^G from  to ' is a local skein relation if it can be written in the form

    ∑_i=1^m α_i ·_P ∘( (D^3,T_i,ω_i,0) 𝕀_)

for some coefficients α_1,…,α_m ∈, for some -colored ribbon set P ⊂ S^2 with at least one point labeled by a projective object, for some morphism _P of _^G from (S^2,P,ϑ,{ 0 }) to ', and for some -colored ribbon graphs T_1,…,T_m ⊂ D^3 from ∅ to P satisfying

    ∑_i=1^m α_i · f_D^3(T_i) ≐ 0

with respect to some embedding f_D^3 : D^3 ↪^2 × I mapping P into ^2 ×{ 1 }. Here the cohomology classes θ and ω_1,…,ω_m are uniquely determined by P and by T_1,…,T_m respectively. On the other hand, we say a formal linear combination of morphisms of _^G from  to ' is a non-local skein relation if it can be written in the form

    (M,T ∪ K,j^*(ω),n) - ψ(⟨ω,ℓ_K ⟩,k) · (M,T,ω,n)

for some k ∈ Z and for some framed knot K ⊂ M ∖ T of color σ(k), where j^* is induced by the inclusion of M ∖ (T ∪ K) into M ∖ T, and where ℓ_K denotes the homology class of K in H_1(M ∖ T;). Then, if = (,P,ϑ,B,) is an object of _^G, and if M is a 3-dimensional cobordism from ∅ to , the admissible skein module (M;) is the quotient, induced by both local and non-local skein relations, of the free vector space (M;) generated by all pairs (T,ω) such that (M,T,ω,0) is a morphism of _^G from ∅ to . The class of a generator (T,ω) of (M;) in (M;) is denoted [T,ω]. The proof of Proposition 4.5 in <cit.> can be adapted to show that if Σ and M are both connected, then (M;) is a finite-dimensional vector space. On the other hand, if M' is a 3-dimensional cobordism from  to ∅, 



we denote with '(M';) the free vector space generated by all pairs (T',ω') such that (M',T',ω',0) is a morphism of _^G from  to ∅. Then, for every k ∈ Z, the degree k state space _^k() of a connected object  of _^G satisfies 


    _^k() ≅(M;^2_-k) / '(M';^2_-k)^⊥

with respect to the pairing ⟨· , ·⟩_^2_-k : '(M';^2_-k) ⊗(M;^2_-k) → defined by

    ⟨ (T',ω') , [T,ω] ⟩_^2_-k := _( (M',T',ω',0) ∘ (M,T,ω,0) ),

where M is a connected 3-dimensional cobordism from ∅ to ⊔ S^2, where M' is a connected 3-dimensional cobordism from ⊔ S^2 to ∅, and where the object 

    ^2_-k = (S^2,P_((+,V_0),(+,σ(-k)),(-,V_0)),ϑ_((+,V_0),(+,σ(-k)),(-,V_0)),B,{ 0 })

of _^G is determined by the -colored ribbon set P_((+,V_0),(+,σ(-k)),(-,V_0))⊂ S^2 composed of three points in standard positions with orientations and colors specified by their subscript for some g_0 ∈ G ∖ X and for some V_0 ∈Θ(_g_0). Remark that an explicit characterization of these quotients can sometimes be achieved. For instance, if P ⊂ S^2 is a -colored ribbon set composed of a single positive point of color F_(P) ∈(_0), then, thanks to Remark 7.2 of <cit.>, the -graded state space _^(^2_P) of the object ^2_P = (S^2,P,0,{ 0 }) of _^G satisfies

    _^(^2_P) ≅_(,F_(P)),

where for all V,V' ∈ we denote with _(V,V') the -graded vector space whose space of degree k vectors is given by _(V,V' ⊗σ(-k)) for every k ∈. See also Proposition 7.16 of <cit.> for a description, in terms of homogeneous colorings of trivalent graphs, of the -graded state space of generic surfaces in _^G.





 §.§ 3-Manifold invariants from finite-dimensional non-degenerate unimodular ribbon Hopf algebras



Next, let us move on to the renormalized Hennings theory. We start by fixing our notation for Hopf algebras, and by recalling some crucial definitions and results. If  is a field, a finite-dimensional ribbon Hopf algebra H is a finite-dimensional vector space over  endowed with a multiplication m : H ⊗ H → H, a unit η : → H, a coproduct Δ : H → H ⊗ H, a counit ε : H →, an antipode S : H → H, an R-matrix R = ∑_i=1^r a_i ⊗ b_i ∈ H ⊗ H, and a ribbon element v in the center of H, see <cit.> for a list of the axioms these structure maps and elements are subject to. We use the notation m(x ⊗ y) = xy for every x ⊗ y ∈ H ⊗ H and η(1) = 1, and we denote with u = ∑_i=1^r S(b_i)a_i ∈ H the Drinfeld element, and with g = uv^-1∈ H the pivotal element associated with the ribbon structure of H. As a consequence of finite-dimensionality, H admits a right integral λ∈ H^* and a left cointegral Λ∈ H which are unique up to scalar, and we can fix a pair satisfying λ(Λ) = 1. The Hopf algebra H is non-degenerate if the stabilization coefficients Δ_-λ := λ(v) and Δ_+λ := λ(v^-1) satisfy Δ_-λΔ_+λ≠ 0, and it is unimodular if the left cointegral Λ is two-sided, meaning that it is also a right cointegral. The category = H-mod of finite-dimensional left H-modules is a ribbon linear category, with evaluation and coevaluation morphisms given by

    [    _V : V^* ⊗ V       →        ;           f ⊗ v       ↦    f(v) ]      [              _V :                                   →           V ⊗ V^*;                                   1                 ↦ ∑_i=1^n v_i ⊗ f_i ]
       [         _V :      V ⊗ V^*            →             ;                     v ⊗ f            ↦ f(ρ_V(g)(v)) ]      [                        _V :                                                       →                     V^* ⊗ V;                                                       1                           ↦ ∑_i=1^n f_i ⊗ρ_V(g^-1)(v_i) ]

for every left H-module V with basis { v_1,…,v_n } and dual basis { f_1,…,f_n }, and with braiding morphisms given by

    [                        c_V,V' :                          V ⊗ V'                               →                          V' ⊗ V;                                                          v ⊗ v'                               ↦ ∑_i=1^r ρ_V'(b_i)(v') ⊗ρ_V(a_i) ]

for all left H-modules V and V'. Thanks to Theorem 1 of <cit.>,  admits an m-trace  on the ideal of projective H-modules (), which is unique up to scalar and uniquely determined by the condition _H(f) = λ(gf(1)) for all f ∈_(H), where H ∈() denotes the regular representation of H. Furthermore,  is non-degenerate.


In <cit.> it is shown that every finite-dimensional non-degenerate ribbon Hopf algebra H gives rise to a topological invariant '_ of admissible pairs (M,T), where M is a closed 3-manifold, and T ⊂ M is a -colored bichrome graph. The latter are -colored ribbon graphs carrying a set of specified edges, and their name comes from the fact that we think about special edges as being red, while the rest of the graph is blue. Red edges can only be colored with the regular representation H, and they can only intersect coupons in a prescribed way: for every coupon of a bichrome graph there exists an integer k ⩾ 0 such that the first k input edges are incoming and red, the first k output edges are outgoing and red, while all the other ones are blue. Such a coupon is colored with a morphism in the k-th stabilized subcategory [k] of , which is the category whose objects have the form [k]V := H^⊗ k⊗ V ∈ for some V ∈, and whose morphisms have the form ∑_i=1^m L_x_i⊗ f_i ∈_([k]V,[k]V') for some left translation L_x_i∈_(H^⊗ k) by x_i∈ H^⊗ k and for some linear map f_i ∈_(V,V'). The ribbon category _λ of -colored bichrome graphs provides a graphical calculus which is formalized by the Hennings-Reshetikhin-Turaev functor F_λ : _λ→ introduced in Proposition 2.5 of <cit.>. By definition, F_λ coincides with the Reshetikhin-Turaev functor in the absence of red edges, it coincides with the Hennings invariant in the absence of blue edges, and it coherently combines the two behaviors for general -colored bichrome graphs. Remark that F_λ yields a notion of skein equivalence between formal linear combinations of -colored bichrome graphs in the same way F_ does for -colored ribbon graphs. This way, _ is naturally identified with the subcategory of _λ whose morphisms are entirely blue. The renormalized Hennings invariant '_ is then defined only for admissible pairs (M,T), which are pairs such that every component of M contains a projective blue edge of T. Its definition uses surgery presentations in S^3, which we interpret as -colored bichrome graphs by arbitrarily choosing orientations, by labeling every component with the regular representation H, and by taking them to be red. The idea is to build '_ out of a renormalized invariant F'_λ of admissible closed -colored bichrome graphs which combines the Hennings-Reshetikhin-Turaev functor F_λ on _λ with the m-trace  on (). We say a closed -colored bichrome graph is admissible if one of his blue edges is projective. The formula

    F'_λ(T) := _V(F_λ(T_V))
 
defines a topological invariant of the admissible closed -colored bichrome graph T thanks to Theorem 2.7 of <cit.>, where V is a projective object of , and where T_V is a cutting presentation of T, meaning an endomorphism of (+,V) in _λ whose trace is T. Then, for a fixed choice of a square root _λ∈ of Δ_- λΔ_+ λ, the formula

    '_(M,T) := _λ^-1-ℓδ_λ^-σ(L) F'_λ (L ∪ T)

defines a topological invariant of the admissible pair (M,T) thanks to Theorem 2.9 of <cit.>, where L ⊂ S^3 is an ℓ-component surgery presentation of M of signature σ(L), and where δ_λ = _λ / Δ_-λ.

