You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We describe a new technique to obtain representations of the braid group $B_n$ from the $\Re$–matrix of a quantum deformed algebra of the one dimensional harmonic oscillator. We consider the action of the $\Re$–matrix not on the tensor product of representations of the algebra, that in the harmonic oscillator case are infinite dimensional, but on the subspace of the tensor product corresponding to the lowest weight vectors.'
author:
- 'M. Tarlini[^1]'
title: Braid group representations from a deformation of the harmonic oscillator algebra
---

Introduction
============

Given a vector space and a matrix acting on it a generator of the Artin braid group $B_n$ [@A] can be represented on the $n$-tensor product of the vector space if and only if the matrix satisfies the Yang-Baxter equation, the original main reason for the quantum groups was to solve this equation [@FRT], and to study links invariants [@KR], [@RT].

In this paper we want to give a new way to find representation of the braid group $B_n$ using a quantum group version [@h1q] of the harmonic oscillator algebra. This Lie algebra has four generators; we consider, besides the ladder operators, the constant operator and the Hamiltonian as generators. The irreducible representations of this algebra are infinite dimensional and it is not possible to use the standard methods based on the quantum $\Re$–matrix formalism without a regularization procedure, for instance a connection between the quantum deformed harmonic oscillator and some links invariants is given in [@GS].

Here we use a different approach which takes into account the fact that lowest (highest) weight spaces (they are finite dimensional) in the $n$-tensor product of representations of the quantum algebra are $B_n$ invariant and they define a representation of the braid group.

This idea has been followed in [@JK] for the $U_q(sl_2)$ quantum algebra, where the authors indicate with ${\bf W}_{n,l}$ the highest weight spaces and prove the isomorphism between ${\bf W}_{n,1}$ and the reduced Burau representation [@Burau],[@KT] of $B_n$ and between ${\bf W}_{n,2}$ and the Lawrence-Krammer-Bigelow representation [@LKB], [@KT].

The advantage of working with the harmonic oscillator algebra is that we are able to construct the $B_n$ representation spaces analogous to the lowest weight spaces ${\bf W}_{n,l}$ very easily. Indeed it is possible to start with different representations of the quantum algebra in each tensor space, and to reach lowest weights space of higher level (corresponding to their parameter $l>2$). We recover in the simplest case, and with all the representations in the tensor product being equal each others, the reduced Burau representation and in a second step the Lawrence-Krammer-Bigelow representation but with a parameter less.

In general following the formula (\[sigmaopo\]) it is possible to quantize a general classical representations of $B_n$.

The paper is organized as follows: in Section 2 we recall the quantum deformation of the harmonic oscillator algebra and its irreducible representation, in Section 3 we write explicitly the $\Re$–matrix and the assignment of the generators $\sigma_i$ of the braid group $B_n$. In Section 4 we describe the lowest weight vectors of the quantum algebra, and in Section 5 we introduce the operator $\opo$ and the reduction of the representations on the $n$-tensor product. In Section 6 we write the general formula (\[sigmaopo\]) that allows to build the representations of the braid group. Finally in Section 7 we give some explicit examples.

Quantum deformed algebra and his irreducible representations {#qda_irreps}
============================================================

In [@3dc; @h1q] we obtained a quantum group deformation of the Lie algebra of the harmonic oscillator by a contraction of the quantum algebra $sl(2)_q$. Here we recall the structure of Hopf algebra of this deformation that we write in the rational form and call $ho_q$.

The associative algebra $ho_q$ is defined over $\C$ with generators $\alpha^\pm\,,\epsilon\,,\qgm\,$, $\qmgm$ and $1$ with the relations $\qpmgm \qmpgm=1$ and with the following commutators, where $q$ is a nonzero parameter: $$\label{hoq}
[\alpha^-,\alpha^+]=\frac{\qg-\qmg}{q-q^{-1}}\equiv[\Gamma]_q\ ,\quad\quad [\epsilon,\alpha^{\pm}]=\pm\,\alpha^\pm\ ,\quad\quad [\qpmgm, \cdot]=0\ .$$

