11institutetext: Amit Verma 22institutetext: Craig School of Business,
Missouri Western State University, Saint Joseph, MO, 64507, United States
Tel.: +1 816-271-4357
Fax: +1 816-271-4508
22email<EMAIL_ADDRESS>33institutetext: Mark Lewis
44institutetext: 44email<EMAIL_ADDRESS>
# QUBO transformation using Eigenvalue Decomposition

Amit Verma  Mark Lewis

(Received: date / Accepted: date)

###### Abstract

Quadratic Unconstrained Binary Optimization (QUBO) is a general-purpose
modeling framework for combinatorial optimization problems and is a
requirement for quantum annealers. This paper utilizes the eigenvalue
decomposition of the underlying $Q$ matrix to alter and improve the search
process by extracting the information from dominant eigenvalues and
eigenvectors to implicitly guide the search towards promising areas of the
solution landscape. Computational results on benchmark datasets illustrate the
efficacy of our routine demonstrating significant performance improvements on
problems with dominant eigenvalues.

###### Keywords:

Quadratic Unconstrained Binary Optimization pseudo-Boolean optimization
eigenvalue decomposition eigenvalues eigenvectors quantum computing maximum
diversity problem

##  1 Introduction

Quadratic Unconstrained Binary Optimization (QUBO) is a popular modeling
framework wherein many optimization problems are transformed into
$max\;x^{t}Qx;x\in\\{0,1\\}$ where $Q$ is a $nXn$ symmetric matrix and $x$
denotes the binary decision vector of size $n$. QUBO has emerged into
significance in the last decade due to advances in quantum annealers which
rely on $0/1$ modeling format (or equivalent Ising spin $-1/+1$). For a
detailed survey of applications using this modeling format, we refer the
readers to Kochenberger et al. (2014).

While the $Q$ matrix is often transformed into an upper diagonal matrix,
eigendecomposition will retain a symmetric format because a symmetric matrix
always leads to real eigenvalues and eigenvectors. According to spectral
decomposition theorem (Halmos (1963)), a symmetric matrix $Q$ can be
decomposed into $\sum_{i=1}^{n}\lambda_{i}c_{i}c_{i}^{t}$ where $\lambda_{i}$
and $c_{i}$ denotes the $i^{\prime}th$ eigenvalue and eigenvector
respectively. Thus, $max\;x^{t}Qx$ can be transformed into
$max\;x^{t}(\sum_{i=1}^{n}\lambda_{i}c_{i}c_{i}^{t})x$. Since each
$\lambda_{i}$ is a scalar, this yields
$max\;\sum_{i=1}^{n}\lambda_{i}(x^{t}c_{i})(c_{i}^{t}x)$ where $x$ and $c_{i}$
are vectors and $x^{t}c_{i}=c_{i}^{t}x$. Thus, we rewrite the final expression
as $max\;\sum_{i=1}^{n}\lambda_{i}(x^{t}c_{i})^{2}$.

The time complexity of the complete eigendecomposition ranges from $O(n^{2})$
to $O(n^{3})$ depending on the structure of the matrix, matrix multiplication
routine and employed algorithm, according to (Pan and Chen (1999)). However,
the top $k$ eigenvalues and associated eigenvectors could be computed more
efficiently using techniques described in Sorensen (1997) and are part of
numerical software packages like ARPACK (Lehoucq et al. (1998)). In this
paper, we will focus on top $k$ eigenvalues and eigenvectors for the $Q$
matrix. As an example the complete eigendecomposition of the following $Q$
matrix is given by:

$\begin{bmatrix}-7&2&2\\\ 2&4&2\\\
2&2&5\end{bmatrix}=-7.56\begin{bmatrix}-0.98\\\ 0.14\\\
0.13\end{bmatrix}\begin{bmatrix}-0.98\\\ 0.14\\\
0.13\end{bmatrix}^{t}+2.44\begin{bmatrix}-0.03\\\ -0.77\\\
0.63\end{bmatrix}\begin{bmatrix}-0.03\\\ -0.77\\\
0.63\end{bmatrix}^{t}+7.12\begin{bmatrix}0.19\\\ 0.61\\\
0.76\end{bmatrix}\begin{bmatrix}0.19\\\ 0.61\\\ 0.76\end{bmatrix}^{t}$

$=\begin{bmatrix}-7.26&1.04&0.96\\\ 1.04&-0.15&-0.14\\\
0.96&-0.14&-0.14\end{bmatrix}+\begin{bmatrix}0.00&0.06&-0.05\\\
0.06&1.45&-1.18\\\
-0.05&-1.18&0.97\end{bmatrix}+\begin{bmatrix}0.26&0.83&1.03\\\
0.83&2.65&3.30\\\ 1.03&3.30&4.11\end{bmatrix}$

The eigenvalues sorted in the descending order of absolute values are given by
$-7.56$, $7.12$, and $2.44$ respectively. The corresponding eigenvectors are
given by $[-0.98,0.14,0.13]^{t}$,$[0.19,0.61,0.76]^{t}$, and
$[-0.03,-0.77,0.63]^{t}$. Thus, we could rewrite the optimization problem
$max\;x^{t}Qx$ as $max\;\sum_{i=1}^{n}\lambda_{i}(x^{t}c_{i})^{2}$ which
reduces to $max\;\sum_{i=1}^{n}\lambda_{i}\sum_{j=1}^{n}(x^{j}c_{i}^{j})^{2}$.
Note that the index $j$ runs over all variables for specific eigenvectors
$c_{i}$. Also, subterms like $(x^{t}c_{i})^{2}$ involves outer product of a
vector by itself and could lead to increase in the overall density of the
resulting sum.

The theoretical properties and applications of eigendecomposition such as
image processing are well studied. According to Smith (2002), eigenvectors can
only be determined for square $nXn$ matrices with $n$ dimensions. All the
eigenvectors of a matrix are orthogonal and normalized such that length is $1$
unit. More importantly, eigenvectors identify the principal components of a
matrix. These components characterize the data, which could be compressed
using some top components. However, information loss occurs in these cases of
dimensionality reduction. The eigenvector with the highest eigenvalue is
defined as the principal component of the dataset. It captures the most
significant relationship amongst the data dimensions.

By sorting the eigenvalues in descending order of absolute values, components
are sorted in order of significance. Dimensionality reduction could be
accomplished by dropping components of lower significance. In this paper, we
emphasize the first $k$ eigenvectors in this sorted list to guide the search
towards areas of importance. Note that the sign of the eigenvalue is also
important since multiplying a vector by $-1$ reverses the direction of the
eigenvector. On the other hand, the magnitude of the eigenvalue could bias the
solution landscape if it is too big. Thus, we limit the impact of the
magnitude of the eigenvalue through a non-negative parameter $M$. To
summarize, we augment the original $Q$ matrix with a penalty or reward term
dependent on $M$, the sign of the eigenvalue, and the outer product of the
eigenvector with itself. A related eigenvalue decomposition based branch and
bound algorithm utilizing semidefinite relaxation was found effective for
Quadratically Constrained Quadratic Programming instances (Lu et al. (2017,
2019)). However, our approach employs a simple transformation of the $Q$
matrix.

Heuristic solvers like tabu search and path relinking benefit from these
augmented terms by implicitly (not directly) guiding the search towards
promising areas in the solution landscape. In the above example, by
considering the top most eigenvalue (sorted in descending order of absolute
values), the contribution term is given by
$1*(-1)*(-0.98x_{1}+0.14x_{2}+0.13x_{3})^{2}$ assuming $M=1$. Note that the
sign of the corresponding eigenvalue $-7.56$ is $-1$. This is equivalent to
adding $1*(-1)*[-0.98,0.14,0.13]^{t}*[-0.98,0.14,0.13]$ to the original $Q$
matrix. Thus, the original $Q$ matrix is transformed into $Q^{\prime}$ as
follows:

$Q^{\prime}=\begin{bmatrix}-7&2&2\\\ 2&4&2\\\
2&2&5\end{bmatrix}-1\begin{bmatrix}-0.98\\\ 0.14\\\
0.13\end{bmatrix}\begin{bmatrix}-0.98\\\ 0.14\\\
0.13\end{bmatrix}^{t}=\begin{bmatrix}-7&2&2\\\ 2&4&2\\\
2&2&5\end{bmatrix}-1\begin{bmatrix}0.96&-0.14&-0.13\\\ -0.14&0.02&0.02\\\
-0.13&0.02&0.02\end{bmatrix}$ $=\begin{bmatrix}-7.96&2.14&2.13\\\
2.14&3.98&1.98\\\ 2.13&1.98&4.98\end{bmatrix}$

##  2 Model

The base problem of interest is $max\;x^{t}Qx$. We propose a transformation of
the base problem into
$max\;x^{t}Q^{\prime}x=max\;x^{t}Qx+\sum_{i=1}^{k}M*sign(\lambda_{i})*(x^{t}c_{i})^{2}$.
The index $i$ is based on the sorted list of absolute eigenvalues in
descending order. Note that the augmented term is quadratic and fits into the
QUBO modeling framework. Thus, $max\;x^{t}Q^{\prime}x$ could be easily handled
by any QUBO solver. We will compare the performance of a tabu-search based
QUBO solver and CPLEX on both the base problem and the transformed problem in
Section 3. Note that our proposed technique classifies as a preprocessing
approach for QUBO similar to Verma and Lewis (2020a).

The magnitude of the penalty coefficient $M$ could be varied to emphasize the
degree of importance. However, a large $M$ biases the solution landscape
heavily towards the direction of the eigenvector. On the other hand, small
values of $M$ are inconsequential and might not lead to any difference in the
search process. Similarly, we can decide the number of top eigenvectors ($k$)
to be considered in the contribution term. The search process will be
overwhelmed if we consider a lot of eigenvectors. Moreover, the eigenvectors
with small eigenvalues could be safely ignored since they do not contain a lot
of information. We will perform sensitivity analysis on both parameters $M$
and $k$ in Section 3.

##  3 Computational Experiments and Results

For computational experiments we utilize the ORLIB datasets (Beasley (1990))
and Palubeckis datasets (Palubeckis (2004)). The former contains datasets with
100, 250, 500 and 1000 nodes, and integral coefficients of the $Q$ matrix are
randomly chosen from the uniform distribution [-100,100]. These datasets are
sparse with linear and quadratic density of 10%. The latter contains larger
and denser (10-100% dense) datasets with 3000, 4000, 5000, 6000 and 7000
nodes. The uniform distribution of the coefficients of the $Q$ matrix is
expected to generate a distribution of eigenvalues that are also uniform,
whereas a non random $Q$ distribution, e.g. clustered $Q$, is expected to have
an eigenvalue distribution with noticeable peaks corresponding to the
clusters. We will investigate computational impact of the transformation on
both types of $Q$ matrices in this section.

The algorithms were implemented in Python 3.6. The experiments were performed
on a 3.40 GHz Intel Core i7 processor with 16 GB RAM running 64 bit Windows 7
OS. Smaller datasets up to 1000 nodes were solved using CPLEX 12.6.3. while
the bigger datasets utilize a path relinking and tabu search based QUBO solver
based on Lewis et al. (2021). In this heuristic, the effect of a single bit-
flip is quickly evaluated in $O(1)$. This enables the selection of the best
variable in a neighborhood with the greatest effect in $O(n)$. This process is
complemented by path relinking, which is implemented as a greedy search of the
restricted solution space defined by the difference bits. Variables selected
to be flipped are assigned to a tabu list to prevent cycling. When stuck in a
local optima, a random restart is executed. We will refer to this heuristic as
PRlocal in the forthcoming discussion.

First, we will investigate the distribution of eigenvalues for one of the
ORLIB instances with 1000 nodes, $bqp\\_1000\\_1$. The results are summarized
in Figure 1. We see a truncated normal distribution with the peak around zero.
This behavior is consistent with other ORLIB and Palubeckis datasets. Note
that these components with smaller absolute eigenvalues could be removed from
the $Q$ matrix according to the associated eigendecomposition in Section 1.
However, we will focus on the components with top $k$ absolute eigenvalues
according to Section 2. Herein, the biggest such value is around 1000.