 


 §.§ 2+1-TQFTs from finite-dimensional factorizable ribbon Hopf algebras

 

A finite-dimensional ribbon Hopf algebra H with R-matrix R = ∑_i=1^r a_i ⊗ b_i is factorizable if the Drinfeld map ψ_H : H^* → H, which is defined by

    ψ_H(f) := ∑_i,j=1^r f(b_ja_i) · a_jb_i
 
for every f ∈ H^*, is an isomorphism. This condition implies both non-degeneracy and unimodularity, see <cit.> and <cit.>. When H is factorizable then, as explained in Section 3 of <cit.>, '_ extends to a 2+1-TQFT _ : _→_ via the universal construction of <cit.>, where _ is the category of admissible cobordisms of dimension 2+1. More precisely, an object of _ is a triple = (,P,), where  is a closed surface, where P ⊂ is a blue -colored ribbon set, and where ⊂ H_1(;) is a Lagrangian subspace. A morphism of _ from (,P,) to (',P',') is an equivalence class of admissible triples = (M,T,n), where M is a 3-dimensional cobordism from  to ', where T ⊂ M is a -colored bichrome graph from P to P', and where n ∈ is a signature defect. A triple (M,T,n) is admissible if every component of M which is disjoint from the incoming boundary ∂_- M contains a projective blue edge of T, and two triples (M,T,n) and (M',T',n') are equivalent if n = n', and if there exists a positive diffeomorphism f : M → M' which preserves boundary identifications and satisfies f(T) = T'. Then, '_ can be extended to an invariant of closed morphisms of _ by setting

    '_(M,T,n) := δ_λ^n '_(M,T) .

Remark that the category _ thus obtained is not rigid, as objects (,P,) such that P does not contain any projective blue point are not dualizable. 


State spaces associated with objects of _ by the TQFT _ can be presented as quotients of admissible skein modules, just like we did in the CGP case, and of course this time only local skein relations are needed. However, they can also be efficiently described in terms of the dual coadjoint H-module X, which is the vector space H equipped with the action ρ_X(h)(x) := h_(2)xS^-1(h_(1)) for all h ∈ H and x ∈ X. Indeed, if V is a left H-module with action ρ_V : H →_(V), we can consider its subspace of H-invariant vectors, which is defined as 

    V^H := { v ∈ V |ρ_V(h)(v) = ε(h) · v   h ∈ H }.

Remark that we have an obvious isomorphism between _(,V) and V^H sending f to f(1). Then, if _g is a closed surface of genus g ∈, and if P_V ⊂_g is a single positive framed blue point of color V ∈, it follows directly from Corollary 3.21 of <cit.> that the state space of the object _g,V = (_g,P_V,) of _ determined by any arbitrary Lagrangian ⊂ H_1(_g;) satisfies

    _(_g,V) ≅(V^* ⊗ X^⊗ g)^H^*.






 §.§ Main results



As we mentioned earlier, this paper contains two main results related
to the non-semisimple constructions we just recalled. The first one
concerns the existence of a family of graded TQFTs, as well as graded
ETQFTs, for the CGP theory. The setting is provided by unrolled
quantum groups at odd roots of unity. More precisely, in Subsection
<ref> we recall the definition, for every
simple complex Lie algebra  of rank n and dimension
2N + n, of a particular quantum deformation, denoted
U^H_q(), of the enveloping algebra U() for
q = e^2 π i/r, where r ≥ 3 is an odd
integer which is required not to be a multiple of 3 when
= _2. These unrolled quantum groups are quite different
from the ones which usually underlie quantum constructions in low
dimensional topology. For instance, they are infinite dimensional:
indeed, they are generated by
{ E_i,F_i,H_i,K_i,K_i^-1| 1 ≤ i ≤ n }, but while
generators E_i and F_i, as well as their induced root vectors, are
set to be nilpotent, generators K_i and K_i^-1 are not required
to be 

quasi-unipotent. 
This produces a representation theory in which
weights are allowed to take arbitrary complex values, instead of
integral ones. As a consequence, we need to be careful when it comes
to defining braidings of representations. Indeed, we need the presence
of generators H_i, which should be thought of a logarithms of
generators K_i. This exponential relation is not set at the level of
the quantum group, but we restrict to representations where it is
satisfied. More precisely, we focus on the full subcategory ^H
of finite-dimensional representations of U^H_q where the
action of generators H_i is diagonalizable, and where the action of
generators K_i is obtained by exponentiating. This category is
non-semisimple, and it was studied in detail in <cit.>: a full
subcategory ^ϑ of ^H was proven to be ribbon,
and the equality ^ϑ = ^H was conjectured. In
<cit.>, it was shown that ^ϑ is non-degenerate
relative pre-modular, and thus yields a quantum invariant
_^H of admissible decorated 3-manifolds. In <cit.>,
the conjecture was proven: ^H is a relative pre-modular
category. Its structure group G is given by ^* / Λ_R,
where  is a Cartan subalgebra of  with root lattice
Λ_R. Its periodicity group Z is given by
Λ_R ∩ (r ·Λ_W), where Λ_W denotes the
weight lattice. Its critical set X is given by
{ [ξ] ∈^* / Λ_R |α∈Φ_+ : 2
⟨α,ξ⟩∈}, where Φ_+ is a set of
positive roots of . The following is our first main result,
which implies, as an immediate consequence, the existence of a
Z-graded ETQFT in dimension 1+1+1 extending the quantum invariant
_^H.



 The category ^H is relative modular.



The second main result of this paper builds a bridge between this
family of quantum invariants and the renormalized Hennings ones coming
from the corresponding small quantum groups, under the additional
assumption that (r,(A)) = 1, where A denotes the Cartan
matrix of . Indeed, in Subsection
<ref> we recall the definition of a more
classical quantum version of U(), denoted U̅_q(),
again for q = e^2 π i/r. These small
quantum groups are far better known: they are finite-dimensional, as
generators K_i and K_i^-1 are set to be

quasi-unipotent, they are
ribbon and factorizable, and thus they yield TQFTs in dimension
2+1. The category  of finite-dimensional representations
of U̅_q is still non-semisimple, but all weights take
integral values. Indeed, we have a very natural forgetful functor
Φ_ from the full subcategory ^H_[0] of
^H whose objects have all weights in Λ_R to
: the image V̅ of an object V of
^H_[0] is simply defined by forgetting the action of
generators H_i. The technical condition (r,(A))=1
ensures

Φ_ is essentially
surjective. Furthermore, Φ_ immediately induces a functor
Φ_ from _^H_[0] to
_: the image T̅ of a morphism T of
_^H_[0] is simply defined by applying the
forgetful functor Φ_ to all its colors.



 If M is a closed 3-manifold and T ⊂ M is an admissible ^H_[0]-colored ribbon graph, then
 
    _^H(M,T,0) = '_(M,T̅).




We point out that each invariant depends on the choice of a square
root of the product of the stabilization coefficients for the
corresponding version of the quantum group. Theorem
<ref> requires a coherent choice of these square roots
in the two theories, otherwise a sign will appear in the
relation. Remark also that a great advantage of the renormalized
Hennings theory is the absence of many technical complications which
characterize the CGP one. For instance, arbitrary surgery
presentations can be used to define and compute quantum invariants
associated with small quantum groups, as we have no computability
condition. Therefore, Theorem <ref> gives a convenient
alternative formulation for the CGP invariants associated with
unrolled quantum groups, at least in the case of 3-manifolds decorated
with the trivial cohomology class.


Let us end this introduction with a final comment 

about our choice for the setting of Theorem
<ref>. When we started writing these results, the
renormalized Hennings construction had only been developed in the case
of finite-dimensional factorizable ribbon Hopf algebras. As a
consequence, a comparison with the CGP construction had to take place
in this context.

The classification of finite-dimensional factorizable quantum groups is
not so easy, see <cit.>, and we choose for simplicity to limit 
ourselves to the smallest set of Cartan generators { K_i | 1 ≤ i ≤ n }
at odd roots of unity.




However, a recent generalization of the renormalized Hennings
construction allows us to consider arbitrary modular categories, in
the non-semisimple sense, as building blocks for non-semisimple
2+1-TQFTs <cit.>. This larger setting encompasses examples of
ribbon categories coming from the representation theory of restricted
quantum groups at even roots of unity. Indeed, while

some of these Hopf algebras might not be braided <cit.>, a
suitable modification of their coalgebra structure produces quasi-Hopf
deformations which admit a ribbon structure
<cit.>. For what concerns unrolled quantum groups,
even roots of unity have been less studied due to several technical
difficulties which produce fascinating but complicated phenomena in
the theory. It would be very interesting to generalize Theorem
<ref> to this setting, and possibly to even larger ones.




§ QUANTUM GROUPS AT ODD ROOTS OF UNITY



In this section we recall definitions of small and unrolled quantum groups associated with arbitrary simple complex Lie algebras , and we prove our first result: categories of finite-dimensional weight representations of unrolled quantum groups at odd roots of unity are relative modular, and can therefore be used to construct ETQFTs in dimension 1+1+1.




 §.§ Small quantum groups



Let  be a simple complex Lie algebra of rank n and dimension 2N + n, let  be a Cartan subalgebra of , let Φ_+ be a set of positive roots of , and let _q be the associated quantum group, over a formal parameter q, introduced in Appendix <ref>. Let us fix an odd integer r ≥ 3 with the further condition that r ≢0 modulo 3 if = _2, and let us specialize q to e^2 π i/r in the De Concini-Kac sense 

<cit.>. Let U̅_q denote the small quantum group of , which is the -algebra obtained from _q by adding relations

    K_μ = 1,    E_α^r = F_α^r = 0

for every μ∈Λ_R ∩ r ·Λ_W and every α∈Φ_+. Then U̅_q inherits from _q the structure of a Hopf algebra, and we denote with U̅_q, with U̅_q _+, and with U̅_q _- the subalgebras of U̅_q generated by { K_i | 1 ≤ i ≤ n }, by { E_i | 1 ≤ i ≤ n }, and by { F_i | 1 ≤ i ≤ n }, respectively. As proved in <cit.>, see also Theorem 30 of <cit.>, a Poincaré-Birkhoff-Witt basis is given by

    {( ∏_k = 1^N F_β_k^c_k) K_μ( ∏_k = 1^N E_β_k^b_k) | 
     [ μ∈Λ_R / (Λ_R ∩ r ·Λ_W),;      0 ≤ b_1,…,b_N < r,;       0 ≤ c_1,…,c_N < r ]},

so U̅_q is finite-dimensional. A pivotal element is given by K_2 ·ρ∈U̅_q where

    ρ := 1/2·∑_k=1^N β_k.