The Hopf algebra coproduct is given by $$\begin{gathered}
\label{hopf}
 \Delta\, \alpha^\pm=\alpha^\pm\otimes\, \qgm + \qmgm\, \otimes\alpha^\pm\ ,\\
 \Delta\, \epsilon =\epsilon\,\otimes 1 +1\otimes\, \epsilon\ , \quad
 \Delta\, \qpmgm =\qpmgm\otimes \qpmgm\ ,
 \end{gathered}$$ with counit of $\alpha^\pm$ and $\epsilon$ equal to zero and counit of $\qpmgm$ equal to one, the antipod is $$\begin{gathered}
 S(\alpha^\pm)=-\alpha^\pm\ , \quad S(\qpmgm)=\qmpgm , \quad S(\epsilon)=-\epsilon\ .\end{gathered}$$

The algebra $ho_q$ has two Casimirs, one is $\qgm$ the other is $$C_q=[\Gamma]_q\; \epsilon -\alpha^+\,\alpha^-\ .$$

The representations of the braid groups will be derived starting from the representations of this algebra. The irreps are labeled by two numbers: $\gamma$ and $c$, we call ${\bf H}^{(\gamma,c)}$ the module freely generated by a set of vectors $h^{(\gamma,c)}_m\,,\ m\in \N_0$.

The representation of $ho_q$ on ${\bf H}^{(\gamma,c)}$ is then $$\begin{aligned}
\label{irreps}
 \alpha^-\cdot h^{(\gamma,c)}_m &=& [\gamma]_q^{1/2}\,m^{1/2}\, h^{(\gamma,c)}_{m-1}\ ,\nonumber\\ 
 \alpha^+\cdot h^{(\gamma,c)}_m &=& [\gamma]_q^{1/2}\,(m+1)^{1/2}\, h^{(\gamma,c)}_{m+1}\ ,\\
 \epsilon\cdot h^{(\gamma,c)}_m &=&(m+c)\ h^{(\gamma,c)}_m\ . \nonumber\end{aligned}$$ With $\qpmgm\cdot h^{(\gamma,c)}_m= q^{\pm\gamma/2}\ h^{(\gamma,c)}_m$ and $C_q\cdot h^{(\gamma,c)}_m=[\gamma]_q\,c\, h^{(\gamma,c)}_m$.

The generators of this representation can be obtained from the vector $h^{(\gamma,c)}_0$ by the action of $\alpha^+$, namely $h^{(\gamma,c)}_m=([\gamma]_q^m m!)^{-1/2}(\alpha^+)^m\, h^{(\gamma,c)}_0$.

We assume $\gamma,c\in \R$, we choose $q\in \R$ ($0<q<1$), then this representation is hermitian with respect to the star involution $(\alpha^\pm)^*=\alpha^\mp\ ,(\epsilon)^*=\epsilon$ and $(\gamma)^*=\gamma$. The star involution is an anti–homomorphism and it fulfills $\Delta \circ *=(*\otimes\,*)\circ \Delta$ and $S\circ *=*\circ S^{-1}$. The scalar product is given by $<h^{(\gamma,c)}_m|h^{(\gamma,c)}_{m'}>=\delta_{m\,m'}$.

The $\Re$–matrix {#Rm}
================

In [@h1q] the quasitriangular $\Re$–matrix is obtained as a contraction limit of the $\Re$–matrix of $sl_q(2)$, the result is $$\label{rmatrix}
\Re=q^{ -({\textstyle \epsilon}\,\otimes\,\Gamma + \Gamma\,\otimes\,{\textstyle \epsilon})} \exp[(q-q^{-1})\,(\qgm\,\otimes\,\qmgm)\,\alpha^-\,\otimes\,\alpha^+]\ .$$ It is a general feature that the $\Re$–matrix it is not an element of the tensor product of the quantum algebra given in the rational form, nevertheless it acts on any tensor product of algebra representations (see for instance Chapter 9 and 10 of the book [@CP] for a general reference). We are interested in the action of $\Re$ on the representations given in (\[irreps\]), this leads us to consider $\Gamma\cdot h^{(\gamma,c)}_m= \gamma\ h^{(\gamma,c)}_m$.

$\Re$ is then an endomorphism on ${\bf H}^{(\gamma_1,c_1)}\,\otimes\, {\bf H}^{(\gamma_2,c_2)}$. If ${\cal P}$ is the permutation on the tensor product, the action of ${\cal P}\, \Re$ gives an action of a generator of the braid group.