Figure 1: Frequency distribution of eigenvalues for $bqp\\_1000\\_1$

Next, we will discuss the performance of the transformed model on CPLEX on
smaller datasets containing fewer than 1000 nodes. We allow 100 seconds for
each experimental run. Further, we utilized default CPLEX settings except
qtolin. We disabled the linearization capability of CPLEX by setting the
qtolin parameter to zero after noticing all the allocated time was invested in
generating linear constraints for the binary quadratic problem. Note that we
use $k=1$ as baseline parameter values since $k=1$ identifies the principal
component having the most significant relationship between data dimensions
(Smith (2002)). Moreover, the leading eigenvector is used prominently in
various research papers (Sarkar and Boyer (1998); Perona and Freeman (1998)).
Our decision on $M=100$ was driven by the fact that eigenvectors are
normalized (the individual entries are less than 1). Hence, the outer product
of the eigenvector by itself would lead to off-diagonal entries, which are the
multiple of two numbers less than 1. Multiplying by $100$ creates a
contribution term which is significant but not very large relative to the
entries in the $Q$ matrix. In other words, we want to perturb the original $Q$
matrix by a small amount such that the search process is not overwhelmed by
the extraneous terms.

For the smaller problems with 100, 250 and 500 nodes, there was no observed
difference in the best solution obtained by CPLEX for both the base problem
and transformed problem described in Section 2. These datasets are smaller
which allows CPLEX to quickly solve these problems within 100 seconds. For the
1000 node instances, we report the average percentage effect of the
transfomation compared to the base problem across all datasets in the
Improvement column of Table 1. In general, as $M$ and $k$ increases, the
performance of the transformation deteriorates.

Next, we analyze the impact of changing the parameters $M$ and $k$ on the
computational performance. For this purpose, we experimented with
$k\in\\{1,2,5,10,20,25\\}$ and $M\in\\{100,200,300,400,500\\}$ in Table 1. We
notice that $M=100$ and $200$ lead to similar performance profiles. However,
as the value of $M$ increases, the performance deteriorates and the
transformation leads to reduction in solution quality. Utilizing up to top
$10$ eigenvectors in the transformation leads to a gain of around 0.02%.
However, this gain diminishes as we consider more eigenvectors in the
transformation.

Table 1: Impact of transformation on 1000 node instances (CPLEX, $M=100,k=1$) | Average Improvement % Over Base Problem
---|---
k | M=100 | M=200 | M=300 | M=400 | M=500
1 | 0.069 | 0.031 | -0.015 | -0.169 | -0.362
2 | 0.076 | 0.055 | -0.031 | -0.236 | -0.955
5 | 0.053 | -0.076 | -0.26 | -0.958 | -2.431
10 | 0.064 | -0.079 | -0.761 | -2.452 | -3.889
20 | 0.056 | -0.426 | -1.431 | -3.017 | -4.343
25 | -0.022 | -0.555 | -1.753 | -3.115 | -4.658

Similar to Verma and Lewis (2020b), we analyze the performance of the solver
on the 1000 node instances through autocorrelation coefficient $\xi$. The
higher value of $\xi$ is related to a lower number of local optima, which
leads to better performance of any local search solver. We observe that the
$\xi$ value for smaller values of $k$ and $M$ is somewhat higher. Thus, the
gain in performance could be explained using a statistic from the search
landscape analysis.

Table 2: Impact on solution landscape for 1000 node instances | Autocorrelation Coefficient ($\xi$)
---|---
k | M=100 | M=200 | M=300 | M=400 | M=500
1 | 356.1 | 356.2 | 354.4 | 355.5 | 354.4
2 | 354.4 | 353.0 | 352.8 | 353.4 | 352.8
5 | 354.3 | 352.7 | 352.5 | 352.6 | 352.0
10 | 352.9 | 352.4 | 352.4 | 352.0 | 349.6
20 | 352.6 | 352.0 | 350.8 | 349.2 | 348.0
25 | 351.1 | 351.8 | 348.5 | 346.7 | 346.4

For the bigger Palubeckis instances, we performed sensitivity analysis on $M$
and $k$ using the baseline values of $M=100$ and $k=1$, and the results are
presented in Figure 2. We observed similar performance profiles based on
Average Improvement % across other bigger instances. Thus,
$k\in\\{1,2,5,10\\}$ and $M\in\\{100,200\\}$ are also acceptable choices for
bigger datasets.

(a) Sensitivity analysis of $k$

(b) Sensitivity analysis of $M$

Figure 2: Results for bigger Palubeckis instances

The results on Average Improvement % across all bigger instances using $M=100$
and $k=1$ are presented in Table 2. The transformation improves or matches the
base problem in 27 out of 31 instances. We want to mention that the additional
contribution term due to eigenvalues i.e.
$M*sign(\lambda_{i})*(x^{t}c_{i})^{2}$ leads to the creation of a dense matrix
because of the inner product. Thus, the transformation is more effective on
Palubeckis instances which are already dense and the adjustment does not
create new off-diagonal entries.

Table 3: Impact of transformation on bigger instances (PRlocal, $M=100,k=1$) Instance Type | Nodes | Avg Improvement | Avg Improvement %
---|---|---|---
ORLIB | 2500 | 2478 | 0.021
Palubeckis | 3000 | 385 | 0.017
| 4000 | 16 | 0.011
| 5000 | 5367 | 0.049
| 6000 | 5018 | 0.065
| 7000 | 3875 | 0.031

Based on our expectation that $Q$ with a few dominant eigenvalues will be more
amenable to this technique, we investigate the impact of the transformation on
such matrices. For this purpose, we study the Maximum Diversity Problem (MDP)
instances discussed in Martí et al. (2013). The optimization problem of
interest $max\sum_{i,j,i\neq
j}d_{ij}x_{i}x_{j}\;s.t.\;\sum_{i}x_{i}=m,\;x_{i}\in\\{0,1\\},\;i\in[1,n]$
comprises of selecting a subset of $m$ elements from $n$ variables such that
the sum of distance (or equivalent diversity measure) between chosen elements
is maximized. The QUBO transformation incorporates the quadratic penalty term
given by $-P(\sum_{i}x_{i}-m)^{2}$ resulting in a $Q$ matrix where $P$
represents a non-negative penalty coefficient. The details of the QUBO
transformation for MDP are discussed in Ghosh (1996). The set of instances
identified as MDPLIB consists of 315 instances belonging to three categories
SOM, GKD and MDG. These categories have small, medium and large size instances
respectively.

The eigenvalue distribution for $SOM\\_a\\_21\\_n100\\_m10$ is detailed next.
We utilize $P=10,m=10$ and $n=100$ for the resulting QUBO model. Figure 3
clearly indicates that there is a dominant eigenvalue around -500 while the
remaining eigenvalues are concentrated around 200. Other MDPLIB instances
follow a similar trend. Thus, the principal component given by the largest
absolute eigenvalue clearly dominates other components. Hence, guiding the
search towards this principle component might have an impact on the
computational performance of the solver.

Figure 3: Frequency distribution of eigenvalues for
$SOM\\_a\\_21\\_n100\\_m10$

Again, we utilize $M=100$, $k=1$ and $t=100$ seconds for these experiments.
Similar to smaller ORLIB QUBO benchmark instances, SOM and GKD have atmost 500
variables. The eigenvalue decomposition has no improvement on solution quality
obtained by PRlocal because of smaller size. On the other hand, larger MDG
instances see a significant increase in solution quality as a result of the
transformation. Note that the Average Improvement % in Table 4 increases as
the size of the QUBO instance increases. Overall, the eigendecomposition
improves or matches the best solution found by the original problem in all but
17 instances. Thus, our technique is found to more effective in $Q$ matrices
with dominant eigenvalues.

Table 4: Impact of transformation on MDPLIB MDG instances (PRlocal, $M=100,k=1$) Instance Type | Nodes | Avg Improvement % Over Base Problem
---|---|---
MDG-a | 500,2000 | 1.0
MDG-b | 500,2000 | 6.0
MDG-c | 3000 | 9.7

##  4 Conclusions and Future Research

We present a learning-based technique that modifies the $Q$ matrix based on
the principal components of the $Q$ matrix obtained from eigenvalue
decomposition. Testing indicates this technique leads to improvement in
solution quality for benchmark QUBO instances especially when there are a few
dominant eigenvalues. Future work involves adjusting $M$ based on the
eigenvalue for each component in the decomposition and dimensionality
reduction of the $Q$ matrix.

##  5 Data Availability Statement

The datasets generated during and/or analyzed during the current study are
available from the corresponding author on reasonable request.

##  6 Conflict of Interest Statement

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

## References

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# Modified symmetrized integral in $G$-coalgebras

Nathan Geer  Utah State University, Department of Mathematics and Statistics,
Logan UT 84341, USA<EMAIL_ADDRESS>,  Ngoc Phu HA  Hung Vuong
University, Faculty of Natural Sciences, Viet Tri, Phu Tho, Viet Nam
<EMAIL_ADDRESS>and  Bertrand Patureau-Mirand  Université Bretagne Sud,
Laboratoire de Mathématiques de Bretagne Atlantique, UMR CNRS 6205, Vannes,
France<EMAIL_ADDRESS>
###### Abstract.

For $G$ a commutative group, we give a purely Hopf $G$-coalgebra construction
of $G$-colored $3$-manifolds invariants using the notion of modified integral.

MSC: 57M27, 17B37.

Key words: Quantum group, ribbon Hopf $G$-coalgebra, Hennings type invariant,
modified symmetrized integral.

##  1\. Introduction

In [11], Ohtsuki defined the structure of a ribbon colored Hopf algebra and he
reveal such a structure based on a non-semisimple example coming from quantum
${\mathfrak{sl}_{2}}$. He showed that a ribbon colored Hopf algebra leads to a
universal link invariant taking values in some quotient of the colored Hopf
algebra. In the example of quantum ${\mathfrak{sl}_{2}}$ this universal
invariant recovers the ADO link invariant given in [1].

A non commutative version of the notion of a ribbon colored Hopf algebra
called a $G$-coalgebra was introduced by Turaev with the goal of producing
Homotopy Quantum Field Theories (HQFTs), see [12]. Virelizier showed that
certain $G$-coalgebras gives rise to universal invariants of a $G$-link (i.e.
a link with a flat connection in a principal $G$-bundle over the complement of
the link). Moreover, he shows that if the $G$-coalgebra has a $G$-trace this
universal invariant leads to a _Hennings-Virelizier invariant_ of 3
dimensional $G$-manifolds (3-manifolds endowed with a flat $G$-bundle), see
[13] and [12, Appendix 7.2]. Moreover, in [14, Theorem 7.4], Virelizier shows
that a $G$-trace exists when the $G$-coalgebra has certain conditions (for
example, when it is semisimple and finite type). Constructing non-trivial
$G$-manifold invariants in the non-semisimple and non-finite type setting
seems to require new techniques which we propose here: we show that in a
special case (when $G$ is commutative and the $G$-coalgebra has certain
conditions including a modified symmetrized integral) Virelizier’s
construction can be re-normalized and leads to non-trivial $G$-manifold
invariants. In particular, we show that unrolled quantum group associated to
${\mathfrak{sl}_{2}}$ gives rise to $G$-coalgebra with modified symmetrized
integral and so an invariant of $G$-manifolds. The link invariant underlying
this example is closely related to the universal link invariant defined by
Ohtsuki in [11]. This link invariant was previously used to produce a
representation theoretic Reshetikhin-Turaev type 3-manifold invariants in [3].
Here we give an Hopf $G$-coalgebra approach to make a Hennings type 3-manifold
invariants.

The main new concept of this paper is the notion of a modified symmetrized
integral on a $G$-coalgebra. This notion is related and inspired by the theory
of modified traces and symmetrized integrals given in [2, 5, 6]. We show that
modified symmetrized integrals exist in the general context of Theorem 3.5.
This general context is inspired and modeled on topological unrolled quantum
groups associated to simple Lie algebras, see [4]. However, the work of [4]
requires technical topological completions where this paper uses more
straightforward algebraic techniques.

The outline and main results of the paper are described as follows. In Section
2, when $G$ is commutative, we describe Virelizier version of the universal
invariant of $G$-links and its corresponding 3-dimensional $G$-manifold
invariant which we call the graded Hennings-Virelizier invariant. In this
context, a $G$-manifolds $(M,\omega)$ is just the data of a 3-manifold with a
cohomology class $\omega\in H^{1}(M,G)$. In Section 3 we give the notion of a
(modified) symmetrized integral and show that such an integral can be used to
renormalize the graded Hennings-Virelizier invariant. In Section 4 we show
that Ohtsuki’s ribbon colored Hopf algebra associated to ${\mathfrak{sl}_{2}}$
leads to an example of the invariants defined in this paper.