Furthermore, if we consider R̅_0 ∈U̅_q ⊗U̅_q given by

    R̅_0 := 1/| Λ_R / (Λ_R ∩ r ·Λ_W) |·∑_μ,μ' ∈Λ_R / (Λ_R ∩ r ·Λ_W) q^- ⟨μ,μ' ⟩· 
     K_μ⊗ K_μ'

and Θ̅∈U̅_q _+ ⊗U̅_q _- given by

    Θ̅ := ∑_b_1,…,b_N=0^r-1( ∏_k=1^N { 1 }_β_k^b_k/[b_k]_β_k! q^b_k(b_k-1)/2) ·( ∏_k=1^N E_β_k^b_k) ⊗( ∏_k=1^N F_β_k^b_k)

then R̅ := R̅_0 Θ̅∈U̅_q ⊗U̅_q is an R-matrix for U̅_q, as proved in <cit.>, see also <cit.>. Next, thanks to Proposition A.5.1 of <cit.>, a right integral λ of U̅_q is given by

    λ( ( ∏_k = 1^N F_β_k^c_k) K_μ( ∏_k = 1^N E_β_k^b_k) ) = q^- 4 ⟨ρ,ρ⟩δ_μ,2 ·ρ∏_k=1^N δ_b_k,r-1∏_k=1^N δ_c_k,r-1.

This formula can be deduced from the one in <cit.> by remarking that Lyubashenko uses Luszitg's coproduct Δ̃ := (ω⊗ω) ∘Δ^∘ω, where ω denotes the involutive algebra automorphism of U̅_q defined by ω(E_i) = F_i and by ω(K_i) = K_i^-1 for all integers 1 ≤ i ≤ n, and by remarking that λ is a right integral for Δ if and only if λ∘ω is a left integral for Δ̃. Finally, thanks to Proposition A.5.2 of <cit.>, a two-sided cointegral Λ of U̅_q satisfying λ(Λ) = 1 is given by

    Λ := ∑_μ∈Λ_R / (Λ_R ∩ r ·Λ_W) q^2 ⟨μ,ρ⟩·( ∏_k = 1^N F_β_k^r-1) K_μ( ∏_k = 1^N E_β_k^r-1).




 The Hopf algebra U̅_q is factorizable and ribbon.



This result is proved in <cit.>, see also <cit.>. We denote with  the ribbon category of finite-dimensional U̅_q-modules, and with  the m-trace on () given by Theorem 1 of <cit.>, which satisfies _U̅(Λ∘ε) = 1 for the regular representation U̅ of U̅_q.



 The renormalized Hennings invariant '_ extends to a TQFT
 
    _ : _→_.






 §.§ Unrolled quantum groups



Let U^H_q denote the unrolled quantum group of , which is the -algebra obtained from _q by adding generators

    { H_i | 1 ⩽ i ⩽ n }

and relations

    [H_i,H_j] = [H_i,K_j] = 0,    [H_i,E_j] = a_ij E_j,    [H_i,F_j] = - a_ij F_j,    E_α^r = F_α^r = 0

for every integer 1 ⩽ i,j ⩽ n and every positive root α∈Φ_+. Then U^H_q can be made into a pivotal Hopf algebra by setting

    Δ(H_i) =  H_i ⊗ 1 + 1 ⊗ H_i,   ε(H_i) = 0,    S(H_i) = -H_i

for every integer 1 ⩽ i ⩽ n, and we denote with U^H_q, with U^H_q _+, and with U^H_q _- the subalgebras of U^H_q generated by { H_i | 1 ⩽ i ⩽ n }, by { E_i | 1 ⩽ i ⩽ n }, and by { F_i | 1 ⩽ i ⩽ n } respectively. For every z ∈ let us introduce the notation

    q^z := e^z 2 π i/r,   { z } := q^z - q^-z.

A U^H_q-module V with action ρ_V : U^H_q →_(V) is a weight module if it is a semisimple U^H_q-module and if for every μ∈^* and every v ∈ V we have

    ρ_V(H_i)(v) = μ(H_i) · v    1 ⩽ i ⩽ n   ⇒  ρ_V(K_i)(v) = q_i^μ(H_i)· v    1 ⩽ i ⩽ n,

where we are identifying  with the corresponding linear subspace of U^H_q in the obvious way. We denote with ^H the full subcategory of the category of finite-dimensional U^H_q-modules whose objects are weight modules. Then ^H can be made into a ribbon category as follows: first of all, a pivotal element is given by K_2 ·ρ^1-r∈ U_q^H, where the choice of the exponent is explained in Remark 4 of <cit.>. Furthermore, if V and V' are objects of ^H, their braiding morphism is given by 

    [                                     c_V,V' :                                       V ⊗ V'                                            →                                       V' ⊗ V;                                                                                    v ⊗ v'                                            ↦ τ_V,V'(R^H_0,V,V'((ρ_V ⊗ρ_V')(Θ^H)(v ⊗ v'))) ]

for the linear maps R^H_0,V,V' : V ⊗ V' → V ⊗ V' and τ_V,V' : V ⊗ V' → V' ⊗ V determined by

    R^H_0,V,V'(v ⊗ v') := q^⟨ν,ν' ⟩· v ⊗ v',   τ_V,V'(v ⊗ v') := v' ⊗ v

for all v ∈ V, v' ∈ V' satisfying

    ρ_V(H_i)(v) = ν(H_i) · v,   ρ_V'(H_i)(v') = ν'(H_i) · v'

for every integer 1 ⩽ i ⩽ n, and for Θ^H = Θ̅∈ U^H_q _+ ⊗ U^H_q _- ≅U̅_q _+ ⊗U̅_q _-. Thanks to Theorem 4 of <cit.>, ^H is a ribbon category.


If we set G := ^* / Λ_R, then ^H supports the structure of a G-category: indeed, for every γ∈^* we can define the homogeneous subcategory ^H_[γ] to be the full subcategory of ^H with objects given by modules whose weights are all of the form γ + μ for some μ∈Λ_R. Furthermore, if we set := Λ_R ∩ (r ·Λ_W), then we have a free realization σ : →^H_[0] mapping every κ∈ to the object σ(κ) ∈^H_[0] given by the vector space  with U^H_q-action specified by

    ρ_σ(κ)(H_i)(1) := κ(H_i),   ρ_σ(κ)(E_i)(1) := 0,   ρ_σ(κ)(F_i)(1) := 0

for every integer 1 ⩽ i ⩽ n.







Now the bilinear map 

    [      ψ :      G ×        →       ^*;           ([γ],κ)        ↦ q^2⟨γ,κ⟩ ]

satisfies c_σ(κ),V∘ c_V,σ(κ) = ψ([γ],κ) ·𝕀_V ⊗σ(κ) for every γ∈^*, every V ∈^H_[γ], and every κ∈. If we consider the critical set

    X := { [ξ] ∈^* / Λ_R |α∈Φ_+ : 2 ⟨α,ξ⟩∈}

then, as explained in Section 7 of <cit.>, the category ^H_[γ] is semisimple for every [γ] ∈ G ∖ X. Therefore, the last relevant piece of structure we are missing is an m-trace. In order to define it, let us introduce typical U^H_q-modules. First of all, we say a vector v_+ of a U^H_q-module V is a highest weight vector if ρ_V(E_i)(v_+) = 0 for every integer 1 ⩽ i ⩽ n. Analogously, we say a vector v_- of V is a lowest weight vector if ρ_V(F_i)(v_-) = 0 for every integer 1 ⩽ i ⩽ n. Then for every weight μ∈^* there exists a simple finite-dimensional weight U^H_q-module V_μ featuring a highest weight vector of weight μ. This module is unique up to isomorphism, and every simple weight U^H_q-module is of this form, see Proposition 33 of <cit.>. Every such module also has a lowest weight vector, and it is called typical if its lowest weight is given by μ - 2(r-1) ·ρ. If we consider the set 

    ^* := {γ∈^* | 2 ⟨α,γ + ρ⟩ + m ⟨α,α⟩∉r  α∈Φ_+,   1 ⩽ m ⩽ r-1 }

then, thanks to Proposition 34 of <cit.>, V_γ is typical if and only if γ∈^*. Remark that if γ∈^* satisfies 2 ⟨α,γ⟩∉ for every α∈Φ, then γ∈^*. This means that if γ∈^* satisfies [γ] ∉X, then V_γ is typical. We also point out that, although [(r-1) ·ρ] ∈ X, the module V_(r-1) ·ρ is always typical, because

    2 ⟨α,(r-1) ·ρ + ρ⟩ + m ⟨α,α⟩ = 2r ⟨α,ρ⟩ + m d_α

is not in r for any integer 1 ≤ m ≤ r-1. Now, thanks to Lemma 7.1 of <cit.> and Theorem 38 of <cit.>, every typical U^H_q-module is projective and ambidextrous. Then, by combining Theorem 3.3.2 of <cit.> with Lemma 17 of <cit.>, there exists a non-zero m-trace on the ideal (^H) of projective objects of ^H which is unique up to scalar. Therefore, we can fix the normalization ^H(V_(r-1) ·ρ) = 1. Thanks to Equation (51) and Lemma 47 of <cit.>, for every μ∈^* we have

    ^H(V_μ) = ∏_k=1^N r {⟨μ - (r-1) ·ρ,β_k ⟩}/{ r⟨μ - (r-1) ·ρ,β_k ⟩}.