We write here the action of ${\cal P}\Re$ on an element $h^{(1)}_m\,\otimes\,h^{(2)}_{m'}\in{\bf H}^{(\gamma_1,c_1)}\,\otimes\, {\bf H}^{(\gamma_2,c_2)}$ , to simplify the notations we use $h_m^{(i)}$ for $h_m^{(\gamma_i,c_i)}$ : $$\label{p_er}
\begin{split}
 {\cal P}\Re\;\cdot h^{(1)}_m\,\otimes\,h^{(2)}_{m'}&= q^{-((m+c_1)\gamma_2 +(m'+c_2)\gamma_1)} \sum^m_{k=0}{\binom {m+k-1}{m-1}}^{1/2} {\binom {m'+k}{m'}}^{1/2}\cdot \\
&\cdot  (1-q^{-2\gamma_1})^{k/2} (q^{2\gamma_2}-1)^{k/2}\,h^{(2)}_{m'+k}\,\otimes\,h^{(1)}_{m-k}\ . 
\end{split}$$

The Artin braid group $B_n$ is defined as the group generated by the $n-1$ generators $\sigma_1,\sigma_2,\cdots\sigma_{n-1}$ and the [*braid relations*]{}: $\sigma_i\sigma_j=
\sigma_j\sigma_i$ for $|i-j|>1$ and $\sigma_i\sigma_{i+i}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ for $i=1,\cdots, n-2$ (see Section 1.1 of [@KT]).

The Yang–Baxter equation $\Re_{12}\Re_{13}\Re_{23}=\Re_{23}\Re_{13}\Re_{12}$ provides the braid relations of the generators $\sigma_i$ of the $B_n$ braid group presentation, then a representation of $B_n$ on ${\bf H}^{\otimes\,n}$ is given by the following identification $$\label{sigma}
 \sigma_j=1^{\otimes\,j-1}\,\otimes\,{\cal P}\,\Re\,\otimes\,1^{\otimes\,n-j-1}\ .$$

The inverse of the $\Re$–matrix and consequently the inverse of $\sigma_i$ are obtained by the expression $\Re^{-1}=(S\,\otimes 1)\,\Re$; from (\[rmatrix\]) it results in the exchange $q\rightarrow q^{-1}$ in $\Re$ and then in $\sigma_i$. In the following section we describe a method to find a finite dimensional space of representation of $B_n$ starting from (\[sigma\]) and the infinite dimensional representations (\[irreps\]) of $ho_q$. The main feature is the fact that the operator ${\cal P}\,\Re$, and hence also the maps $\sigma_i$, commute with the action of the $ho_q$ algebra on the tensor product, this action is given by the coproduct.

Lowest weight spaces of $ho_q$ {#sec_weight_spaces}
==============================

The action of the generators of $ho_q$ on the $n$–tensor product of representations is given by the iterated coproduct defined as $\Delta^{(n)} = 
(\Delta^{(n-1)}\,\otimes  1) \Delta$ with $n\geq 2$ and $\Delta^{(2)}\equiv\Delta$.

\[weight\_space\] The weight space ${\bf W}^{(\gamma,e)}$ corresponding to the weights $\gamma$ and $e$ is defined as ${\rm ker}(\Delta^{(n)}\Gamma -\gamma)\cap 
{\rm ker}(\Delta^{(n)}\epsilon -e)\subset 
\bigoplus_{\perm\in S_n}\bigotimes_{i=1}^n {\bf H}^{(\gamma_{\perm(i)},c_{\perm(i)})}$, where $\perm\in S_n$ is a permutation of $(1,\cdots,n)$.

The generator $\epsilon$ and $\Gamma$ are primitive then ${\bf W}^{(\gamma,e)}$ is the span of vectors of the form $$\label{vector_n}
h^{\perm(1)}_{m_1}\,\otimes h^{\perm(2)}_{m_2}\cdots\otimes\,h^{\perm(n)}_{m_n}\quad  \text{with} \quad \left\{
\aligned  \gamma_1+\cdots+\gamma_n&=\gamma\ , \\m_1+\cdots +m_n&=e-c_1-\cdots -c_n\ .
\endaligned \right.$$ Here we used the short notation $h^{(i)}_m$ for $h^{(\gamma_i,c_i)}_m$.

Then we have that $e\geq \sum_i^n c_i$ and $e - \sum_i^n c_i$ is a nonnegative integer, moreover the value $\gamma$ is fixed by the $\gamma_i$.

In the following we fix the representations ${\bf H}^{(\gamma_i,c_i)}$.