### Acknowledgments

N.G. is partially supported by NSF grants DMS-1664387 and DMS-2104497. He
would also like to thank the Max Planck Institute for Mathematics in Bonn for
its hospitality during work on this paper. N. P. H. would like to thank the
fundamental research program of Hung Vuong University.

##  2\. Virelizier invariant of $G$-manifolds: the abelian case

Let $\mathbb{K}$ be an integral domain.

###  2.1. Ribbon Hopf $G$-coalgebra over abelian group

The notion of a Ribbon Hopf $G$-coalgebra is a tool used to produce homotopy
R-matrices and examples of non commutative HQFTs. In this section we give a
simplified version of a Hopf $G$-coalgebra in the quasi-triangular case which
coincide with Ohtsuki’s notion of colored Hopf algebra [11]. In particular the
group $G$ is commutative and we use additive notation.

A Hopf $G$-coalgebra is a family ${\mathbf{H}}=\\{{\mathbf{H}}_{a}\\}_{a\in
G}$ of $\mathbb{K}$-algebras (with a product
$m_{a}:{\mathbf{H}}_{a}\otimes{\mathbf{H}}_{a}\to{\mathbf{H}}_{a}$ and a unit
$\eta_{a}:\mathbb{K}\to{\mathbf{H}}_{a}$ for each $a\in G$) endowed with a
comultiplication
$\Delta=\\{\Delta_{a,b}:{\mathbf{H}}_{a+b}\to{\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b}\\}_{a,b\in
G}$, a counit $\varepsilon:{\mathbf{H}}_{0}\to\mathbb{K}$ and an antipode
$S=\\{S_{a}:{\mathbf{H}}_{a}\to{\mathbf{H}}_{-a}\\}_{a\in G}$ which satisfy
(see [11, 12, 14]):

  1. (1)

$\Delta$ is coassociative: for any $a,b,c\in G$,

$\left(\Delta_{a,b}\otimes\operatorname{Id}_{{\mathbf{H}}_{c}}\right)\circ\Delta_{a+b,c}={\left(\operatorname{Id}_{{\mathbf{H}}_{a}}\otimes\Delta_{b,c}\right)}\circ\Delta_{a,b+c},$

  2. (2)

$\varepsilon$ is a counit:
${\left(\operatorname{Id}_{{\mathbf{H}}_{a}}\otimes\varepsilon\right)}\circ\Delta_{a,0}={\left(\varepsilon\otimes\operatorname{Id}_{{\mathbf{H}}_{a}}\right)}\circ\Delta_{0,a}=\operatorname{Id}_{{\mathbf{H}}_{a}}$,

  3. (3)

$S$ is a family of bijective antipode: for any $a\in G$,

$m_{a}\circ{\left(S_{-a}\otimes\operatorname{Id}_{{\mathbf{H}}_{a}}\right)}\circ\Delta_{-a,a}=\eta_{a}\circ\varepsilon=m_{a}\circ{\left(\operatorname{Id}_{{\mathbf{H}}_{a}}\otimes
S_{-a}\right)}\circ\Delta_{a,-a},$

  4. (4)

$\Delta$ and $\varepsilon$ are algebra maps (for the standard algebra
structure of ${\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b}$ and $\mathbb{K}$).

Let us now recall some definitions from [14, 8].

######  Definition 2.1 (pivotal structure).

Let ${\mathbf{H}}=\\{{\mathbf{H}}_{a}\\}_{a\in G}$ be a Hopf $G$-coalgebra.

  1. (1)

A _$G$ -grouplike_ element is a family $\\{x_{a}\in{\mathbf{H}}_{a}\\}_{a\in
G}$ such that $\Delta_{a,b}(x_{a+b})=x_{a}\otimes x_{b}$, for all $a,b\in G$
and $\varepsilon(x_{0})=1$.

  2. (2)

A _pivot_ for ${\mathbf{H}}$ is a $G$-grouplike element $\\{g_{a}\\}_{a\in G}$
such that, for all $a\in G$ for any $x\in{\mathbf{H}}_{a}$,
$S_{-a}S_{a}(x)=g_{a}xg_{a}^{-1}$. If ${\mathbf{H}}$ has a pivot we say it is
a _pivotal_ Hopf $G$-coalgebra.

######  Definition 2.2 (quasitriangular structure).

Let
$\tau:{\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b}\to{\mathbf{H}}_{b}\otimes{\mathbf{H}}_{a}$
be the linear map defined by $x\otimes y\mapsto y\otimes x$. A Hopf
$G$-coalgebra ${\mathbf{H}}=\\{{\mathbf{H}}_{a}\\}_{a\in G}$ is
quasitriangular if it has an R-matrix, which is a family of invertible
elements
${\mathcal{R}}=\\{{\mathcal{R}}_{a,b}\in{\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b}\\}_{a,b\in
G}$ satisfying for any $a,b,c\in G$

  1. (1)

${\mathcal{R}}_{a,b}.\Delta_{a,b}(x)=\tau{\left(\Delta_{b,a}(x)\right)}.{\mathcal{R}}_{a,b}$
for any $x\in{\mathbf{H}}_{a+b}$,

  2. (2)

${\left(\operatorname{Id}_{{\mathbf{H}}_{a}}\otimes\Delta_{b,c}\right)}{\mathcal{R}}_{a,b+c}={\left({\mathcal{R}}_{a,c}\right)}_{1b3}{\left({\mathcal{R}}_{a,b}\right)}_{12c}$,

  3. (3)

${\left(\Delta_{a,b}\otimes\operatorname{Id}_{{\mathbf{H}}_{c}}\right)}{\mathcal{R}}_{a+b,c}={\left({\mathcal{R}}_{a,c}\right)}_{1b3}{\left({\mathcal{R}}_{b,c}\right)}_{a23}$

where ${\left({\mathcal{R}}_{a,c}\right)}_{1b3}=r_{a}\otimes 1_{b}\otimes
s_{c},\ {\left({\mathcal{R}}_{a,b}\right)}_{12c}=r_{a}\otimes s_{b}\otimes
1_{c}$ and ${\left({\mathcal{R}}_{b,c}\right)}_{a23}=1_{a}\otimes r_{b}\otimes
s_{c}$ uses the notation $r_{a}\otimes s_{b}$ (with an implied summation) for
the element ${\mathcal{R}}_{a,b}\in{\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b}$.

In [14], Virelizier defines a ribbon Hopf $G$-coalgebra using a ribbon element
(a similar style of definition is given in [11]). Here we give an equivalent
definition in terms of the pivotal structure and a twist element.

######  Definition 2.3 (ribbon structure).

A pivotal quasitriangular Hopf $G$-coalgebra is _ribbon_ if for any $a\in G$
the element $\theta_{a}=m_{a}(\tau((g_{a}\otimes 1_{a})R_{a,a}))$ satisfies

(1) $\theta_{a}=m_{a}{\left((1_{a}\otimes g_{a}^{-1})R_{a,a}\right)}.$

######  Definition 2.4 (symmetrized integral).

A _symmetrized integral_ for a pivotal Hopf $G$-coalgebra is a family of
linear forms ${\mu}=\\{{\mu}_{a}\in{\mathbf{H}}_{a}^{*}\\}_{a\in G}$ such that

(2) $\displaystyle{\left({\mu}_{a}\otimes g_{b}\right)}\Delta_{a,b}(x)$
$\displaystyle={\mu}_{a+b}(x)1_{b}\text{ for }x\in{\mathbf{H}}_{a+b},$ (3)
$\displaystyle{\mu}_{a}(xy)$ $\displaystyle={\mu}_{a}(yx)\ \text{for}\
x,y\in{\mathbf{H}}_{a},$ (4) $\displaystyle{\mu}_{-a}{\left(S_{a}(x)\right)}$
$\displaystyle={\mu}_{a}(x)\ \text{for}\ x\in{\mathbf{H}}_{a}.$

A symmetrized integral is twist non degenerate if the scalar

$\mu_{0}(g_{0}\theta_{0})\mu_{0}(g_{0}^{-1}\theta_{0}^{-1})$

is non zero.

It is known (see [14]) that if ${\mathbf{H}}$ is ribbon and finite type (i.e.
$\dim({\mathbf{H}}_{a})<\infty$ for all $a\in G$), then ${\mathbf{H}}$ has a
unique up to a scalar symmetrized integral closely related to the right
integral. However, the finiteness is not a necessary conditions as we shall
see in Section 4.

###  2.2. The universal invariant of $G$-links

Let ${\mathbf{H}}=\\{{\mathbf{H}}_{a}\\}_{a\in G}$ be a ribbon Hopf
$G$-coalgebra and let
$\text{H\\!H}_{0}({\mathbf{H}}_{a})={\mathbf{H}}_{a}/[{\mathbf{H}}_{a},{\mathbf{H}}_{a}]$
where $[{\mathbf{H}}_{a},{\mathbf{H}}_{a}]$ is the subspace of
${\mathbf{H}}_{a}$ spanned by $xy-yx$ for $x,y\in{\mathbf{H}}_{a}$.

######  Definition 2.5.

A $G$-link is a couple $(L,\omega)$ where $L$ is an oriented framed link
embedded in $\mathbb{R}^{3}$ and $\omega$ is a map from the set of components
of $L$ to $G$. To reduce some technicalities, we will consider $G$-links with
ordered components.

Let $L$ be a $G$-link and denote the color of its $i^{th}$-component by
$a_{i}\in G$. The universal invariant of $L$ is an element
$J(L,\omega)\in\bigotimes_{i}\text{H\\!H}_{0}({\mathbf{H}}_{a_{i}})$ obtained
as follows. Let $D$ be a regular planar diagram of $L$ where each component
has a marked point. For each crossing, cup and cap put beads colored with
elements of the algebra determined by the following diagrams:

$\framebox{$\begin{array}[]{c}\\\
\quad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=17.22217pt}}\end{array}\put(-16.0,1.0){\mbox{\tiny$1_{a_{i}}$}}\qquad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=17.22217pt}}\end{array}\put(-17.0,3.0){\mbox{\tiny$1_{a_{i}}$}}\qquad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=17.22217pt}}\end{array}\put(-35.0,0.0){\mbox{\tiny$g_{a_{i}}$}}\qquad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=17.22217pt}}\end{array}\put(-33.0,2.0){\mbox{\tiny${g^{-1}_{a_{i}}}$}}\\\
\\\
\qquad\qquad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\put(-28.0,-3.0){\mbox{\tiny$r_{a_{i}}$}}\put(-3.0,-3.0){\mbox{\tiny$s_{a_{j}}$}}\qquad\qquad\qquad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\put(-63.0,11.0){\mbox{\tiny$S_{-a_{j}}(r_{-a_{j}}$})}\put(-3.0,10.0){\mbox{\tiny$s_{a_{i}}$}}\qquad\qquad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\put(-28.0,-3.0){\mbox{\tiny$r_{a_{i}}$}}\put(-3.0,-3.0){\mbox{\tiny$S_{-a_{j}}(s_{-a_{j}})$}}\qquad\qquad\qquad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\put(-28.0,11.0){\mbox{\tiny$r_{a_{j}}$}}\put(-3.0,10.0){\mbox{\tiny$s_{a_{i}}$}}\\\
\\\ \end{array}$}$

where the $i$ and $j$ refer to the ordering of the components of $L$ and as
above we write $r_{a_{i}}\otimes s_{a_{j}}$ for ${\mathcal{R}}_{a_{i},a_{j}}$.
The four remaining possible orientations at a crossing are obtained by
reversing simultaneously the orientation of the two strands in the figure
above. For each component of $D$ (starting at the marked point) we multiply
these colored beads with the following rules:

(5)
$\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=43.05542pt}}\end{array}\put(-12.0,8.0){\mbox{\tiny$x$}}\put(-12.0,-9.0){\mbox{\tiny$y$}}\quad\equiv\quad\hskip
4.30554pt\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=43.05542pt}}\end{array}\put(-15.0,2.0){\mbox{\tiny$xy$}}\qquad\text{
and
}\qquad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=43.05542pt}}\end{array}\put(-12.0,12.0){\mbox{\tiny$x$}}\put(-12.0,-5.0){\mbox{\tiny$y$}}\quad\equiv\quad\hskip
4.30554pt\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=43.05542pt}}\end{array}\put(-15.0,2.0){\mbox{\tiny$yx$}}\quad.$

Also beads can freely move around cap and cup with any orientation

(6)
$\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=17.22217pt}}\end{array}\put(-30.0,-1.0){\mbox{\tiny$x$}}\quad\equiv\quad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=17.22217pt}}\end{array}\put(-10.0,-1.0){\mbox{\tiny$x$}}\qquad\text{
and
}\qquad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=17.22217pt}}\end{array}\put(-30.0,4.0){\mbox{\tiny$x$}}\quad\equiv\quad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=17.22217pt}}\end{array}\put(-10.0,4.0){\mbox{\tiny$x$}}\quad.$

and it can pass through a crossing. For the $i^{th}$ component, from this
process we obtain an element in ${\mathbf{H}}_{a_{i}}$ and consider its image
in
$\text{H\\!H}_{0}({\mathbf{H}}_{a_{i}})={\mathbf{H}}_{a_{i}}/[{\mathbf{H}}_{a_{i}},{\mathbf{H}}_{a_{i}}]$.
We define the collection of these images as
$J(D,\omega)\in\bigotimes_{i=1}^{n}\text{H\\!H}_{0}({\mathbf{H}}_{a_{i}})$
where $n$ is the number of components of $L$. The following theorem is a
graded version of Lawrence-Ohtsuki universal invariant ([10, 11]).