For all μ,ν∈^*, if f_μ,ν^+ := F_^H(T_μ,ν^+) and f_μ,ν^- := F_^H(T_μ,ν^-) for the ^H-colored ribbon graphs T_μ,ν^+ and T_μ,ν^- represented in Figure <ref>, Proposition 45 of <cit.> gives

    ^H_V_ν(f_μ,ν^+) = r^N q^2 ⟨μ - (r-1) ·ρ,ν - (r-1) ·ρ⟩,

and, since V_μ^* ≅ V_2(r-1) ·ρ - μ, it also gives

    ^H_V_ν(f_μ,ν^-) = r^N q^-2 ⟨μ - (r-1) ·ρ,ν + (1-r) ·ρ⟩
     = r^2N^H_V_ν(f_μ,ν^+)^-1.


 





 §.§ Relative modularity



In this subsection we will prove the category ^H is relative modular, and thus yields a -graded TQFT. In order to do this, we will first need a preliminary definition. We say an endomorphism f ∈_^H(V) of an object V of ^H_[0] is transparent in ^H_[0] if for all objects U,W ∈^H_[0] we have 

    𝕀_U ⊗ f = c_V,U∘ (f ⊗𝕀_U) ∘ c_U,V,   
     f ⊗𝕀_W = c_W,V∘ (𝕀_W ⊗ f) ∘ c_V,W.




 If f ∈_^H(V) is transparent in ^H_[0], then there exist some integer m and some morphisms g_i ∈_^H(V,σ(κ_i)) and h_i ∈_^H(σ(κ_i),V) for every integer 1 ⩽ i ⩽ m such that
 
    f = ∑_i=1^m h_i ∘ g_i.




 If v_+ is a highest weight vector of V_(r-1) ·ρ and v is a weight vector of V then c_V_(r-1) ·ρ,V(v_+ ⊗ v) is proportional to v ⊗ v_+ because
 
    ρ_V_(r-1) ·ρ( ∏_k=1^N E_β_k^b_k)(v_+) = 0

 for all integers 0 ≤ b_1,…,b_N < r whose sum is strictly positive. Furthermore, c_V,V_(r-1) ·ρ(f(v) ⊗ v_+) is proportional to v_+ ⊗ f(v) because f is transparent in ^H. But now
 
    {ρ_V_(r-1) ·ρ( ∏_k=1^N F_β_k^b_k)(v_+) | 0 ≤ b_1,…,b_N < r }

 is a basis of V_(r-1) ·ρ thanks to Proposition 34 of <cit.>. This means that
 
    ρ_V ( ∏_k=1^N E_β_k^b_k)(f(v)) = 0

 for every weight vector v ∈ V and for all integers 0 ≤ b_1,…,b_N < r whose sum is strictly positive.
 
 
 Analogously, if v_- is a lowest weight vector of V_(r-1) ·ρ and v is a weight vector of V then  c_V,V_(r-1) ·ρ(v ⊗ v_-) is proportional to v_- ⊗ v because
 
    ρ_V_(r-1) ·ρ( ∏_k=1^N F_β_k^b_k)(v_-) = 0

 for all integers 0 ≤ b_1,…,b_N < r whose sum is strictly positive.  Furthermore, c_V_(r-1) ·ρ,V(v_- ⊗ f(v)) is proportional to f(v) ⊗ v_- because f is transparent in ^H. But now
 
    {ρ_V_(r-1) ·ρ( ∏_k=1^N E_β_k^b_k)(v_-) | 0 ≤ b_1,…,b_N < r }

 is a basis of V_(r-1) ·ρ thanks to Proposition 34 of <cit.>. This means that
 
    ρ_V ( ∏_k=1^N F_β_k^b_k)(f(v)) = 0

 for every weight vector v ∈ V and for all integers 0 ≤ b_1,…,b_N < r whose sum is strictly positive.
 
 
 Now, since K_i - K_i^-1 = (q_i-q_i^-1) · [E_i,F_i] for every integer 1 ⩽ i ⩽ n, we get the equality ρ_V(K_i)(f(v)) - ρ_V(K_i^-1)(f(v)) = 0 for every v ∈ V, which implies 
 
    ρ_V(K_i)^2(f(v)) = f(v).

 Then, if f(v) is a weight vector of weight κ, this tells us that 2 ⟨κ,α_i ⟩∈ r for every integer 1 ⩽ i ⩽ n. Since r is odd and coprime with d_i for every integer 1 ⩽ i ⩽ n, this means precisely that κ∈. Therefore, each weight vector of f determines a 1-dimensional submodule which is isomorphic to σ(κ) for some κ∈. Since f is a direct sum of its weight spaces, this means that
 
    f ≅⊕_i=1^m σ(κ_i)

 for some integer m ⩾ 1 and some κ_1, …, κ_m ∈. Let π_i ∈_^H( f,σ(κ_i)) and ι_i ∈_^H(σ(κ_i), f) denote the corresponding projection and injection morphisms for every integer 1 ⩽ i ⩽ m. We can factorize f = ι_f ∘π_f where π_f ∈_^H(V, f) is naturally induced by f and ι_f ∈_^H( f,V) denotes inclusion. Then the result follows by setting g_i := π_i ∘π_f and h_i := ι_f ∘ι_i for every integer 1 ⩽ i ⩽ m.



Let us complete { 0 }⊂Λ_R to a set
_r ⊂Λ_R of representatives of equivalence classes
in Λ_R /. Similarly, let us choose a set
I_G ∖ X⊂^* of representatives of
equivalence classes in G ∖ X. If for every
γ∈ I_G ∖ X we define
I_[γ] := {γ} + _r, then

    Θ(_[γ]) := { V_μ∈_[γ]|μ∈ I_[γ]}

is a set of representatives of -orbits of isomorphism classes of simple objects of _[γ]. We have now everything in place to prove Theorem <ref>.






 We know ^H is a non-degenerate relative pre-modular category thanks to Theorem 7.2 of <cit.> and to Theorem 4 of <cit.>. Therefore, we only need to prove that ^H satisfies the relative modularity condition of Definition <ref>. We will do this by showing the skein equivalence of Figure <ref> for every [γ] ∈ G ∖ X, for every μ∈ I_G ∖ X, and for every ν∈ I_[μ], so let f_[γ],μ,ν denote the morphism of ^H obtained by applying the Reshetikhin-Turaev functor F_^H to the ^H-colored ribbon graph represented in the left hand part of Figure <ref>, ignoring the coefficient. Thanks to the handle slide property, we have the skein equivalence of Figure <ref>,
  
 which means f_[γ],μ,ν is transparent in ^H_[0]. Now, thanks to Lemma <ref>, we have
 
    f_[γ],μ,ν = ∑_i=1^m h_[γ],μ,ν,i∘ g_[γ],μ,ν,i

 with g_[γ],μ,ν,i∈_^H(V_μ⊗ V_ν^*,σ(κ_i)), with h_[γ],μ,ν,i∈_^H(σ(κ_i),V_μ⊗ V_ν^*), and with κ_i ∈ for every integer 1 ≤ i ≤ m. But now h_[γ],μ,ν,i = 0 unless μ =ν and κ_i = 0, because
 
    _^H (V_μ⊗ V_ν^*,σ(κ_i)) ≅_^H (V_μ,V_ν⊗σ(κ_i))

 and because _r is a set of representatives of equivalence classes in Λ_R /. This means that f_[γ],μ,μ factors through the tensor unit . But now, since V_μ is simple, both _^H (V_μ⊗ V_μ^*,) and _^H (, V_μ⊗ V_μ^*) are 1-dimensional. This means f_[γ],μ,μ is a scalar multiple of _V_μ∘_V_μ. In order to compute the proportionality coefficient let us compare the m-traces of _V_μ∘_V_μ and of f_[γ],μ,μ. The first m-trace is easily seen to be
 
    ^H_V_μ⊗ V_μ^*(_V_μ∘_V_μ) = ^H_V_μ(𝕀_V_μ) = ^H(V_μ).

 Indeed, this follows immediately from the partial trace property of ^H. On the other hand, if
 
    f_[γ],μ,μ = ∑_ν∈{γ} + _r^H(V_ν) · f_ν,μ,μ,

 where f_ν,μ,μ is obtained from f_[γ],μ,μ by replacing the label Ω_[γ] of the meridian with V_ν, then, the second m-trace is given by
 
    ^H_V_μ⊗ V_μ^*(f_[γ],μ,μ) 
         = ∑_ν∈{γ} + _r^H(V_ν) ^H_V_μ⊗ V_μ^*(f_ν,μ,μ) 
       = ∑_ν∈{γ} + _r^H(V_ν) ^H_V_ν(f_μ,ν^- ∘ f_μ,ν^+) 
       = ∑_ν∈{γ} + _r^H(V_ν) ^H(V_ν)^-1^H_V_ν(f_μ,ν^-) ^H_V_ν(f_μ,ν^+) 
       = r^2N| _r |,

 where the morphisms f_μ,ν^- and f_μ,ν^+ are represented in Figure <ref>. Indeed, the second equality follows from both the cyclicity and the partial trace properties of ^H, using isotopy, the third equality follows from the fact that V_ν is simple, and the fourth equality follows from the very last equation of Subsection <ref>.




 The CGP invariant _^H extends to a -graded TQFT
 
    _^H^ : _^H^G →_^.






 §.§ Projective generators and forgetful functor



In this subsection we prove some key technical results which will be later used for the proof Theorem <ref>. We say an object P of  is a projective generator of  if for every object V of () there exist some integer m and some morphisms f_i ∈_(V,P) and g_i ∈_(P,V) for every integer 1 ⩽ i ⩽ m such that

    𝕀_V = ∑_i=1^m g_i ∘ f_i.

Remark that a natural choice for a projective generator of  is the regular representation U̅ of U̅_q. Analogously, we say an object P of ^H_[0] is a relative projective generator of ^H_[0] if for every object V of (^H_[0]) there exist some integer m, some κ_i ∈, and some morphisms f_i ∈_^H(V,P ⊗σ(κ_i)) and g_i ∈_^H(P ⊗σ(κ_i),V) for every integer 1 ⩽ i ⩽ m such that

    𝕀_V = ∑_i=1^m g_i ∘ f_i.