Note that the Casimir $C_q$ is not primitive therefore a generic vector in ${\bf W}^{(\gamma,e)}$ is not an eigenvector of $\Delta^{(n)} C_q$, in particular of the term $\Delta^{(n)}\alpha^+\,\Delta^{(n)}\alpha^-$.

\[lowest\_weight\] The lowest weight space corresponding to $e$ is the space ${\bf V}^{(e)}\subset {\bf W}^{(\gamma,e)}$ of vectors $v_0^{(e)}$ such that $\Delta^{(n)} \alpha^- \cdot v_0^{(e)}=0$.

For every vector $v_0^{(e)}\in{\bf V}^{(e)}$ we have that $$\label{deltaC}
 \Delta^{(n)} C_q\cdot\, v_0^{(e)}= (\frac{\Delta^{(n)}\, q^\Gamma-\Delta^{(n)}\, q^{-\Gamma}}{q-q^{-1}}\; 
 \Delta^{(n)}\epsilon -\Delta^{(n)}\alpha^+\,\Delta^{(n)}\alpha^-)\cdot\,v_0^{(e)}=[\gamma]_q\,e\;v_0^{(e)}\ .$$ From now on we will indicate the eigenvalue of $\Delta^{(n)} C_q$ neglecting the $[\gamma]_q$ factor, namely we consider $e$ as the eigenvalue of $v_0^{(e)}$. Starting from a vector $v_0^{(e)}\in {\bf V}^{(e)}$ one can built a representation of $ho_q$ by applying the operator $\Delta^{(n)}\alpha^+$, namely we define $$\label{vi}
v_m^{(e)}=([\gamma]^m_q\, m!)^{-1/2}(\Delta^{(n}\alpha^+)^m \, v_0^{(e)}\ ,$$ we notice that $v_m^{(e)}\in {\bf W}^{(\gamma,e+m)}$ is not an element of ${\bf V}^{(e+m)}$, indeed $\Delta^{(n)} \alpha^-\cdot\; v_m^{(e)}=[\gamma]_q^{1/2}\,m^{1/2}\, v_{m-1}^{(e)}\neq0$. The eigenvalue of the Casimir on $v_m^{(e)}$ is $e$ .

The relations (\[irreps\]) are fulfilled with the substitutions: $c\rightarrow e$, $h^{(c)}_m\rightarrow v_m^{(e)}$, $\alpha^+\rightarrow \Delta^{(n)} \alpha^+$, $\alpha^-\rightarrow \Delta^{(n)} \alpha^-$, $\epsilon\rightarrow \Delta^{(n)} \epsilon$ and the scalar product is $<v_m^{(e)}|v_{m'}^{(e)}>=\delta_{m\,m'}$.

For $n=2$ the lowest value for $e$ is $c_1+c_2$. The lowest weight space ${\bf V}^{(c_1+c_2)}$ is generated by $v_0^{(c_1+c_2)}=h_0^{(1)}\otimes h_0^{(2)}$ (and the permutation $h_0^{(2)}\otimes h_0^{(1)}$), namely $\Delta\epsilon\cdot\, v_0^{(c_1+c_2)}=(c_1+c_2)\; v_0^{(c_1+c_2)}$ and $\Delta\alpha^-\cdot\, v_0^{(c_1+c_2)}=0$.

The next value for $e$ is $c_1+c_2+1$ then we write $v_0^{(c_1+c_2+1)}=a\;h_0^{(1)}\otimes h_1^{(2)} + b\;h_1^{(1)}\otimes h_0^{(2)}$ so that $\Delta\epsilon\cdot\, v_0^{(c_1+c_2+1)}=(c_1+c_2+1)\; v_0^{(c_1+c_2+1)}$. Imposing the lowest weight relation we get $v_0^{(c_1+c_2+1)}$ proportional to $q^{\gamma_2/2}\,[\gamma_1]_q^{1/2}\;h_0^{(1)}\otimes h_1^{(2)} - q^{-\gamma_1/2}\,[\gamma_2]_q^{1/2}\;h_1^{(1)}\otimes h_2^{(2)}$. We have that ${\bf V}^{(c_1+c_2+1)}$ is generated by this $v_0^{(c_1+c_2+1)}$ and the permutation obtained by the exchange $\gamma_1 \leftrightarrow \gamma_2$ and $(1)\leftrightarrow (2)$.