######  Theorem 2.6.

The element $J(D,\omega)$ is a diffeomorphism invariant of the $G$-link
$(L,\omega)$. We call this invariant the _universal ${\mathbf{H}}$ invariant
of $L$_ and denote it by $J(L,\omega)$.

###### Proof.

First, because we quotient by the subspace
$[{\mathbf{H}}_{a_{i}},{\mathbf{H}}_{a_{i}}]$ for each $i$, the element
$J(D,\omega)$ does not depend on the choice of the marked points on each
component. We need to check that if $D$ and $D^{\prime}$ are two regular
planar diagram of $L$ then $J(D,\omega)=J(D^{\prime},\omega)$. It is well
known that $D$ and $D^{\prime}$ are related by isotopy in $\mathbb{R}^{2}$ and
a finite sequence of Reidemeister II and III moves and the move:

(7)
$\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\quad\longleftrightarrow\quad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}$

Thus, to prove the theorem we need to show these moves can be translated into
identities satisfied by the ribbon Hopf $G$-coalgebra ${\mathbf{H}}$. To show
the assignment is invariant under isotopies in $\mathbb{R}^{2}$ it is enough
to show it satisfies the move:

$\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\
\longleftrightarrow\
\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\
\longleftrightarrow\
\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\qquad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\
\longleftrightarrow\
\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\
\longleftrightarrow\
\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}$

This move follows directly from the definition of the universal invariant on
cups and caps.

From Proposition 1.4 of [11], the quasitriangular structure of ${\mathbf{H}}$
implies

(8) ${\mathcal{R}}_{a,b}^{-1}=(1_{a}\otimes
S^{-1}_{b}){\mathcal{R}}_{a,-b}=(S_{-a}\otimes 1_{b}){\mathcal{R}}_{-a,b},$
(9)
${\left({\mathcal{R}}_{a,b}\right)}_{12c}{\left({\mathcal{R}}_{a,c}\right)}_{1b3}{\left({\mathcal{R}}_{b,c}\right)}_{a23}={\left({\mathcal{R}}_{b,c}\right)}_{a23}{\left({\mathcal{R}}_{a,c}\right)}_{1b3}{\left({\mathcal{R}}_{a,b}\right)}_{12c},$
(10) $(S_{a}\otimes S_{b}){\mathcal{R}}_{a,b}={\mathcal{R}}_{-a,-b}.$

These relations imply that the Reidemeister II and III moves with all
orientations of strands hold.

Finally, we need to prove Move (7). Since ${\mathbf{H}}$ is ribbon then the
condition in Definition 2.3 implies that the universal invariant satisfies the
move:

$\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}\
\longleftrightarrow\
\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=34.44434pt}}\end{array}$

After applying this move to the first loop in the first diagram of Equation
(7) we can apply Reidemeister II and III moves to unknot the diagram. ∎

We now give two well known properties of the universal invariant, analogous
propositions have been proved in related situations, see [13] and [4].

######  Proposition 2.7.

Let $(L,\omega)$ and $(L^{\prime},\omega^{\prime})$ be $G$-links such that
$(L^{\prime},\omega^{\prime})$ is be obtained from $(L,\omega)$ by reversing
the orientation of the $i$th component and changing the corresponding value of
$\omega$ to its opposite. Then $J(L^{\prime},\omega^{\prime})$ is obtained
from $J(L,\omega)$ by applying the antipode to the $i$th factor.

###### Proof.

This proposition was proved in a special case of certain unrolled quantum
groups in [4, Lemma 5.4]. The proof of [4, Lemma 5.4] works word for word in
the context of this paper after adding the grading on the antipode map,
pivotal element and the R-matrix. ∎

######  Proposition 2.8.

Let $(L,\omega)$ be a $G$-link and $(L^{\prime},\omega^{\prime})$ be obtained
from $L$ by replacing the $i$th component colored by $a+b\in G$ by two
parallel copies colored by $a$ and $b$ respectively. Then
$J(L^{\prime},\omega^{\prime})$ is obtained from $J(L,\omega)$ by applying the
coproduct $\Delta_{a,b}$ to the $i$th factor.

###### Proof.

To prove this proposition it is enough to check on elementary diagrams
consisting of a cup, cap or crossing. For a crossing the relations (2) and (3)
of Definition 2.2 imply the desired equality of beads. For a cup of cap the
equality on beads follows from the fact that the pivotal element is grouplike.
∎

###  2.3. Invariant of $G$-manifolds

From now on we assume $\mathbb{K}$ is an algebraic closed field. A _$G$
-manifold_ is a pair $(M,\omega)$ where $M$ is an oriented closed $3$-manifold
and $\omega\in H^{1}(M,G)$. A surgery presentation of a $G$-manifold $M$ is a
$G$-link $L$ in $S^{3}$ such that $S^{3}_{L}$ is homeomorphic to $M$. Note, we
put an ordering on the components of $L$. Such a link is $G$-colored where the
color of the $i^{th}$ component of $L$ is defined by the value of $\omega$ on
its oriented meridian in $S^{3}\setminus L\subset M$ and denoted by $a_{i}$.

In [13], Virelizier proves the following theorem, we give a sketch of the
proof because similar ideas will be used below when we re-normalize this
invariant. Let ${\mathbf{H}}$ be a ribbon Hopf $G$-coalgebra with a twist non-
degenerate symmetrized integral (we assume
$\mu_{0}(g_{0}^{-1}\theta_{0}^{-1})^{-1}=\mu_{0}(g_{0}\theta_{0})=\delta$ for
some non zero element $\delta\in\mathbb{K}$, which is always possible, up to
rescaling, since $\mathbb{K}$ has square roots).

######  Theorem 2.9.

For each $G$-manifold $(M,\omega)$ the assignment

${\mathrm{HV}}(M,\omega)=\delta^{-s}\bigotimes_{i=1}^{n}\mu_{a_{i}}{\left(J(L,\omega)\right)}\in\mathbb{K},$

is an invariant of $G$-manifolds where $s\in\mathbb{Z}$ is the signature of
the linking matrix of $L$.

###### Proof.

Any two presentations of $M$ are related by isotopy, orientation reversal of
components of $L$ and colored Kirby I and II moves which are a stabilization
with $\pm 1$ framed $0$-colored unknot and sliding along a component of $L$,
respectively. The colored Kirby II move is represented by

$\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=51.6665pt}}\end{array}\put(-38.0,4.0){\mbox{\tiny$a+b$}}\put(0.0,0.0){\mbox{\tiny$b$}}\quad\leftrightarrow\quad\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=51.6665pt}}\end{array}\put(-22.0,4.0){\mbox{\tiny$a$}}\put(2.0,-20.0){\mbox{\tiny$b$}}\put(-13.0,-18.0){\mbox{\tiny$g_{b}$}}\;\;\;.$

To prove the theorem we will show ${\mathrm{HV}}$ is unchanged under these
operations.

Since ${\mu}\circ S={\mu}$ then Proposition 2.7 implies ${\mathrm{HV}}$ is
invariant by orientation reversal of a component of $L$. A Kirby I move
increases the signature of $L$ by $\pm 1$ and also multiplies the invariant by
$\mu_{0}((g_{0}\theta_{0})^{\pm 1})=\delta^{\pm 1}$. Thus, ${\mathrm{HV}}$ is
globally unchanged under a Kirby I move. Finally, invariance by a colored
Kirby II move follow from Proposition 2.8 and the first property of the
symmetrized integral. Remark that the invariance under this move only uses
that the components colored with $a+b$ and $a$ on the picture and are
evaluated with the symmetrized integral. ∎

We call ${\mathrm{HV}}$ the _graded Hennings-Virelizier invariant_. When the
grading is trivial (i.e. $G=0$) then ${\mathrm{HV}}$ is the Hennings invariant
defined in [9]. As we will see in Section 4 there is a Ribbon Hopf
$\mathbb{C}/2\mathbb{Z}$-coalgebra ${\mathbf{U}}$, associated to
${\mathfrak{sl}_{2}}$, with a twist non-degenerate symmetrized integral. The
degree zero restriction of the invariant ${\mathrm{HV}}$ associated to
${\mathbf{U}}$ is the Hennings invariant associated to usual quantum group of
${\mathfrak{sl}_{2}}$. However, in this example ${\mathrm{HV}}$ vanishes in
non-integral degree. Next we will discuss how to modify the integral so that
one can produce a non-zero invariant where it previously vanished.

##  3\. The modified symmetrized integral and modified invariant

###  3.1. Ambidexterity of the symmetrized integral

Let ${\mathbf{H}}$ be a finite type pivotal Hopf $G$-coalgebra with a
symmetrized integral ${\mu}$.

For a linear endomorphism $f$ of a finite dimensional $\mathbb{K}$-vector
space $V\otimes W$ we denote by
$\operatorname{ptr}^{\mathbb{K}}_{W}(f)\in\operatorname{End}_{\mathbb{K}}(V)$
the right partial trace of $f$ given by
$\operatorname{ptr}^{\mathbb{K}}_{W}(f)(v)=\sum_{i}(\operatorname{Id}_{V}\otimes
w_{i}^{*})(f(v\otimes w_{i}))$ for any basis $\\{w_{i}\\}$ of $W$ with dual
basis $\\{w_{i}^{*}\\}$. We define similarly the left partial trace of $f$
denoted
$\operatorname{ptr}^{\mathbb{K}}_{V}(f)\in\operatorname{End}_{\mathbb{K}}(W)$.
Remark that denoting by $\operatorname{tr}^{\mathbb{K}}$ the usual linear
trace, we have
$\operatorname{tr}^{\mathbb{K}}_{V}(\operatorname{ptr}^{\mathbb{K}}_{W}(f))=\operatorname{tr}^{\mathbb{K}}_{W}(\operatorname{ptr}^{\mathbb{K}}_{V}(f))=\operatorname{tr}^{\mathbb{K}}_{V\otimes
W}(f)\in\mathbb{K}$.

Let $L_{x}:{\mathbf{H}}_{a}\to{\mathbf{H}}_{a}$ be the left multiplication by
$x\in{\mathbf{H}}_{a}$.

######  Proposition 3.1.

Let $a,b\in G$ and
$f\in\operatorname{End}_{\mathbb{K}}({\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b})$
be a morphism of ${\mathbf{H}}_{a+b}$-module (i.e. $f(\Delta_{a,b}(x).y\otimes
z)=\Delta_{a,b}(x)f(y\otimes z)$). Then

(11)
${\mu}_{a}(\operatorname{ptr}^{\mathbb{K}}_{{\mathbf{H}}_{b}}{\left((\operatorname{Id}\otimes
L_{g_{b}})f\right)}(1_{a}))={\mu}_{b}(\operatorname{ptr}^{\mathbb{K}}_{{\mathbf{H}}_{a}}{\left((L_{g_{a}^{-1}}\otimes\operatorname{Id})f\right)}(1_{b}))$

###### Proof.

The proof is based on the theory of modified trace: The guiding principle is
that by [2, 8] the symmetrized integral is the modified trace and we transpose
the partial trace property of modified trace to a property for the symmetrized
integral.