Next, we apply the following result to ^H.



 Let  be an abelian relative pre-modular G-category over an algebraically closed field . Then  has enough projectives, and every indecomposable projective object is a projective cover of a simple object.




 Choose some g ∈ G ∖ X and some simple object V ∈_g. By definition _g is semisimple, and so V is projective and has epic evaluation. Then for every W ∈ the morphism
 
    𝕀_W⊗_V∈_(W ⊗ V ⊗ V^*,W).

 is an epimorphism from a projective object to W. Thus  has enough projectives. Next, for the second statement, let P be an indecomposable projective object of _h for some h ∈ G. Since X is small in G, there exists some g ∈ G satisfying g+h ∈ G ∖ X. Then, let V be a simple object of _g. The algebra _(P) embeds into _ (P ⊗ V) ⊂_N() with N bounded by the number of simple summands of P ⊗ V. Then 
 Fitting's Lemma applies to _(P), i.e. its elements are either nilpotent or  isomorphisms. We can now prove that P has a unique maximal proper subobject. To see this, let us suppose by contradiction M_1 and M_2 are distinct maximal proper subobjects. Then we have an epimorphism p ∈_^H(M_1 ⊕ M_2,P) which admits a section (f_1,f_2) ∈_^H(P,M_1 ⊕ M_2), because P is projective. If i_j ∈_^H(M_j,P) denotes the inclusion and f̃_j := i_j ∘ f_j, then f̃_1 + f̃_2 = 𝕀_P. Since their sum is the identity of P, they commute, and f̃_1 and f̃_2 cannot be simultaneously nilpotent. Thanks to Fitting's Lemma, f̃_j is an isomorphism for some j. This implies that P is a retract of M_j, and thus P is a proper subobject of P. Then P ⊗ V is a proper subobject of itself, but P ⊗ V is also semisimple, which is a contradiction. 



We will denote by P_μ a projective cover of V_μ for any μ∈^*.






























Thanks to Proposition <ref>, if _r denotes the set of representatives of equivalence classes in Λ_R / of Figure <ref>, then

    := ⊕_μ∈_r P_μ

is by construction a relative projective generator of ^H_[0].



 _( _^H(,σ(κ)) ) = _( _^H(σ(κ),) ) = δ_0,κ for every κ∈.




 Since the vector space of U^H_q-module morphisms from a projective indecomposable weight U^H_q-module to its unique simple quotient is 1-dimensional, and since {σ(κ) ∈^H_[0]|κ∈} is the -orbit of the tensor unit ,

 we have
 
    _( _^H(,σ(κ)) ) = δ_0,κ.

 Furthermore, since the dual of an indecomposable projective weight U^H_q-module is also indecomposable and projective, see for instance Proposition
6.1.3 of <cit.>, Proposition <ref> implies that each indecomposable projective weight U^H_q-module P has an unique simple submodule which is dual to the simple quotient of P^*. Since ^H is unimodular, see <cit.> and Theorem 3.1.3 of <cit.>, P_0 is the unique indecomposable projective weight U^H_q-module that constains the tensor unit = σ(0) as a submodule, and similarly P_0 ⊗σ(κ) is the unique indecomposable projective weight U^H_q-module that contains σ(κ) as a submodule. Hence we have
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    _( _^H(σ(κ),) )= δ_0,κ.




The following result establishes a cutting property for Kirby meridians which is analogous to Lemma 3.6 of <cit.>.



 There exist generators ∈_^H(,) and ∈_^H(,) satisfying ^H_(∘) = 1 which realize the skein equivalence of Figure <ref>.







 Thanks to Lemma <ref>, and thanks to the non-degeneracy of ^H, the composition of a non-trivial morphism of _^H(,) with a non-trivial morphism of _^H(,) has non-zero m-trace. Therefore, let us fix a pair of generators ∈_^H(,) and ∈_^H(,) satisfying ^H_(∘) = 1, and let us prove they realize the skein equivalence of Figure <ref>. If h_[γ] is the morphism of ^H obtained by applying the Reshetikhin-Turaev functor F_^H to the ^H-colored ribbon graph represented in the left hand part of Figure <ref>, the handle slide property yields the skein equivalence represented in Figure <ref>.
 
 This means that, thanks to Lemmas <ref> and <ref>, the morphism h_[γ] factors through the tensor unit . But then, since _^H (,) and _^H (,) are both 1-dimensional, we must have h_[γ] = α·∘ for some α∈. Then, let us show α = r^2N| _r |. Proposition <ref> implies _^H(P_μ,) is δ_μ,0-dimensional for every μ∈_r. 
 
 Moreover, for every μ∈^*, the projective cover P_0 of V_0 = is a direct summand of V_μ⊗ V_μ^* of multiplicity 1. In other words, we have
 
    𝕀_V_μ⊗ V_μ^* = ∑_i=0^m ι_μ,i∘π_μ,i

 for some weights μ_1,…,μ_m ∈Λ_R satisfying μ_0 = 0 and μ_i ≠ 0 for all integers 1 ≤ i ≤ m, and for some U^H_q-module morphisms π_μ,i∈_^H(V_μ⊗ V_μ^*,P_μ_i) and ι_μ,i∈_^H(P_μ_i,V_μ⊗ V_μ^*). Remark that this implies
 
    _V_μ∘_V_μ = _V_μ∘_V_μ∘ι_μ,0∘π_μ,0.

 Therefore, let us fix some μ∈ I_G ∖ X, and let us consider the U^H_q-module morphisms f_μ := ι_μ,0∘π_0 ∈_^H(,V_μ⊗ V_μ^*) and g_μ := ι_0 ∘π_μ,0∈_^H(V_μ⊗ V_μ^*,) determined by the projection π_0 ∈_^H(,P_0) and the injection ι_0 ∈_^H(P_0,). Thanks to Figure <ref>, we have the skein equivalence of Figure <ref>.
 
 Then, let us compute the m-trace of h_[γ] in two different ways. On one hand, we have
 
    ^H_(h_[γ])    = α^H_(∘) = α.

 On the other hand, we have
 
    ^H_(h_[γ])    = ^H(V_μ)^-1 r^2N| _r | ^H_( g_μ∘_V_μ∘_V_μ∘ f_μ) 
       = ^H(V_μ)^-1 r^2N| _r | ^H_V_μ⊗ V_μ^*(_V_μ∘_V_μ∘ f_μ∘ g_μ) 
       = ^H(V_μ)^-1 r^2N| _r | ^H_V_μ⊗ V_μ^*(_V_μ∘_V_μ∘ι_μ,0∘π_μ,0) 
       = ^H(V_μ)^-1 r^2N| _r | ^H_V_μ⊗ V_μ^*(_V_μ∘_V_μ) 
       = ^H(V_μ)^-1 r^2N| _r | ^H_V_μ(𝕀_V_μ) 
       = r^2N| _r |,

 where the second and fifth equalities follow from the cyclicity and the partial trace properties of ^H respectively. 


 
Let us consider the forgetful functor Φ_ : ^H_[0]→ which forgets the action of H_i for all integers 1 ⩽ i ⩽ n. If V is an object of ^H_[0] we denote with V̅ its image under Φ_, and if f is a morphism of ^H_[0] we denote with f̅ its image under Φ_. This induces a ribbon functor Φ_ : _^H_[0]→_ from the category of ^H_[0]-colored ribbon graphs to the category of -colored ribbon graphs. If T is a morphism of _^H_[0] we denote with T̅ its image under Φ_. 



 The forgetful functor Φ_ : ^H_[0]→ is ribbon, and it satisfies
 
    Φ_∘ F_^H_[0] = F_∘Φ_.





 First of all, remark that K_2 ·ρ^1-r = K_2 ·ρ in U̅_q. Then the result follows immediately from the equality 
 
    R^H_0,V,V' = (ρ_V  ⊗ρ_V')(R̅_0)

 for all V, V' ∈^H_[0], where R^H_0,V,V' : V ⊗ V' → V ⊗ V' is defined in Subsection <ref>, and where R̅_0 ∈U̅_q ⊗U̅_q is defined in Subsection <ref>. To show the claim, remark
 
    ∑_μ' ∈Λ_R / Z q^⟨μ,μ' ⟩ = | Λ_R / Z | δ_μ,0

 for every μ∈Λ_R. This means that
 
    (ρ_V ⊗ρ_V')(R̅_0)(v ⊗ v')
         = 1/| Λ_R / Z |·∑_μ,μ' ∈Λ_R / Z q^- ⟨μ,μ' ⟩·ρ_V(K_μ)(v) ⊗ρ_V'(K_μ')(v') 
       = 1/| Λ_R / Z |·∑_μ,μ' ∈Λ_R / Z q^⟨μ,ν⟩ + ⟨μ',ν' ⟩ - ⟨μ,μ' ⟩· v ⊗ v' 
       = ∑_μ∈Λ_R / Zδ_μ,ν' q^⟨μ,ν⟩· v ⊗ v' 
       = q^⟨ν,ν' ⟩· v ⊗ v'

 for all v ∈ V, v' ∈ V' satisfying
 
    ρ_V(H_i)(v) = ν(H_i) · v,   ρ_V'(H_i)(v') = ν'(H_i) · v'

 for every integer 1 ⩽ i ⩽ n.



The forgetful functor Φ_ preserves the property of being projective.



 If P is a projective object of ^H_[0], then P̅ is a projective object of .




 The typical U^H_q-module V_(r-1) ·ρ introduced in Subsection <ref> generates (^H_[0]) thanks to Lemma 17 of <cit.>. Then P must be a direct summand of a tensor product V_(r-1) ·ρ⊗ W for some W ∈^H_[0]. Now the proof of Lemma 7.1 of <cit.> can be repeated to show the image V̅_(r-1) ·ρ of V_(r-1) ·ρ under the forgetful functor Φ_ is projective. This means V̅_(r-1) ·ρ generates (), and thus P̅, which is a direct summand of V̅_(r-1) ·ρ⊗W̅, is projective.