The operator $\opo$ and the reduction of the tensor product of representations
==============================================================================

We start with $n=2$, we take the representations ${\bf H}^{(\gamma_1,c_1)}$ and ${\bf H}^{(\gamma_2,c_2)}$. The idea is to find an operator $\opo\in ho_q\,\otimes\,ho_q$ that commutes with $\Delta \alpha^-$ and that works as a ladder operator on $\Delta\epsilon$ so that $\opo$ maps vectors of ${\bf V}^{(e)}$ in vectors of ${\bf V}^{(e+1)}$. The following proposition derives from a straightforward computation:

The operator $$\label{opo}
 \opo=q^{-\Gamma/2}\,[\Gamma]_q^{-1/2}\alpha^+\,\otimes\,[\Gamma]_q^{1/2} - [\Gamma]_q^{1/2} \,\otimes\,q^{\Gamma/2}\,[\Gamma]_q^{-1/2}\alpha^+$$ satisfies the following commutators: $$\label{commopo}
[\Delta \alpha^-,\opo]=0\;, \qquad [\Delta \epsilon,\opo]=\opo\ ,$$ and obviously $$[\Delta \Gamma,\opo]=0\;,\qquad [\Delta \alpha^+,\opo]=0\ .$$

Following Definition \[lowest\_weight\] the space ${\bf V}^{(c_1+c_2)}$ is the lowest weight space with $e=c_1+c_2$ and $\gamma=\gamma_1+\gamma_2$, where $\gamma_i\,,c_i$ define the Casimirs eigenvalues on the representations ${\bf H}^{(\gamma_i,c_i)}$.

From the vector $v_0^{(c_1+c_2)}=h^{(1)}_{0}\otimes h^{(2)}_{0}\in{\bf V}^{(c_1+c_2)}$ we can obtain $v_0^{(c_1+c_2+1)}=[\gamma]_q^{-1/2}\opo\,v_0^{(c_1+c_2)}$, with the eigenvalue of the Casimir $\Delta C_q$ given by $c_1+c_2+1$ and $v_1^{(c_1+c_2)}=[\gamma]_q^{-1/2}\,\Delta\alpha^+\,v_0^{(c_1+c_2)}$, with the eigenvalue of the Casimir given by $c_1+c_2$.

One can start from the vector ${v'}_0^{(c_1+c_2)}=h^{(2)}_{0}\otimes h^{(1)}_{0}\in{\bf V}^{(c_1+c_2)}$ and repeat the same construction.

The vectors $v_0^{(c_1+c_2+1)}$ and $v_1^{(c_1+c_2)}$ together with ${v'}_0^{(c_1+c_2+1)}$ and ${v'}_1^{(c_1+c_2)}$ have the eigenvalue of $\Delta\epsilon$ equal to $c_1+c_2+1$ and they span ${\bf W}^{(\gamma,c_1+c_2+1)}=\langle \; h_0^{(\perm(1))}\otimes h_1^{(\perm(2))},\,h_1^{(\perm(1))}\otimes h_0^{(\perm(2))}\,;\ \perm\in S_2\;\rangle$, namely the weight space with $e=c_1+c_2+1$.

In general ${\bf W}^{(\gamma,c_1+c_2+N)}$ is spanned by the $2(N+1)$ vectors $h_m^{\perm(1)}\otimes h_{m'}^{\perm(2)}$ for $m+m'=N$. It can be reduced in the combination of vectors of the representations with Casimir from $c$ to $c+N$ of the form $v_i^{(c_1+c_2+j)}$ with $i+j=N\ (i,j\geq0)$ obtained from $v_0^{(c_1+c_2)}=h_0^{\perm(1)}\otimes h_0^{\perm(2)}$ using (\[vi\]) and $$\label{v0}
v_0^{(c_1+c_2+k)}=([\gamma]^k_q\, k!)^{-1/2}\opo^{\,k} \, v_0^{(c_1+c_2)}\ .$$

From the commutation relation $[\opo^*,\opo]=\Delta [\Gamma]_q$ and $[\Delta\alpha^-,\opo]=0$ we can prove that these vectors verify $$<v_i^{(c_1+c_2+j)}|v_{i'}^{(c_1+c_2+j')}>=\delta_{i\,i'}\delta_{j\,j'}\ .$$ The rising of $c_1+c_2$ by $j$ steps using $\opo$ and the rising from $0$ to $i$ using $\Delta\alpha^+$ can be made in any order because of the commutativity of $\opo$ with $\Delta\alpha^+$.