In this proof we use the notation and result of [8]. Let
${\operatorname{\mathscr{C}}}$ be the pivotal category of left
${\mathbf{H}}$-modules (an object of ${\operatorname{\mathscr{C}}}$ is a
finite dimensional $\mathbb{K}$-vector space equipped with a structure of left
${\mathbf{H}}_{a}$-module for some $a\in G$). For $a\in G$, the action of left
multiplication on ${\mathbf{H}}_{a}$ makes ${\mathbf{H}}_{a}$ into a
projective object of ${\operatorname{\mathscr{C}}}$. A _cyclic trace_ on the
subcategory of projective object $\operatorname{\mathsf{Proj}}$ of
${\operatorname{\mathscr{C}}}$ is a family of linear function
${\left\\{\operatorname{\mathsf{t}}_{P}:\operatorname{End}_{\operatorname{\mathscr{C}}}(P)\to\mathbb{K}\right\\}}_{P\in\operatorname{\mathsf{Proj}}}$
satisfying
$\operatorname{\mathsf{t}}_{P}(fg)=\operatorname{\mathsf{t}}_{Q}(gf)$ for any
$f:Q\to P$ and $g:Q\to P$. By [2, Proposition 2.4] and Definition 2.4 (2),
there is a unique cyclic trace $\operatorname{\mathsf{t}}$ such that for any
$a\in G$ and
$f\in\operatorname{End}_{{\operatorname{\mathscr{C}}}}({\mathbf{H}}_{a})$ we
have $\operatorname{\mathsf{t}}_{{\mathbf{H}}_{a}}(f)={\mu}_{a}(f(1_{a}))$.
Definition 2.4 (1) implies that ${\mu}$ is a right symmetrized $G$-integral,
thus by [8, Theorem 1.1], $\operatorname{\mathsf{t}}$ is a right modified
trace thus satisfies the right partial trace property which implies that for
any
$f\in\operatorname{End}_{\operatorname{\mathscr{C}}}({\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b})$,

$\operatorname{\mathsf{t}}_{{\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b}}(f)=\operatorname{\mathsf{t}}_{{\mathbf{H}}_{a}}(\operatorname{ptr}^{\operatorname{\mathscr{C}}}_{{\mathbf{H}}_{b}}(f))$

where $\operatorname{ptr}^{\operatorname{\mathscr{C}}}_{{\mathbf{H}}_{b}}$ is
the partial trace in ${\operatorname{\mathscr{C}}}$ which uses the pivotal
structure of ${\operatorname{\mathscr{C}}}$. Now Definition 2.4 (3) implies
that ${\mu}$ is also a left symmetrized $G$-integral (i.e. $({\mathbf{H}},g)$
is unibalanced) thus $\operatorname{\mathsf{t}}$ satisfies the left partial
trace property:
$\operatorname{\mathsf{t}}_{{\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b}}(f)=\operatorname{\mathsf{t}}_{{\mathbf{H}}_{b}}(\operatorname{ptr}^{\operatorname{\mathscr{C}}}_{{\mathbf{H}}_{a}}(f))$.
Since in ${\operatorname{\mathscr{C}}}$ the left and right categorical partial
trace are given by the vector space partial trace precomposed with the action
of $g_{a}^{-1}$ and $g_{b}$ respectively, combining the two equalities gives
the desired identity. ∎

######  Remark 3.2.

Even if the grading is trivial and ${\mathbf{H}}$ is an ordinary unibalanced
unimodular Hopf algebra, we do not know a proof without using modified traces
of Proposition 3.1 which shows an unexpected result about the integral.

###  3.2. The modified symmetrized integral

Let

$\Delta_{a_{1},...,a_{n}}:{\mathbf{H}}_{\sum_{i=1}^{n}a_{i}}\to{\mathbf{H}}_{a_{1}}\otimes\cdots\otimes{\mathbf{H}}_{a_{n}}$

be the $(n-1)$-fold coproduct. For example,

$\Delta_{a_{1},a_{2},a_{3}}=\left(\Delta_{a_{1},a_{2}}\otimes\operatorname{Id}_{{\mathbf{H}}_{a_{3}}}\right)\circ\Delta_{a_{1}+a_{2},a_{3}}={\left(\operatorname{Id}_{{\mathbf{H}}_{a_{1}}}\otimes\Delta_{a_{2},a_{3}}\right)}\circ\Delta_{a_{1},a_{2}+a_{3}}.$

Let $C({\mathbf{H}}_{a_{1}}\otimes\cdots\otimes{\mathbf{H}}_{a_{n}})$ be the
subspace of $\otimes_{i=1}^{n}{\mathbf{H}}_{a_{i}}$ of all elements which
commute with the image of $\Delta_{a_{1},...,a_{n}}$, in other words all
$x\in\otimes_{i=1}^{n}{\mathbf{H}}_{a_{i}}$ such that

$x\Delta_{a_{1},...,a_{n}}(y)=\Delta_{a_{1},...,a_{n}}(y)x$

for all $y\in{\mathbf{H}}_{\sum_{i=1}^{n}a_{i}}$ (note here if $n=1$ then
$\Delta_{a_{1}}=\operatorname{Id}_{{\mathbf{H}}_{a_{1}}}$).

######  Lemma 3.3.

Recall $L_{x}:{\mathbf{H}}_{a}\to{\mathbf{H}}_{a}$ is the left multiplication
by $x\in{\mathbf{H}}_{a}$. Then

${\mu}_{a_{1}}L_{g_{a_{1}}^{-1}}\otimes\operatorname{Id}(C({\mathbf{H}}_{a_{1}}\otimes\cdots\otimes{\mathbf{H}}_{a_{n}}))\subset
C({\mathbf{H}}_{a_{2}}\otimes\cdots\otimes{\mathbf{H}}_{a_{n}})$ $\text{and
}\operatorname{Id}\otimes{\mu}_{a_{n}}L_{g_{a_{n}}}(C({\mathbf{H}}_{a_{1}}\otimes\cdots\otimes{\mathbf{H}}_{a_{n}}))\subset
C({\mathbf{H}}_{a_{1}}\otimes\cdots\otimes{\mathbf{H}}_{a_{n-1}})$

###### Proof.

We prove the first inclusion, the second is similar. In this proof we use
implied summation when using coproducts. Let
$y\in{\mathbf{H}}_{\sum_{i=2}^{n}a_{i}}$ we use the following notation:

$\otimes_{i=0}^{n}y_{i}=\Delta_{-a_{1},a_{1},a_{2},...,a_{n}}(y)=\left((\Delta_{-a_{1},a_{1}}\otimes\operatorname{Id}_{\otimes_{i=2}^{n}{\mathbf{H}}_{a_{i}}})\Delta_{0,a_{2},...,a_{n}}\right)(y)$

is an element of
${\mathbf{H}}_{-a_{1}}\otimes{\mathbf{H}}_{a_{1}}\otimes(\otimes_{i=2}^{n}{\mathbf{H}}_{a_{i}})$.
Then the defining property of the antipode imply

$1_{{\mathbf{H}}_{a_{1}}}\otimes(\Delta_{a_{2},...,a_{n}}(y))=S_{-a_{1}}(y_{0})y_{1}\otimes(\otimes_{i=2}^{n}y_{i})=y_{1}S^{-1}_{a_{1}}(y_{0})\otimes(\otimes_{i=2}^{n}y_{i}).$

We use this equation in the first and last equalities of the following
calculation: for $\otimes_{i=1}^{n}x_{i}\in
C(\otimes_{i=1}^{n}{\mathbf{H}}_{a_{i}})$ with $x_{i}\in{\mathbf{H}}_{a_{i}}$
we have

$\displaystyle{\mu}_{a_{1}}(g^{-1}_{a_{1}}x_{1})(\Delta_{a_{2},...,a_{n}}(y)).(\otimes_{i=2}^{n}x_{i})$
$\displaystyle={\mu}_{a_{1}}(g_{a_{1}}^{-1}S_{-a_{1}}(y_{0})y_{1}x_{1})(\otimes_{i=2}^{n}y_{i}).(\otimes_{i=2}^{n}x_{i})$
$\displaystyle={\mu}_{a_{1}}(g_{a_{1}}^{-1}S_{-a_{1}}(y_{0})x_{1}y_{1})(\otimes_{i=2}^{n}x_{i}).(\otimes_{i=2}^{n}y_{i})$
$\displaystyle={\mu}_{a_{1}}(S_{a_{1}}^{-1}(y_{0})g_{a_{1}}^{-1}x_{1}y_{1})(\otimes_{i=2}^{n}x_{i}).(\otimes_{i=2}^{n}y_{i})$
$\displaystyle={\mu}_{a_{1}}(g_{a_{1}}^{-1}x_{1}y_{1}S_{a_{1}}^{-1}(y_{0}))(\otimes_{i=2}^{n}x_{i}).(\otimes_{i=2}^{n}y_{i})$
$\displaystyle={\mu}_{a_{1}}(g_{a_{1}}^{-1}x_{1})(\otimes_{i=2}^{n}x_{i}).(\Delta_{a_{2},...,a_{n}}(y))$

where the second equality comes from the fact that $\otimes_{i=1}^{n}x_{i}$ is
in $C(\otimes_{i=1}^{n}{\mathbf{H}}_{a_{i}})$, the third from the property of
the pivot and the forth from the cyclic property of the integral. ∎

######  Definition 3.4.

Let $X\subset G$ and set $G^{\prime}=G\setminus X$. A modified symmetrized
integral on $G^{\prime}$ is a family of $\mathbb{K}$-linear maps

${\mu}^{\prime}=\\{{\mu}^{\prime}_{a}:\
C({\mathbf{H}}_{a})\to\mathbb{K}\\}_{a\in G^{\prime}}$

satisfying for any $a,b\in G^{\prime}$

${\mu}_{a}L_{g_{a}^{-1}}\otimes{\mu}^{\prime}_{b}={\mu}^{\prime}_{a}\otimes{\mu}_{b}L_{g_{b}}\text{
on }C({\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b}).$

######  Theorem 3.5.

Assume $\mathbb{K}$ is an algebraically closed field. Let ${\mathbf{H}}_{a}$
be finite dimensional and semi-simple for all $a\in G^{\prime}$. Then there
exists a family ${\left\\{z_{a}\in C({\mathbf{H}}_{a})\right\\}}_{a\in
G^{\prime}}$ such that

${\mu}_{a}(x)=\operatorname{tr}_{{\mathbf{H}}_{a}}^{\mathbb{K}}(L_{z_{a}x})\text{
for all }x\in{\mathbf{H}}_{a}.$

Furthermore, there is a modified symmetrized integral on $G^{\prime}$ defined
by

${\mu}^{\prime}_{a}(z)={\mu}_{a}(z_{a}z)=\operatorname{tr}_{{\mathbf{H}}_{a}}^{\mathbb{K}}(L_{z_{a}^{2}z})\text{
for all }z\in C({\mathbf{H}}_{a}).$

###### Proof.

Since $\mathbb{K}$ is an algebraically closed field then for each $a\in
G^{\prime}$ the algebra ${\mathbf{H}}_{a}$ is a product of matrix algebra.
Since any linear form on $M_{n}(\mathbb{K})$ can be realized by
$M\mapsto\operatorname{tr}^{\mathbb{K}}_{M_{n}(\mathbb{K})}(L_{N}L_{M})$ for a
unique $N\in M_{n}(\mathbb{K})$ we have ${\mu}_{a}$ is equal to
$x\mapsto\operatorname{tr}^{\mathbb{K}}_{{\mathbf{H}}_{a}}(L_{z_{a}x})$ for
some unique $z_{a}\in{\mathbf{H}}_{a}$. Property (2) of Definition 2.4 implies
that $z_{a}\in C({\mathbf{H}}_{a})$.