In particular, the image  of the relative projective generator  of ^H_[0] defined in Subsection <ref> is projective.



 _( _(,) ) = _( _(,) ) = 1.




 Let V be a weight U^H_q-module in ^H_[0], and remark that its image V̅ under Φ_ coincides with V as a vector space. Let us consider the space V̅^U̅_q of U̅_q-invariants vectors of V̅. Remark that _(,V̅) is naturally isomorphic to V̅^U̅_q, simply by identifying every morphism f ∈_(,V̅) with the image f(1) ∈V̅^U̅_q. We claim the subspace V^U̅_q of V formed by vectors of V̅^U̅_q is a U^H_q-submodule of V. Indeed, this follows from the commutation relations satisfied by the additional generators H_1, …, H_n. But now remark that every weight vector of V^U̅_q with respect to the action of U^H_q determines a split 1-dimensional submodule of V^U̅_q which is isomorphic to σ(κ) for some κ∈, as all 1-dimensional weight U^H_q-modules in ^H_[0] are. This means that
 
    V^U̅_q ≅⊕_i=1^_(V̅^U̅_q )σ(κ_i)

 with κ_i ∈ for every integer 1 ⩽ i ⩽_(V̅^U̅_q ). Thus we get 
 
    _(V̅^U̅_q ) ≤_( ⊕_κ∈_^H( σ(κ),V ) ),

and the converse inequality follows from the equality Φ_(σ(κ)) = for every κ∈.


 First, let us consider V =. Thanks to Lemma <ref>, we have 
 
    _( ⊕_κ∈_^H( σ(κ),) ) = 1.
  
 Thus, the space ^U̅_q is 1-dimensional. This means 
 
    _( _(,) ) = 1.

 Next, let us consider V = ^*. Thanks to Lemma <ref>, we have 
 
    _( ⊕_κ∈_^H( σ(κ),^* ) ) = _( ⊕_κ∈_^H( ,σ(-κ) ) ) = 1.
  
 Thus, the space (^*)^U̅_q is 1-dimensional. This means
 
    _( _(,) ) = _( _(,^*) ) = 1.




The last result we will need requires an additional hypothesis.



 If (r,(A)) = 1, then  is a projective generator of .











  Let us show that every indecomposable projective
  U̅_q-module P is isomorphic to a direct summand of
  . Let V be the unique simple quotient of P. Its
  highest weight is a ring homomorphism
  φ : U̅_q → assigning to each K_i the root
  of unity by which it acts on the highest weight vector of V. Since
  r is coprime with d_1, …, d_n defined in Appendix
  <ref>, there exists some ω∈Λ_W
  satisfying φ(K_i)=q^ω,α_i for every
  integer 1 ≤ i ≤ n. Furthermore, since r is coprime also
  with (A), the matrix (d_ia_ij)_1 ≤ i,j ≤ n is
  invertible modulo r. In particular, there exist
  ω̃_i ∈Λ_R such that
  ω̃_i,α_j≡δ_ij modulo r for
  all integers 1 ≤ i,j ≤ n. Then, if we set
 
    ω̃ := ∑_i=1^n ⟨ω,α_i ⟩·ω̃_i ∈Λ_R,

 the simple weight U^H_q-module V_ω̃∈^H_[0] satisfies Φ_(V_ω̃) ≅ V,
 
 
 because Φ_(V_ω̃) and V have the same highest weight. Moreover, if P_ω̃ is a projective cover of V_ω̃, then P_ω̃⊗σ(κ) is isomorphic to a direct summand of  for some κ∈. But now, since Φ_(P_ω̃⊗σ(κ)) is projective, it must contain P as a direct summand, so this proves our statement. 














§ EQUALITY OF 3-MANIFOLD INVARIANTS



The goal for this section is to prove Theorem <ref>. We will use as a key ingredient the fact that meridians labeled with Kirby colors have the cutting property with respect to the relative projective generator  of ^H_[0], while red meridians labeled with the regular representation have the cutting property with respect to its image  in . The proof will require a comparison of all the ingredients that correspond to each other in the two theories. Since it is part of the hypotheses of Theorem  <ref>, we will suppose throughout this section that (r,(A)) = 1.




 §.§ Stabilized surgery presentations



In this subsection we introduce special surgery presentations of admissible decorated closed 3-manifolds which are tailored for the comparison between the CGP and the renormalized Hennings invariants. We start with a preliminary comment about our notation: we will always denote with F_ the restriction of F_λ to -colored blue ribbon graphs, and similarly we will always denote with F'_ the restriction of F'_λ to admissible closed -colored blue ribbon graphs, in order to stress the absence of red edges. Next, we need to compare the m-trace ^H on (^H) with the m-trace  on (). 



Both ^H and  are unique up to scalar, but the chosen normalizations do not agree, as they are determined by the conditions

    ^H_(∘) = 1,   _U̅(Λ∘ε) = 1

respectively, where  is the relative projective generator of ^H_[0] introduced in Subsection <ref>, where ∈_^H(,) and ∈_^H(,) are the morphisms introduced in Lemma <ref>, where U̅ is the regular representation of U̅_q, and where ε and Λ are the counit and the cointegral respectively.

Nevertheless, ∘Φ_ clearly defines an m-trace
on (^H_[0]): indeed, it obviously satisfies the cyclicity property, and the partial trace property is just a consequence of the fact that partial
traces commute with the ribbon functor Φ_ (see also Corollary 2.8 of
<cit.> for a similar statment in the setting of
finite dimensional Hopf algebras).

Therefore, 
there exists a non-zero coefficient α∈^* such that

    ^H^H_[0] = α·∘Φ_,

meaning that ^H_V(f) = α_V̅(f̅) for every V ∈(^H_[0]) and every f ∈_^H(V). 




 The renormalized invariants F'_^H_[0] and F'_ satisfy
 
    F'_^H_[0] = α· F'_∘Φ_,

 meaning that F'_^H_[0](T) = α F'_(T̅) for every closed admissible ^H_[0]-colored ribbon graph T, where α is the coefficient introduced in Remark <ref>.




 If a closed admissible ^H_[0]-colored ribbon graph T admits a projective edge of color V, and if T_V is a cutting presentation of T,




 then we have
 
    ^H_V(F_^H_[0](T_V)) = α_V̅(Φ_(F_^H_[0](T_V))) = α_V̅(F_(Φ_(T_V))),

 where the second equality follows from Lemma <ref>.



Now, let us recall the formulas defining the CGP and the renormalized Hennings invariants in the setting of Theorem <ref>. If (M,T,0,0) is a closed connected morphism of _^H^G, if L = L_1 ∪…∪ L_ℓ⊂ S^3 is a surgery presentation of M, and if we replace (T,0) with some (T̃,ω̃) obtained by projective stabilization of sufficiently generic index ensuring L becomes computable, as explained in Subsection <ref> and, in greater detail, in Section 3.2 of <cit.>, then we have

    _^H(M,T,0,0) = _Ω^-1-ℓδ_Ω^-σ(L) F'_^H(L ∪T̃).

On the other hand, if (M,T̅,0) is the closed connected morphism of _ obtained by applying the functor Φ_ to T ⊂ M, then we have

    '_(M,T̅,0) = _λ^-1-ℓδ_λ^-σ(L) F'_λ(L ∪T̅).

The surgery presentation L is used in different ways by the two constructions. In the first case, L is labeled with Kirby colors as prescribed by ω̃, and thus it is not a morphism in the domain of Φ_. In the second case, L is taken to be red, and thus it is not a morphism in the image of Φ_. In order to compare the two formulas, we introduce special morphisms of  which encode these two different procedures.


First, for all weights μ,ν∈^* satisfying [μ] = [ν] ∈ G ∖ X, the tensor product W_μ,ν := V_μ⊗ V_ν^* is an object of ^H_[0]. Therefore, since U̅ is a projective generator of , we can fix a decomposition

    𝕀_W̅_μ,ν = ∑_i=1^m_μ,ν g_μ,ν,i∘ f_μ,ν,i

for some morphisms f_μ,ν,i∈_(W̅_μ,ν,U̅) and g_μ,ν,i∈_(U̅,W̅_μ,ν). Let us also set

    d_μ,ν := (p_μ,ν⊗𝕀_W_μ,ν) ∘ (𝕀_V_ν⊗_V_μ⊗𝕀_V_ν^*) ∘ s_ν∈_^H(,W_μ,ν^* ⊗ W_μ,ν),

where s_ν∈_^H(,W_ν,ν) is a morphism satisfying _V_ν∘ s_ν =, and where p_μ,ν∈_^H(V_ν⊗ V_μ^*,(V_μ⊗ V_ν^*)^*) is the isomorphism coming from the pivotal structure of ^H. Now, let us fix once and for all a weight γ∈^* satisfying [γ] ∈ G ∖ X. Then we denote with h_Ω∈_(,U̅^* ⊗U̅) the morphism

    h_Ω := ∑_μ∈{γ} + _r∑_i=1^m_μ,γ^H(V_μ) · ((g_μ,γ,i)^* ⊗ f_μ,γ,i) ∘d̅_μ,γ.



Next, we denote with f_λ⊗ 1∈_(U̅,U̅^* ⊗U̅) the unique morphism which sends the generator 1 ∈U̅ to λ⊗ 1 ∈U̅^* ⊗U̅, where λ is the right integral of U̅_q, and we consider a morphism s_∈_(,U̅⊗) satisfying (ε⊗𝕀_) ∘ s_ = 𝕀_. Then we denote with h_λ∈_(,U̅^* ⊗U̅) the morphism

    h_λ := ( f_λ⊗ 1⊗) ∘ s_.