For the second assertion we need to show that ${\mu}^{\prime}_{a}$ is a
modified symmetrized integral on $G^{\prime}$. Let $a,b\in G^{\prime}$ and
$x=x^{\prime}\otimes x^{\prime\prime}\in
C({\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b})$ (summation symbol omitted). Then
let $\tilde{x}=(z_{a}\otimes z_{b}).x\in
C({\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b})$ so that $L_{\tilde{x}}$ is an
endomorphism of the ${\mathbf{H}}_{a+b}$-module
${\mathbf{H}}_{a}\otimes{\mathbf{H}}_{b}$ as in Proposition 3.1. Applying the
proposition, we get following equivalences of equalities:

$\displaystyle{\mu}_{b}(\operatorname{ptr}^{\mathbb{K}}_{{\mathbf{H}}_{a}}{\left((L_{g_{a}^{-1}}\otimes\operatorname{Id})L_{\tilde{x}}\right)}(1_{b}))$
$\displaystyle={\mu}_{a}(\operatorname{ptr}^{\mathbb{K}}_{{\mathbf{H}}_{b}}{\left((\operatorname{Id}\otimes
L_{g_{b}})L_{\tilde{x}}\right)}(1_{a}))$
$\displaystyle{\mu}_{b}(\operatorname{ptr}^{\mathbb{K}}_{{\mathbf{H}}_{a}}(L_{g_{a}^{-1}z_{a}x^{\prime}})z_{b}x^{\prime\prime})$
$\displaystyle={\mu}_{a}(z_{a}x^{\prime}\operatorname{ptr}^{\mathbb{K}}_{{\mathbf{H}}_{b}}(L_{g_{b}z_{b}x^{\prime\prime}}))$
$\displaystyle\operatorname{tr}^{\mathbb{K}}_{{\mathbf{H}}_{a}}(L_{g_{a}^{-1}z_{a}x^{\prime}}){\mu}_{b}(z_{b}x^{\prime\prime})$
$\displaystyle={\mu}_{a}(z_{a}x^{\prime})\operatorname{tr}^{\mathbb{K}}_{{\mathbf{H}}_{b}}(L_{g_{b}z_{b}x^{\prime\prime}})$
$\displaystyle{\mu}_{a}L_{g_{a}^{-1}}\otimes{\mu}^{\prime}_{b}(x)$
$\displaystyle={\mu}^{\prime}_{a}\otimes{\mu}_{b}L_{g_{b}}(x)$

and the proposition follows. ∎

######  Remark 3.6.

The central elements $z_{a}$ are determined via a basis
${\left\\{z_{i}\right\\}}$ of the center $C({\mathbf{H}}_{a})$ in which
$z_{i}$ is the element corresponding to the identity matrix of
$i{\textsuperscript{th}}$-factor in the decomposition
${\mathbf{H}}_{a}\cong\prod_{i}\operatorname{End}(V_{i})$,
$V_{i}\simeq\mathbb{K}^{n_{i}}$ are irreducible representations of
${\mathbf{H}}_{a}$. Remark that
$\operatorname{tr}^{\mathbb{K}}_{{\mathbf{H}}_{b}}(L_{z_{i}})=\dim_{\mathbb{K}}(\operatorname{End}(V_{i}))=n_{i}^{2}$.
In particular, if $\mathbb{K}$ has characteristic $0$ then
$z_{a}=\sum_{i}\frac{d_{i}}{n_{i}}z_{i}$ where $d_{i}$ is an element of
$\mathbb{K}$ (note $d_{i}$ can be interpreted as the modified dimension of
$V_{i}$, see [8]). Note ${\mu}_{a}(z_{i})=n_{i}d_{i}$ then
${\mu}^{\prime}_{a}(z_{i})=d_{i}^{2}$ and
${\mu}^{\prime}_{a}(1)={\mu}_{a}(z_{a})=\sum_{i}d_{i}^{2}$.

###  3.3. The modified invariant

For this section let ${\mathbf{H}}$ be a ribbon Hopf $G$-coalgebra with a
twist non-degenerate symmetrized integral and a modified symmetrized integral
$\mu^{\prime}=(\mu^{\prime}_{a})_{a\in G^{\prime}}$.

An admissible $G$-manifold is a $G$-manifold with the requirement that the
cohomology class $\omega\in H^{1}(M,G)\simeq\operatorname{Hom}(H_{1}(M),G)$
has a value in $G^{\prime}$ i.e. $\operatorname{Im}(\omega)\cap
G^{\prime}\neq\varnothing$. A surgery presentation $L$ of a $G$-manifold is
called computable if there exists a component $L_{i}$ of $L$ such that
$\omega(m_{i})\in G^{\prime}$ where $m_{i}$ is a meridian of $L_{i}$.

######  Lemma 3.7.

Let $M$ be an admissible $G$-manifold. There exists a computable surgery
presentation of $M$.

###### Proof.

Consider any link presentation $L$ of $M$. Since
$\operatorname{Im}(\omega)\subset G$ is generated by its values on the
meridians of $L$, there exists a sequence $a_{1},\ldots a_{k}\in G$ such that
$\sum_{j=1}^{k}a_{j}\in G^{\prime}$ with $a_{j}=\omega(m_{i_{j}})$ where
$m_{i}$ is a meridian of the $i^{th}$-component of $L$. We will show that a
series of Kirby moves can be applied to $L$ to obtain the desired computable
presentation as follows. Isotope $L$ so that all the strands with the colors
$a_{j}$ are in a small ball as in the left side of the diagram in Equation
(12). Now do two Kirby I moves to create both a $+1$ and $-1$ framed
$0$-colored unknot. Sliding one of these unknots over the other we obtain a
Hopf link where one of the components is zero framed. Sliding the strands
colored with the $a_{j}$’s over this zero framed component we obtain the right
side of Equation (12):

(12)
$\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=43.05542pt}}\end{array}\put(-62.0,-2.0){\mbox{\tiny$a_{1}$}}\put(-50.0,-2.0){\mbox{\tiny$a_{2}$}}\put(-13.0,-2.0){\mbox{\tiny$a_{k}$}}\longrightarrow\begin{array}[]{c}\raisebox{-4.0pt}{\psfig{height=43.05542pt}}\end{array}\put(-120.0,-10.0){\mbox{\tiny$0$}}\put(-6.0,-25.0){\mbox{\tiny$\sum_{j}a_{j}$}}.$

∎

Let $L$ be a $G$-link with $n$ components. A $k$-string $G$-link $T$ with
$\ell$ closed component is a $(k,k)$-pure framed oriented tangle with $k+\ell$
ordered components colored by elements of $G$. Its closure $\hat{T}$ is a
$G$-link in $S^{3}$ with $k+\ell$ components obtained by taking the braid
closure (which is well defined up to isotopy). We define as in Section 2.2 a
universal invariant $J(T)$ by multiplying the beads along components. The
proof of the following proposition follows in the same way as the proof of
Theorem 2.6.

######  Proposition 3.8.

Let $T$ as above, let $H^{T}_{i}={\mathbf{H}}_{a_{i}}$ if the ith component of
$T$ is open and $H^{T}_{i}=\text{H\\!H}_{0}({\mathbf{H}}_{a_{i}})$ if the ith
component of $T$ is closed. The element
$J(T)\in\bigotimes_{i=1}^{n}{H^{T}_{i}}$ is an isotopy invariant of $T$.
Furthermore, let
$cl:{\mathbf{H}}_{a_{i}}\to\text{H\\!H}_{0}({\mathbf{H}}_{a_{i}})$ be the map
that send $x$ to the class of $g_{a_{i}}x$ then we have

$cl^{\otimes k}(J(T))=J(\hat{T})$

Suppose $(M,\omega)$ is an $G$-manifold with surgery presentation $L$. Let
$T_{j}$ be a $1$-string $G$-link whose closure is $L$. Then the universal
invariant of $T_{j}$ is given by

$J{\left(T_{j},\omega\right)}\in\bigotimes_{i=1}^{j-1}\text{H\\!H}_{0}({\mathbf{H}}_{a_{i}})\otimes{\mathbf{H}}_{a_{j}}\otimes\bigotimes_{i=j+1}^{n}\text{H\\!H}_{0}({\mathbf{H}}_{a_{i}}).$

######  Lemma 3.9.

We have

${\left(\bigotimes_{i=1}^{j-1}{\mu}_{a_{i}}\otimes\operatorname{Id}_{{\mathbf{H}}_{a_{j}}}\otimes\bigotimes_{i=j+1}^{n}{\mu}_{a_{i}}\right)}{\left(J{\left(T_{j},\omega\right)}\right)}\in
C({\mathbf{H}}_{a_{j}}).$

###### Proof.

Consider a $n$-string $G$-link $T$ whose braid closure of the $n-1$ right-most
strands produces $T_{j}$. Then one can prove (see for example [4, Theorem
5.1]) that $J(T)\in C(\bigotimes_{i=1}^{n}{\mathbf{H}}_{a_{i}})$. Then
applying $n-1$ times Lemma 3.3 we have the above property. ∎

######  Remark 3.10.

Applying ${\mu}_{a_{i}}$ to $g_{a_{i}}$ times the element of the previous
lemma produces $\delta^{s}{\mathrm{HV}}(M,\omega)$. Suppose now that for $a\in
G^{\prime}$, $H_{a}$ is semi-simple and all its irreducible representations
have zero quantum dimension (where the quantum dimension is defined as the
trace on the action of $g_{a}$). Then it follows that $g_{a}C(H_{a})$ is in
the kernel of ${\mu}_{a}$. In this situation, we get that
${\mathrm{HV}}(M,\omega)=0$ for any admissible $G$-manifold $(M,\omega)$. This
motivate the renormalized invariant of Theorem 3.11.

Now let $(M,\omega)$ be an admissible $G$-manifold with computable surgery
presentation $L$. Suppose the coloring of the $j^{th}$ component of $L$ is in
$G^{\prime}$ then set

${\mathrm{HV}}^{\prime}{\left(M,\omega\right)}=\delta^{-s}\bigotimes_{i=1}^{j-1}{\mu}_{a_{i}}\otimes{\mu}^{\prime}_{a_{j}}\otimes\bigotimes_{i=j+1}^{n}{\mu}_{a_{i}}{\left(J{\left(T_{j},\omega\right)}\right)}.$

######  Theorem 3.11.

${\mathrm{HV}}^{\prime}{\left(M,\omega\right)}$ is an invariant of the
admissible $G$-manifold $(M,\omega)$.

###### Proof.

Given a computable $G$-link $L$ with one marked point on one component colored
by an element of $G^{\prime}$, consider the complement of a small 3-ball
centered at the marked point. It is diffeomorphic to a $1$-string $G$-link
whose closure is $L$. Let denote by ${\mathrm{HV}}^{\prime}(L)$ the image of
this $1$-string $G$-link by the map of Lemma 3.9. The scalar
${\mathrm{HV}}^{\prime}(L)$ only depends of the isotopy class of $L$. Property
of $\mu^{\prime}$ implies that moving the marked point from one
$G^{\prime}$-colored component to an other does not change
${\mathrm{HV}}^{\prime}(L)$. The proof of Theorem 2.9 implies the scalar
${\mathrm{HV}}^{\prime}(L)$ is also invariant by Kirby I moves between
computable links. Moreover, it is also invariant by any Kirby II moves when
one slides any strand of a component that does not have the marked point. We
call such Kirby II moves admissible.

Let $L^{0},L^{N}$ be two computable $G$-link with marked points $m_{0}$ and
$m_{N}$, respectively. Then $L^{0}$ and $L^{N}$ are related by a sequence of
ambient isotopies and Kirby I and II moves which correspond to a sequence of
$G$-links $L^{1},L^{2},\cdots,L^{N-1}$. Here it is possible that these Kirby
moves are not admissible (two problems can happen: no $G^{\prime}$ color or
one of the moves could slide over the strand with the marked point). However,
we will show this sequence can be used to produce a new sequence that only
contains admissible Kirby moves. For this we perform a series of admissible
Kirby moves similar to the moves depicted in Equation (12): close to a strand
of $L^{0}$ colored with an element of $G^{\prime}$ do two Kirby I moves and a
Kirby II move to create a Hopf link with a zero and a $+1$ framed component.
Slide the $G^{\prime}$ colored strand of $L^{0}$ over the zero framed
component to create an link with a new $G^{\prime}$ colored unknot (as in the
right side of Equation (12)). Move the marked point $m_{0}$ to this unknot to
create a new admissible link presentation contains $L^{0}$ as a sublink. Now
we can perform the same Kirby moves ignoring the newly created Hopf link to
get a sequence of admissible Kirby moves
$\overline{L^{0}}\to\overline{L^{1}},\cdots,\overline{L^{N}}$. By construction
$\overline{L^{N}}$ is the link $L^{N}$ with a Hopf link stabilization as
depicted in the right side of Equation (12) with a marked point on the zero
framed component of the Hopf link. Move this marked point to the marked point
$m_{N}$ on the sublink $L^{N}$. Preform the reverse admissible Kirby moves to
remove the Hopf link of $\overline{L^{N}}$ and obtain $L^{N}$. Since all moves
are admissible we have that
${\mathrm{HV}}^{\prime}(L^{0})={\mathrm{HV}}^{\prime}(L^{N})$. ∎

##  4\. Examples from quantum groups

Here we show the unrolled quantum group associated to ${\mathfrak{sl}_{2}}$
gives rise to ribbon Hopf coalgebra with a modified symmetrized integral. We
first describe a Hopf algebra that embed into the topological unrolled quantum
group $\widehat{U_{\xi}^{H}}{\mathfrak{sl}_{2}}$ of [4].
Let $\ell\geq 3$ be an integer, let $\ell^{\prime}=\ell/\gcd(\ell,2)$ and
$\xi={\operatorname{e}}^{\frac{2\pi i}{\ell}}$. We use the notations
$\xi^{x}=\exp(2i\pi x/\ell),\ {\left\\{x\right\\}}=\xi^{x}-\xi^{-x}$ and
${\left\\{n\right\\}}!={\left\\{n\right\\}}{\left\\{n-1\right\\}}...{\left\\{1\right\\}}$.

Let $\mathcal{U}^{K}$ be the $\mathbb{C}$-algebra with generators $E$, $F$,
$K^{\alpha}$ where $\alpha\in\mathbb{C}$ with relations: for any
$\alpha,\beta\in\mathbb{C},$

$\displaystyle K^{\alpha}K^{\beta}$ $\displaystyle=K^{\alpha+\beta},$
$\displaystyle K^{\alpha}EK^{-\alpha}$ $\displaystyle=\xi^{2\alpha}E,$
$\displaystyle K^{\alpha}FK^{-\alpha}$ $\displaystyle=\xi^{-2\alpha}F,$
$\displaystyle K^{0}$ $\displaystyle=1,$ $\displaystyle[E,F]$
$\displaystyle=\frac{K-K^{-1}}{\xi-\xi^{-1}},$ $\displaystyle
E^{\ell^{\prime}}=$ $\displaystyle 0=F^{\ell^{\prime}}.$

The algebra $\mathcal{U}^{K}$ is a Hopf algebra where the coproduct, counit
and antipode are defined by

$\displaystyle\Delta(E)$ $\displaystyle=1\otimes E+E\otimes K,$
$\displaystyle\varepsilon(E)$ $\displaystyle=0,$ $\displaystyle S(E)$
$\displaystyle=-EK^{-1},$ $\displaystyle\Delta(F)$
$\displaystyle=K^{-1}\otimes F+F\otimes 1,$ $\displaystyle\varepsilon(F)$
$\displaystyle=0,$ $\displaystyle S(F)$ $\displaystyle=-KF,$
$\displaystyle\Delta(K^{\alpha})$ $\displaystyle=K^{\alpha}\otimes K^{\alpha}$
$\displaystyle\varepsilon(K^{\alpha})$ $\displaystyle=1,$ $\displaystyle
S(K^{\alpha})$ $\displaystyle=K^{-\alpha}.$

The Hopf algebra $\mathcal{U}^{K}$ is pivotal with pivot
$g=K^{1-\ell^{\prime}}$. Let $G=\mathbb{C}/2\mathbb{Z}$. For ${a}\in G$, set
${\mathbf{U}}_{a}=\mathcal{U}^{K}/(K^{\ell/2}-\xi^{\ell a/2})$. The Hopf
algebra structure of $\mathcal{U}^{K}$ induces a structure of pivotal Hopf
$G$-coalgebra on the family
${\mathbf{U}}={\left\\{{\mathbf{U}}_{a}\right\\}}_{a\in G}$. We denote the
coproduct, counit and antipode of ${\mathbf{U}}$ by
$\Delta={\left\\{\Delta_{a,b}\right\\}}_{a,b\in G},\ \varepsilon,$ and
$S={\left\\{S_{{a}}\right\\}}_{a\in G}$, respectively.

In [11], Ohtsuki proved a version of the following theorem for a colored Hopf
algebra closely related to ${\mathbf{U}}$.

######  Theorem 4.1.

The Hopf $G$-coalgebra ${\mathbf{U}}$ is ribbon with $R$-matrix

$\displaystyle{\mathcal{R}}_{a,b}={\mathcal{H}}_{a,b}\check{{\mathcal{R}}}_{a,b}\in{\mathbf{U}}_{a}\otimes{\mathbf{U}}_{b},$
with
$\displaystyle\check{{\mathcal{R}}}_{a,b}=\sum_{n=0}^{\ell^{\prime}-1}\frac{{\left\\{1\right\\}}^{2n}}{{\left\\{n\right\\}}!}\xi^{\frac{n(n-1)}{2}}E^{n}\otimes
F^{n}\in{\mathbf{U}}_{a}\otimes{\mathbf{U}}_{b}$ and where
$\displaystyle{\mathcal{H}}_{a,b}=\frac{1}{\ell^{\prime}}\sum_{m_{1},m_{2}=0}^{\ell^{\prime}-1}\xi^{-2(m_{1}+\beta)(m_{2}+\alpha)}K^{\beta+m_{1}}\otimes
K^{\alpha+m_{2}}\in{\mathbf{U}}_{a}\otimes{\mathbf{U}}_{b}$

does not depends of $\alpha,\beta\in\mathbb{C}$ congruent to
$\frac{a}{2},\frac{b}{2}$ respectively modulo $\mathbb{Z}$.

###### Proof.

In this proof we use the topological ribbon Hopf algebra
$\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}$ defined in [4, Examples
2.3 and 2.7 and Section 3.2]. For $a\in\mathbb{C}/2\mathbb{Z}$, let
$I_{a}=(\xi^{\ell H}-\xi^{\ell
a/2})\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}$ the ideal of
$\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}$. The assignment $E\mapsto
E$, $F\mapsto F$ and $K^{\alpha}\mapsto\xi^{2\alpha H}$ extends to an
embedding of the Hopf algebra $\mathcal{U}^{K}$ into
$\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}$ which induces an
embedding of (topological) Hopf $G$-coalgebra
$\\{{\mathbf{U}}_{a}\to\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}/I_{a}\\}_{a\in
G}$. We will use this embedding to define a ribbon structure on ${\mathbf{U}}$
as follows.

The $R$-matrix element of $\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}$
is given by

${\mathcal{R}}=\xi^{{2}{H\otimes
H}}\sum_{n=0}^{\ell^{\prime}-1}\frac{{\left\\{1\right\\}}^{2n}}{{\left\\{n\right\\}}!}\xi^{\frac{n(n-1)}{2}}E^{n}\otimes
F^{n}\in\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}\widehat{\otimes}\,\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}.$

We get the $R$-matrix of the Hopf $G$-coalgebra ${\mathbf{U}}$ by applying the
discrete Fourier transform in [7, Proposition 4.6] as follow: Let
$a,b\in\mathbb{C}/2\mathbb{Z}$ and $\alpha$, $\beta\in\mathbb{C}$ such that
$a\equiv 2\alpha,b\equiv 2\beta$ modulo $2\mathbb{Z}$. The function
$\xi^{2(H-\alpha)\otimes(H-\beta)}$ is $\ell^{\prime}$-periodic on
$(a,b)=(\alpha,\beta)+\mathbb{Z}^{2}$ thus it can be expressed as a polynomial
in $K\otimes 1=\xi^{2H\otimes 1}$ and $1\otimes K=\xi^{1\otimes 2H}$.

$\mathcal{F}_{(a,b)}{\left(\xi^{2(H-\alpha)\otimes(H-\beta)}\right)}=\sum_{m_{1},m_{2}=0}^{\ell^{\prime}-1}c_{m_{1}m_{2}}K^{m_{1}}\otimes
K^{m_{2}},$

where

$\displaystyle c_{m_{1}m_{2}}$ $\displaystyle=\frac{1}{\ell^{\prime
2}}\sum_{i_{1},i_{2}=0}^{\ell^{\prime}-1}\xi^{-2m_{1}(\alpha+i_{1})-2m_{2}(\beta+i_{2})}\xi^{2i_{1}i_{2}}$
$\displaystyle=\frac{1}{\ell^{\prime
2}}\xi^{-2m_{1}\alpha-2m_{2}\beta}\sum_{i_{1}=0}^{\ell^{\prime}-1}\xi^{-2m_{1}i_{1}}\sum_{i_{2}=0}^{\ell^{\prime}-1}\xi^{2(i_{1}-m_{2})i_{2}}$
$\displaystyle=\frac{1}{\ell^{\prime
2}}\xi^{-2m_{1}\alpha-2m_{2}\beta}\sum_{i_{1}=0}^{\ell^{\prime}-1}\xi^{-2m_{1}i_{1}}\ell^{\prime}\delta_{0,i_{1}-m_{2}}$
$\displaystyle=\frac{1}{\ell^{\prime}}\xi^{-2m_{1}\alpha-2m_{2}\beta}\xi^{-2{m_{1}m_{2}}}.$

The third equality holds as $\xi^{2}$ is a primitive $\ell^{\prime}$ root of
unity. Let

${\mathcal{H}}_{a,b}=\frac{1}{\ell^{\prime}}\sum_{m_{1},m_{2}=0}^{\ell^{\prime}-1}\xi^{-2(m_{1}+\beta)(m_{2}+\alpha)}K^{\beta+m_{1}}\otimes
K^{\alpha+m_{2}}\in{\mathbf{U}}_{a}\otimes{\mathbf{U}}_{b},$

then (see [4, Proposition 4.3]), $\xi^{2H\otimes
H}={\mathcal{H}}_{a,b}\in\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}/I_{a}\widehat{\otimes}\,\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}/I_{b}$.
Hence, the image of ${\mathcal{R}}$ in
$\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}/I_{a}\widehat{\otimes}\,\widehat{\mathcal{U}_{\xi}^{H}}{\mathfrak{sl}_{2}}/I_{b}$
belongs to ${\mathbf{U}}_{a}\otimes{\mathbf{U}}_{b}$ and is equal to
${\mathcal{R}}_{a,b}$.

Since ${\mathcal{R}}$ is the universal $R$-matrix of the topological ribbon
Hopf algebra $\widehat{\mathcal{U}_{\xi}^{H}}$ with pivot
$K^{1-\ell^{\prime}}$, the properties for $({\mathcal{R}}_{a,b})_{a,b\in G}$
defined in Definition 2.2 and Equation (1) are direct transposition of the
properties of ${\mathcal{R}}$. ∎

Now we consider a finite type sub Hopf $G$-coalgebra of ${\mathbf{U}}$: For
$a\in\mathbb{C}/{2\mathbb{Z}}$, let ${\widetilde{\mathbf{U}}}_{a}$ be the
subalgebra of ${\mathbf{U}}_{a}$ generated by $E$,$F$ and $K$. By the PBW
theorem this algebra is finite with dimension ${\ell^{\prime}}^{3}$ and has a
basis $\\{E^{n_{E}}F^{n_{F}}K^{n_{K}}\\}_{0\leq
n_{E},n_{F},n_{K}<\ell^{\prime}}$. The family
${\widetilde{\mathbf{U}}}=\\{{\widetilde{\mathbf{U}}}_{a}\\}_{a\in G}$ forms a
finite type unimodular pivotal sub Hopf $G$-coalgebra of ${\mathbf{U}}$ (which
is not ribbon), see [8]. Its symmetrized integral $\widetilde{\mu}$ is given
in the PBW basis by sending all vectors to $0$ except
$\widetilde{\mu}_{a}(E^{\ell^{\prime}-1}F^{\ell^{\prime}-1})=\eta$ for some
arbitrary non zero constant independent of $a\in G$.

By Theorem 3.5, there exists a unique $z_{a}\in
C({\widetilde{\mathbf{U}}}_{a})$ such that
$\widetilde{\mu}_{a}(x)=\operatorname{tr}^{\mathbb{C}}_{{\mathbf{U}}_{a}}(L_{z_{a}x})$
and there is a modified symmetrized integral ${\widetilde{\mu}}^{\prime}$
given for ${a}\in G\setminus\\{\overline{0},\overline{1}\\}$ by
${\widetilde{\mu}}^{\prime}_{a}(x)=\operatorname{tr}^{\mathbb{C}}_{{\mathbf{U}}_{a}}(L_{z_{a}^{2}x})=\widetilde{\mu}_{a}(z_{a}x)$.