This allows us to review the recipe for the computation of the two invariants. If (M,T,0,0) is a closed morphism of _^H^G, if e ⊂ T is a projective edge of color V, and if L = L_1 ∪…∪ L_ℓ⊂ S^3 is a surgery presentation of M, then let us fix disjoint paths γ_j ⊂ S^3 ∖ (L ∪ T) connecting e to L_j for every integer 1 ≤ j ≤ℓ. Before starting, we perform special projective stabilizations both on e ⊂ T and on e̅⊂T̅ at the intersection point with γ_j for every integer 1 ≤ j ≤ℓ, as shown in Figure <ref>, 

where s_V ∈_^H(V,⊗ V) is a section of ⊗𝕀_V, where s_V̅∈_(V̅,⊗V̅) is its image under Φ_, and where s_γ∈_^H(,W_γ,γ) is a morphism satisfying _V_γ∘ s_γ =. Next, we isotope the s_γ-colored and the -colored coupons along the path γ_j until the intersection point with L_j. This is our initial configuration.


Let us start from the CGP invariant. First, we need to slide every V_γ-colored edge along the corresponding component L_j, so to turn the surgery presentation L into a computable one. Proposition 3.1 of <cit.> implies the choice of the weight γ∈^* satisfying [γ] ∈ G ∖ X is inconsequential. This produces the ^H-colored ribbon graph represented in the top-left corner of Figure <ref>.  Up to skein equivalence of ^H-colored ribbon graphs, we can replace a tubular neighborhood of L_j as shown in the top-right corner of Figure <ref>, thus obtaining a ^H_[0]-colored ribbon graph. This means we can apply the functor Φ_ which, up to skein equivalence of -colored ribbon graphs, produces the -colored ribbon graph represented in the bottom-left corner of Figure <ref>. Again up to skein equivalence of -colored ribbon graphs, we can replace a tubular neighborhood of L_j as shown in the bottom-right corner of Figure <ref>. The resulting -colored ribbon graph is denoted (L ∪T̅)_h_Ω, and is said to be obtained from the surgery presentation L and from the admissible ^H_[0]-colored ribbon graph T by Ω-stabilization along the paths γ_1,…,γ_ℓ. By construction, using Lemma <ref>, we have 

    _^H(M,T,0,0) = α_Ω^-1-ℓδ_Ω^-σ(L) F'_( (L ∪T̅)_h_Ω).



Let us move on to discuss the renormalized Hennings invariant. First, we need to interpret every component L_j as a red edge, and to label it with the regular representation U̅. This produces the -colored bichrome graph represented in the left-hand part of Figure <ref>. 

Up to skein equivalence of -colored bichrome graphs, we can turn every red component blue by replacing a tubular neighborhood of L_j as shown in the right-hand side of Figure <ref>. The resulting -colored ribbon graph is denoted (L ∪T̅)_h_λ, and is said to be obtained from the surgery presentation L and from the admissible ^H_[0]-colored ribbon graph T by λ-stabilization along the paths γ_1,…,γ_ℓ. By construction, using Lemma 3.8 of <cit.>, we have 

    '_(M,T̅,0) = _λ^-1-ℓδ_λ^-σ(L) F'_( (L ∪T̅)_h_λ).



Remark that, since (L ∪T̅)_h_Ω and (L ∪T̅)_h_λ were obtained starting from surgery presentations computing _^H and '_ through operations which do not alter the values of F'_^H and of F'_λ respectively, this means that both F'_ ( (L ∪T̅)_h_Ω) and F'_ ( (L ∪T̅)_h_λ) are independent of the choice of the paths γ_1, …, γ_ℓ. 




 §.§ Stabilization coefficients







































































































































































































Next, we need to compare _Ω with _λ, δ_Ω with δ_λ, and h_Ω with h_λ. In order to do this, we will prove a key technical result. Let (^1 ×^1)_(-,) be the object of _ defined by

    (^1 ×^1)_(-,) := ( S^1 × S^1,P_(-,),),

where the blue -colored ribbon set P_(-,) is given by a single framed point with negative orientation and color , and where the Lagrangian subspace  is generated by the homology class of the curve { (1, 0) }× S^1. Analogously, let ^2_((-,),(-,),(+,)) be the object of _ defined by

    ^2_((-,),(-,),(+,)) := ( S^2,P_((-,),(-,),(+,)),{ 0 }),

where the blue -colored ribbon set P_((-,),(-,),(+,)) is given by three framed points, two with negative, one with positive orientation, and all with color . Let us also consider the morphism (^3 ∖^3)_ : (^1 ×^1)_(-,)→^2_((-,),(-,),(+,)) of _ defined by

    (^3 ∖^3)_ := ( D^3 ∖ N^3,T_,0 ),

where N^3 ⊂ D^3 is an open tubular neighborhood of the curve { 0 }×1/2· S^1 ⊂ D^3, and where the -colored framed tangle T_ is represented in Figure <ref>.





 The linear map
 
    _( (^3 ∖^3)_) : _( (^1 ×^1)_(-,)) →_( ^2_((-,),(-,),(+,)))

 is injective.



 As we will show, the proof follows rather directly from the surjectivity of
 
    '_( (^3 ∖^3)_) : '_( ^2_((-,),(-,),(+,))) →'_( (^1 ×^1)_(-,)).

 Indeed, a vector of the form
 
    ∑_i=1^m α_i ·[ (^3 ∖^3)_∘_i ] ∈_( ^2_((-,),(-,),(+,)))

 for some α_1,…,α_m ∈ and some [_1],…,[_m] ∈_( (^1 ×^1)_(-,)) is trivial if and only if
 
    ∑_i=1^m α_i ⟨',(^3 ∖^3)_∘_i ⟩_^2_((-,),(-,),(+,)) = 0

 for every ['] ∈'_( ^2_((-,),(-,),(+,))), where for every object  of _ the linear map
 
    ⟨·, ·⟩_ : '_() ⊗_() →

 denotes the non-degenerate pairing induced by the universal construction in Section 3.3 of <cit.>. Then, since
 
    ⟨',(^3 ∖^3)_∘⟩_^2_((-,),(-,),(+,)) = ⟨' ∘ (^3 ∖^3)_,⟩_(^1 ×^1)_(-,)

 for every [] ∈_( (^1 ×^1)_(-,)) and every ['] ∈'_( ^2_((-,),(-,),(+,))), the injectivity of 
 
    _( (^3 ∖^3)_) : _( (^1 ×^1)_(-,)) →_( ^2_((-,),(-,),(+,)))

 is equivalent to the surjectivity of 
 
    '_( (^3 ∖^3)_) : '_( ^2_((-,),(-,),(+,))) →'_( (^1 ×^1)_(-,)).

 
 
 
 In order to prove that '_((^3 ∖^3)_) is surjective we remark that, as soon as an object = (,P,) of _ features a projective blue point of P in every connected component of , the proof of Proposition 3.13 of <cit.> can be repeated to show that the linear map 
 
    [     π'_ :    '(M';)         →      '_();                  T'         ↦ [M',T',0] ]

 is surjective for every connected 3-dimensional cobordism M' from Σ to ∅. This means that every vector in '_((^1 ×^1)_(-,)) can be described by a linear combination of -colored bichrome graphs inside (D^3 ∖ N^3) ∪_S^2D̅^̅3̅ from P_(-,) to ∅. Let T' be such a -colored bichrome graph. Up to isotoping its -colored edge intersecting P_(-,), we can make sure there is a projective edge of T' piercing the solid torus N^3. But now, thanks to Proposition <ref>,  is a projective generator of . This means that, up to skein equivalence, we can insert a pair of coupons joined by a -colored edge. As a direct consequence, T' is skein equivalent to a -colored ribbon graph like the one represented in Figure <ref> for some f' ∈_(⊗^*,). Clearly every vector of this form lies in the image of '_((^3 ∖^3)_). 


Let us consider now the morphisms (^1 ×^2)_h_Ω : ∅→ (^1 ×^1)_(-,) and (^1 ×^2)_h_λ : ∅→ (^1 ×^1)_(-,) of _ defined by

    (^1 ×^2)_h_Ω := (S^1 × D^2,T_h_Ω,0), 
    
     (^1 ×^2)_h_λ := (S^1 × D^2,T_h_λ,0)

where the -colored ribbon graphs T_h_Ω and T_h_λ are represented in the left hand part and in the right hand part of Figure <ref> respectively. 






 The morphisms (^1 ×^2)_h_Ω and (^1 ×^2)_h_λ of _ satisfy
 
    [(^1 ×^2)_h_Ω] = α· [(^1 ×^2)_h_λ] ∈_( (^1 ×^1)_(-,)),

 where α is the coefficient introduced in Remark <ref>. Furthermore, there exist compatible choices for the coefficients _Ω and _λ yielding
 
    _Ω = α_λ,   δ_Ω = δ_λ.





 We start by proving [(^1 ×^2)_h_Ω] and [(^1 ×^2)_h_λ] are linearly dependent in _( (^1 ×^1)_(-,)). This is done by using Lemma <ref>. Indeed, on one hand, Lemma <ref> gives the equality
 
    [(^3 ∖^3)_) ∘ (^1 ×^2)_h_Ω] = r^2N| _r | · [(D^3,T_D^3,0)] ∈_( ^2_((-,),(-,),(+,))),

 where the -colored ribbon graph T_D^3 is represented in Figure <ref>.
 
 On the other hand, U̅ is a projective generator of , which means
 
    𝕀_ = ∑_i=1^m g_,i∘ f_,i

 for some morphisms f_,i∈_(,U̅) and g_,i∈_(U̅,). Then, thanks to Lemma 3.6 of <cit.> combined with Lemma <ref>, we know there exists a non-zero coefficient β∈^* giving the skein equivalence of Figure <ref>.
 
 This gives
 
    [(^3 ∖^3)_) ∘ (^1 ×^2)_h_λ] = β· [(D^3,T_D^3,0)] ∈_( ^2_((-,),(-,),(+,))).

 Therefore, we get
 
    β· [(^1 ×^2)_h_Ω] = r^2N| _r | · [(^1 ×^2)_h_λ].
 
 
 Next, this relation allows us to compare the stabilization coefficients. Indeed, if T_±Ω and T_±λ denote the -colored ribbon graphs represented in Figure <ref>,
 
 then, by definition of h_Ω, we have
 
    F'_(T_±Ω) = Δ_±Ω_(∘) = αΔ_±Ω^H_(∘) = αΔ_±Ω,

 and analogously, by definition of h_λ, we have
 
    F'_(T_±λ) = Δ_±λ_(∘) = αΔ_±λ^H_(∘) = αΔ_±λ.

 This means
 
    '_(S^3,T_±Ω,0) = α_λ^-1Δ_±Ω,   '_(S^3,T_±λ,0) = α_λ^-1Δ_±λ.

 But now, thanks to the previous equality, we have 
 
    β'_(S^3,T_±Ω,0) = r^2N| _r | '_(S^3,T_±λ,0).

 This gives
 
    βΔ_±Ω = r^2N| _r | Δ_±λ.

 In particular, combining this equality with the explicit value of Δ_-ΩΔ_+Ω given by Figure <ref>, we can choose
 
    _Ω = r^N √(| _r |),   _λ = β/r^N √(| _r |).

 This immediately implies 
 
    δ_Ω = _Ω/Δ_- Ω = _λ/Δ_- λ = δ_λ.

 
 





























 
 
 
 Furthermore, we can now compute the equality
 
    β = r^2N| _r |/α.

 Indeed, let us consider the object ^2_(+,) of _ defined by (S^2,P_(+,),{ 0 }), where the blue -colored ribbon set P_(+,) is given by a single framed point with positive orientation and color , and let us consider the closed morphism ^2_(+,)×^1 of _ defined by (S^2 × S^1,P_(+,),0). The strategy will be to compute the renormalized Hennings invariant of ^2_(+,)×^1 in two different ways. On one hand, the isomorphism _( ^2_(+,))≅_(,) gives
 
    '_( ^2_(+,)×^1 ) = _( _( ^2_(+,)) ) = _( _(,) ) = 1,

 where the last equality follows from Lemma <ref>. On the other hand, we can choose a surgery presentation of S^2 × S^1 composed of a single unknot of framing 0. This choice determines the -colored bichrome graph H_ given by a positive Hopf link of framing 0, with one red component colored with U̅ and one blue component colored with . Therefore, we get
 
    '_( ^2_(+,)×^1 ) 
         = _λ^-2 F'_(H_) 
       = r^2N| _r | _(∘)/β
       = r^2N| _r | ^H_(∘)/αβ
       = r^2N| _r |/αβ.






 §.§ Proof of Theorem <ref>



We are now ready to prove Theorem <ref>. As explained in the proof of Lemma <ref>, using the explicit value of Δ_-ΩΔ_+Ω given by Figure <ref>, we can choose the square roots _Ω and _λ to be of the form

    _Ω = r^N √(| _r |),   _λ = r^N √(| _r |)/α,
 
Also, let us set

    δ := δ_Ω = δ_λ.




 If M is a closed 3-manifold, and T ⊂ M is an admissible ^H_[0]-colored ribbon graph, then let e ⊂ T be a projective edge of color V, let L = L_1 ∪…∪ L_ℓ⊂ S^3 be a surgery presentation of M, let γ_j ⊂ S^3 ∖ (L ∪ T) be disjoint paths connecting e to L_j for every integer 1 ≤ j ≤ℓ, and let (L ∪T̅)_h_Ω and (L ∪T̅)_h_λ be -colored ribbon graphs obtained by Ω-stabilization and by λ-stabilization along γ_1, …, γ_ℓ, as explained in Subsection <ref>. Then we have
 
    _^H(M,T,0,0)    = α_Ω^-1-ℓδ^-σ(L) F'_((L ∪T̅)_h_Ω) 
       = α^-ℓ_λ^-1-ℓδ^-σ(L) F'_((L ∪T̅)_h_Ω) 
       = α^-ℓ_λ^-ℓδ^-σ(L)'_(S^3,(L ∪T̅)_h_Ω,0) 
       = _λ^-ℓδ^-σ(L)'_(S^3,(L ∪T̅)_h_λ,0) 
       = _λ^-1-ℓδ^-σ(L) F'_((L ∪T̅)_h_λ) 
       = '_(M,T̅,0),

 where the second and the fourth equalities follow from Lemma <ref>.








§ QUANTUM GROUPS



In this appendix we collect some standard definitions related to quantum groups, see <cit.> for more details. Let  be a simple complex Lie algebra of rank n and dimension 2N + n, let B be its Killing form, let  be a Cartan subalgebra of , let Φ be the corresponding root system, let Φ_+ be a choice of a set of positive roots of , and let {α_1, …, α_n } be an ordering of its set of simple roots. Let A = ( a_ij )_1 ⩽ i,j ⩽ n be the corresponding Cartan matrix, which is the integral matrix given by

    a_ij := 2 B^*(α_i,α_j)/B^*(α_i,α_i),

where B^* is the symmetric bilinear form on ^* determined by the restriction of B to  under the isomorphism which identifies a vector H ∈ with the linear form B(H,·) ∈^*, and let { H_1,…, H_n } be the basis of  determined by α_j (H_i) = a_ij for all integers 1 ⩽ i,j ⩽ n. For every α∈Φ_+ we set

    d_α := B^*(α,α)/min{ B^*(α_i, α_i) | 1 ⩽ i ⩽ n }

and for every integer 1 ⩽ i ⩽ n we use the short notation d_i := d_α_i. We denote with ⟨·,·⟩ the symmetric bilinear form on ^* determined by ⟨α_i,α_j ⟩ = d_i a_ij for all integers 1 ⩽ i,j ⩽ n, and we denote with ω_1,…,ω_n the corresponding fundamental dominant weights, which are the vectors of ^* determined by the condition ⟨ω_i,α_j ⟩ = d_i δ_ij for every i,j = 1,…,n. We denote with Λ_R the root lattice, which is the subgroup of ^* generated by simple roots, and we denote with Λ_W the weight lattice, which is the subgroup of ^* generated by fundamental dominant weights. If q is a formal parameter, then for every α∈Φ_+ we set q_α := q^d_α, for all k ⩾ℓ∈ we define

    { k }_α := q_α^k - q_α^-k,    [k]_α := { k }_α/{ 1 }_α,
        [k]_α! := [k]_α[k-1]_α⋯[1]_α, 
    kℓ_α := [k]_α!/[ℓ]_α![k-ℓ]_α!,

and for every integer 1 ⩽ i ⩽ n we use the short notation 

    q_i := q_α_i,   { k }_i := { k }_α_i,    [k]_i := [k]_α_i,
        [k]_i! := [k]_α_i!,   kℓ_i := kℓ_α_i.

Let _q denote the quantum group of , which is the (q)-algebra with generators

    { K_i,K_i^-1,E_i,F_i | 1 ⩽ i ⩽ n }

and relations

    K_i K_i^-1 = K_i^-1 K_i = 1,    [K_i,K_j] = 0, 
    
     K_i E_j K_i^-1 = q_i^a_ij· E_j,   
     K_i F_j K_i^-1 = q_i^- a_ij· F_j, 
    
     [E_i,F_j] = δ_ij·K_i - K_i^-1/q_i-q_i^-1

for all integers 1 ⩽ i,j ⩽ n and

    ∑_k=0^1 - a_ij (-1)^k1-a_ijk_i · E_i^k E_j E_i^1-a_ij-k = 0, 
    ∑_k=0^1 - a_ij (-1)^k1-a_ijk_i · F_i^k F_j F_i^1-a_ij-k = 0,

for all integers 1 ⩽ i,j ⩽ n with i ≠ j. Then _q can be made into a Hopf algebra by setting

    Δ(K_i)    = K_i ⊗ K_i,    ε(K_i)    = 1,     S(K_i)    = K_i^- 1, 
    Δ(E_i)    = E_i ⊗ K_i + 1 ⊗ E_i,    ε(E_i)    = 0,     S(E_i)    = -E_i K_i^-1, 
    Δ(F_i)    = F_i ⊗ 1 + K_i^-1⊗ F_i,    ε(F_i)    = 0,     S(F_i)    = - K_i F_i

for all integers 1 ⩽ i ⩽ n. For every

    μ = ∑_i=1^n m_i ·α_i ∈Λ_R

we use the notation

    K_μ := ∏_i=1^n K_i^m_i,

and for every α∈Φ_+ we define root vectors E_α and F_α as follows: first, we consider the Weyl group W of  associated with , which is the subgroup of (^*) generated by reflections

    [          s_i :             ^*              →             ^*;                           α_j              ↦ α_j - a_ij·α_i ]

for every integer 1 ⩽ i ⩽ n. Next, we consider the unique element w_0 ∈ W corresponding to a word of maximal length in the generators. The choice of a decomposition w_0 = s_i_1∘⋯∘ s_i_N determines a total order on the set of positive roots

    Φ_+ = {α_i_1,s_i_1(α_i_2),…,(s_i_1∘⋯∘ s_i_N-1)(α_i_N) }.

Then, for every integer 1 ⩽ i ⩽ n, we consider the automorphism T_i of _q determined by 

    T_i(K_j)    := K_j K_i^-a_ij, 
    
     T_i(E_j)    := 
     
      - F_iK_i - K_i^-1E_i    i = j, 
    ∑_k=0^-a_ij (-1)^kq_i^a_ij + k/[k]_i![-a_ij-k]_i!· E_i^k E_j E_i^-a_ij-k    i ≠ j,
     
    
     T_i(F_j)    := 
     
      - K_i^-1E_i - F_iK_i    i = j, 
    ∑_k=0^-a_ij (-1)^-a_ij-kq_i^k/[k]_i![-a_ij-k]_i!· F_i^k F_j F_i^-a_ij-k    i ≠ j.

Now, for every integer 1 ⩽ k ⩽ N, we set 

    β_k := (s_i_1∘⋯∘ s_i_k-1)(α_i_k) ∈Φ_+

and

    E_β_k := (T_i_1∘⋯∘ T_i_k-1)(E_i_k) ∈_q ,    
     F_β_k := (T_i_1∘⋯∘ T_i_k-1)(F_i_k) ∈_q .




 



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