######  Theorem 4.2.

The pivotal Hopf $G$-coalgebra ${\left({\mathbf{U}},g\right)}$ has a
symmetrized integral ${\mu}$ given by

(13) ${\mu}_{a}:\left|\begin{array}[]{l}E^{n_{E}}F^{n_{F}}K^{n_{K}}\mapsto
0\text{ if }\begin{array}[t]{l}{n_{E}}\neq\ell^{\prime}-1\text{ or
}{n_{F}}\neq\ell^{\prime}-1\\\ \text{ or
}{n_{K}}\in\mathbb{C}\setminus\frac{\ell}{2}\mathbb{Z},\end{array}\\\
E^{\ell^{\prime}-1}F^{\ell^{\prime}-1}K^{n\frac{\ell}{2}}\mapsto\xi^{n\frac{\ell}{2}a}\eta,\end{array}\right.$

for an arbitrary non zero constant $\eta$. Furthermore ${\mathbf{U}}$ has a
modified symmetrized integral on
$G^{\prime}=\mathbb{C}/2\mathbb{Z}\setminus\\{\overline{0},\overline{1}\\}$
given by ${\mu}^{\prime}_{a}={\mu}_{a}\circ L_{z_{a}}$.

###### Proof.

For $a\in\mathbb{C}/{2\mathbb{Z}}$, let
$M_{a}=\bigoplus_{\alpha\in]0,1[+i\mathbb{R}}K^{\alpha}{\widetilde{\mathbf{U}}}_{a}$.
Then ${\mathbf{U}}_{a}={\widetilde{\mathbf{U}}}_{a}\oplus M_{a}$ with
$S_{a}(M_{a})\subset M_{-a}$ and $\Delta_{a,b}(M_{a+b})\subset M_{a}\otimes
M_{b}$. Let $\pi_{a}:{\mathbf{U}}_{a}\to{\widetilde{\mathbf{U}}}_{a}$ be the
projection with kernel $M_{a}$ and define
${\mu}_{a}=\widetilde{\mu}_{a}\circ\pi_{a}$. Then for any
$x=h+m\in{\mathbf{U}}_{a+b}={\widetilde{\mathbf{U}}}_{a+b}\oplus M_{a+b}$,

$\displaystyle{\mu}_{a}\otimes g_{b}(\Delta_{a,b}x)$
$\displaystyle={\mu}_{a}\otimes g_{b}(\Delta_{a,b}h)+{\mu}_{a}\otimes
g_{b}(\Delta_{a,b}m)$ $\displaystyle=\widetilde{\mu}_{a}\otimes
g_{b}(\Delta_{a,b}h)=\widetilde{\mu}_{a+b}(h)1_{b}={\mu}_{a+b}(x)1_{b}$

where the second equality comes from
$\pi_{a}\otimes\operatorname{Id}(\Delta_{a,b}m)=0$. This proves point (1) of
Definition 2.4. Point (3) follows for $x=h+m\in{\mathbf{U}}_{a}$ from

${\mu}_{-a}(S_{a}(x))=\widetilde{\mu}_{-a}(S_{a}(h))+{\mu}_{-a}(S_{a}(m))=\widetilde{\mu}_{a}(h)+0={\mu}_{a}(x).$

Finally we take advantage of the $\mathbb{Z}$-grading in ${\mathbf{U}}$ by the
weights where the weights of $E$, $F$ and $K^{\alpha}$ are respectively $2,-2$
and $0$. Remark that ${\mu}$ vanishes on homogeneous elements of non zero
weight. Let $\alpha\in[0,1[+i\mathbb{R}$ and $\beta\in]-1,0]+i\mathbb{R}$ and
fix $x\in K^{\alpha}{\mathbf{U}}_{a}$ and $y\in K^{\beta}{\mathbf{U}}_{a}$
with homogeneous weights $w_{x}$ and $w_{y}$ respectively. Then
${\mu}_{a}(xy)={\mu}_{a}(yx)=0$ unless $\beta=-\alpha$ and $w_{y}=-w_{x}$. In
this last case $K^{\alpha}$ commutes with $xy$ so using that
$K^{-\alpha}x,yK^{\alpha}\in{\widetilde{\mathbf{U}}}_{a}$ we have

${\mu}_{a}(xy)=\widetilde{\mu}_{a}((K^{-\alpha}x)(yK^{\alpha}))=\widetilde{\mu}_{a}((yK^{\alpha})(K^{-\alpha}x))={\mu}_{a}(yx).$

We now check that the family ${\left\\{{\mu}^{\prime}_{a}\right\\}}_{a\in
G\setminus{\left\\{\overline{0},\overline{1}\right\\}}}$ satisfies Definition
3.4. Since $z_{a}M_{a}\subset M_{a}$ and $g_{a}^{\pm 1}M_{a}\subset M_{a}$,
for $x\in{\mathbf{U}}_{a},y\in{\mathbf{U}}_{b}$ with $x\in M_{a}$ or $y\in
M_{b}$ then we have ${\mu}_{a}L_{g_{a}^{-1}}\otimes{\mu}^{\prime}_{b}(x\otimes
y)=0={\mu}^{\prime}_{a}\otimes{\mu}_{b}L_{g_{b}}(x\otimes y)$. On the other
hand, if $x\in{\widetilde{\mathbf{U}}}_{a},y\in{\widetilde{\mathbf{U}}}_{b}$
then

$\displaystyle{\mu}_{a}L_{g_{a}^{-1}}\otimes{\mu}^{\prime}_{b}(x\otimes y)$
$\displaystyle=\widetilde{\mu}_{a}L_{g_{a}^{-1}}\otimes\widetilde{\mu}^{\prime}_{b}(x\otimes
y)$
$\displaystyle=\widetilde{\mu}^{\prime}_{a}\otimes\widetilde{\mu}_{b}L_{g_{b}}(x\otimes
y)$ $\displaystyle={\mu}^{\prime}_{a}\otimes{\mu}_{b}L_{g_{b}}(x\otimes y).$

∎

As in the proof of Theorem 4.1, the discrete Fourier transform on the twist of
$\widehat{\mathcal{U}_{\xi}^{H}}$

$\theta=\sum_{n=0}^{\ell^{\prime}-1}\xi^{\frac{n^{2}+3n}{2}}\frac{{{\left\\{1\right\\}}}^{2n}}{{\left\\{n\right\\}}!}\xi^{2H^{2}}K^{\ell^{\prime}-1-n}E^{n}F^{n}$

and on its inverse

$\theta^{-1}=\sum_{n=0}^{\ell^{\prime}-1}(-1)^{n}\xi^{-\frac{n^{2}+3n}{2}}\frac{{{\left\\{1\right\\}}}^{2n}}{{\left\\{n\right\\}}!}\xi^{-2H^{2}}K^{n+1-\ell^{\prime}}E^{n}F^{n}$

gives a formula for the twists $\theta_{0}^{\pm}$ in ${\mathbf{U}}_{0}$ such
that

${\mu}_{0}(g\theta_{0})=\overline{{\mu}_{0}(g^{-1}\theta_{0}^{-1})}=\lambda\frac{(-1)^{\ell+1}\xi^{-1}}{{\ell^{\prime}}^{2}}{{\left\\{1\right\\}}}^{2\ell^{\prime}-2}\sum_{k=0}^{\ell^{\prime}-1}\xi^{2k^{2}+2k}$

where we assume the normalization factor $\lambda$ is in $\mathbb{R}^{*}$.
Finally, the last Gauss sum in the previous equation is known to vanish if and
only if $\ell\in 8\mathbb{Z}$. Thus, the Hopf
$\mathbb{C}/2\mathbb{Z}$-coalgebra ${\mathbf{U}}$ lead to an invariant of
$\mathbb{C}/2\mathbb{Z}$-manifolds ${\mathrm{HV}}$ and a modified invariant
${\mathrm{HV}}^{\prime}$ of admissible $\mathbb{C}/2\mathbb{Z}$-manifolds.

We finish with a simple example which shows that ${\mathrm{HV}}^{\prime}$ is a
non-trivial renormalization of ${\mathrm{HV}}$.

######  Proposition 4.3.

Assume $\ell\in 2\mathbb{Z}\setminus 8\mathbb{Z}$. Let $M=S^{2}\times S^{1}$
and $\omega$ be a cohomology class on $M$ which takes value
$a\in\mathbb{C}/2\mathbb{Z}$ on a generator of $H_{1}(M)$. Then
${\mathrm{HV}}(M,\omega)=0$. However, if
$a\notin\\{\overline{0},\overline{1}\\}$ then

${\mathrm{HV}}^{\prime}(M,\omega)=\left\\{\begin{array}[]{cl}-2\frac{{\left\\{1\right\\}}^{4\ell^{\prime}-4}}{{\left\\{\ell^{\prime}a\right\\}}^{2}}\eta^{2}&\text{if
}\ell>4\\\
2\frac{{\left\\{a\right\\}}^{2}}{{\left\\{2a\right\\}}^{2}}\eta^{2}&\text{if
}\ell=4\;.\end{array}\right.$

###### Proof.

A surgery presentation of $M$ is given by an unknot colored by $a$. The
universal invariant is then $g_{a}=\xi^{-\ell^{\prime}a}K$ and ${\mu}_{a}$
vanishes on it. Let now $\alpha\in\mathbb{C}$ be in the class of
$a\in\mathbb{C}/2\mathbb{Z}\setminus\\{\overline{0},\overline{1}\\}$. To
compute ${\mathrm{HV}}^{\prime}$ we open the unknot to obtain an open straight
strand. The universal invariant is then $1_{a}$ and one should compute

${\mathrm{HV}}^{\prime}(M,\omega)={\mu}^{\prime}_{a}(1_{a})={\mu}_{a}(z_{a}).$

Now ${\widetilde{\mathbf{U}}}_{a}$ has $\ell^{\prime}$ irreducible
representations of dimensions $\ell^{\prime}$:
$\\{V_{\xi^{\alpha+2k}}:k=0\cdots\ell^{\prime}-1\\}$ where $\xi^{\alpha+2k}$
is the highest weight of $V_{\xi^{\alpha+2k}}$ (i.e. the eigenvalue of $K$ on
the kernel of $E$ in $V_{\xi^{\alpha+2k}}$). Let us write
$z_{a}=\sum_{k=0}^{\ell^{\prime}-1}\frac{d_{\alpha+2k}}{\ell^{\prime}}z_{\alpha+2k}$
as in Remark 3.6. As shown by the second author in [8] (see also [2]),
$d_{\alpha+2k}=\operatorname{tr}^{\mathbb{C}}(L_{z_{a}z_{\alpha+2k}})=\widetilde{\mu}_{a}(z_{\alpha+2k})$
is (a normalization of) the modified dimension of $V_{\xi^{\alpha+2k}}$. The
explicit computation done for an even root of unity gives that

$d_{\alpha+2k}=d_{0}\frac{\ell^{\prime}{\left\\{\alpha+2k\right\\}}}{{\left\\{\ell^{\prime}\alpha\right\\}}}\text{
where
}d_{0}=\frac{{\left\\{1\right\\}}^{2\ell^{\prime}-2}}{{\ell^{\prime}}^{3}}\eta.$

Then for $\ell>4$,

$\displaystyle{\mu}^{\prime}(1_{a})$
$\displaystyle=\sum_{k=0}^{\ell^{\prime}-1}d_{\alpha+2k}^{2}=\sum_{k=0}^{\ell^{\prime}-1}d_{0}^{2}\ell^{\prime
2}\dfrac{\xi^{2(a+2k)}+\xi^{-2(a+2k)}-2}{{\left\\{\ell^{\prime}a\right\\}}^{2}}$
$\displaystyle=\dfrac{d_{0}^{2}\ell^{\prime
2}}{{\left\\{\ell^{\prime}a\right\\}}^{2}}{\left(-2\ell^{\prime}+\xi^{2a}\sum_{k=0}^{\ell^{\prime}-1}\xi^{2k}+\xi^{-2a}\sum_{k=0}^{\ell^{\prime}-1}\xi^{-2k}\right)}$
$\displaystyle=-2\dfrac{{\ell^{\prime}}^{3}d_{0}^{2}}{{\left\\{\ell^{\prime}a\right\\}}^{2}}=-2\frac{{\left\\{1\right\\}}^{4\ell^{\prime}-4}}{{\ell^{\prime}}^{3}{\left\\{\ell^{\prime}a\right\\}}^{2}}\eta^{2},$

where we used that the last two sum vanish for $\ell>4$. ∎

The example of this section is the Hopf $G$-coalgebra version of the
invariants of [4].

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