We have then the following proposition:

\[reduction\] For $n=2$ we define ${\bf U}^{(j,N)}\subset {\bf W}^{(\gamma,c_1+c_2 +N)}$ as the space spanned by the vectors $\langle\; (\Delta\alpha^+)^{(N-j)}\;\opo^{\,j}\,v_0^{(c_1+c_2)}\; \rangle$, with $v_0^{(c_1+c_2)}\in {\bf V}^{(c_1+c_2)}$. The vectors in ${\bf U}^{(j,N)}$ are eigenvectors of $\Delta C_q$ and $\Delta \epsilon $ with eigenvalue respectively $c_1+c_2+j$ and $c+N$. We have that $$\label{eq_reduction}
{\bf W}^{(\gamma,c_1+c_2+N)}=\bigoplus_{j=0}^N{\bf U}^{(j,N)}\ .$$

We treat now the case of a generic $n$.

One can repeat what was done for $n=2$ using the operators $$\Delta^{(n)}\alpha^+ \quad \text{and}\quad \opo_k=1^{\otimes k-1}\otimes\underbrace{\opo}_{k,k+1}\otimes\; 1^{\otimes n-k-1}\,,$$

$\Delta^{(n)}\alpha^-$, $\Delta^{(n)}\epsilon$ and $\opo_k$ still satisfy the commutation relations (\[commopo\]) for every $k$.

Fixing the Casimirs $c_i$ of the representations in the tensor product we call ${\bf c}$ the sum of them: ${\bf c}=c_1+c_2+\cdots +c_n$.

The number ${\cal N}_{n,N}$ of vectors of the form (\[vector\_n\]) (apart the action of ${\cal S}\in S_n$) that span ${\bf W}^{(\gamma,{\bf c}+N)}$ is the number of different ways to add (taking into account the order) $n$ nonnegative integers $(m_1,m_2,\cdots,m_n)$ to get $N$, namely $${\cal N}_{n,N}=\binom{n+N-1}{n-1} \,,$$ the vectors obtained by the action of degree $j$ monomials $\opo_1^{j_1}\opo_2^{j_2}\cdots \opo_{n-1}^{j_{n-1}}$ on $v_0^{({\bf c})}$ with nonnegative $j_i$ and $j=j_1+\cdots +j_{n-1}$ are eigenspaces of constant Casimir equal to ${\bf c}+j$. They are in a number equal to $${\cal M}_{n,j}=\binom{n+j-2}{n-2} \,.$$ We have to act with $(\Delta^{(n)}\alpha^+)^{N-j}$ on the vectors with constant Casimir ${\bf c}+j$ to get vectors in ${\bf W}^{(\gamma,{\bf c} +N)}$, namely with $\Delta^{(n)}\epsilon$ equal to ${\bf c}+N$ . If we sum up all the value of $j$ from $0$ to $N$ we recover ${\cal N}_{n,N}\,$: $${\cal N}_{n,N}=\sum_{j=0}^N {\cal M}_{n,j}\ .$$

For example let us built the vectors of ${\bf W}^{(\gamma,{\bf c}+3)}$ for $n=3$ and ${\bf c}=c_1+c_2+c_3$, (they have then the eigenvalue of $\Delta^{(3)}\epsilon$ equal to ${\bf c}+3$):

  --------------------------------------------------------- ------------- ---------------------
                                                     States    Casimir     ${\cal N}_{3,3}=10$
                                                                          
                $(\Delta^{(3)}\alpha^+)^3\,v_0^{({\bf c})}$  ${\bf c}+0$   ${\cal M}_{3,0}=1$
        $(\Delta^{(3)}\alpha^+)^2\,\opo_1\,v_0^{({\bf c})}$  ${\bf c}+1$  
        $(\Delta^{(3)}\alpha^+)^2\,\opo_2\,v_0^{({\bf c})}$  ${\bf c}+1$  
    $\Delta^{(3)}\alpha^+\,\opo_1\,\opo_2\,v_0^{({\bf c})}$  ${\bf c}+2$  
        $\Delta^{(3)}\alpha^+\,(\opo_1)^2\,v_0^{({\bf c})}$  ${\bf c}+2$  
        $\Delta^{(3)}\alpha^+\,(\opo_2)^2\,v_0^{({\bf c})}$  ${\bf c}+2$  
                              $(\opo_1)^3\,v_0^{({\bf c})}$  ${\bf c}+3$  
                      $(\opo_1)^2\,\opo_2\,v_0^{({\bf c})}$  ${\bf c}+3$  
                      $\opo_1\,(\opo_2)^2\,v_0^{({\bf c})}$  ${\bf c}+3$  
                              $(\opo_2)^3\,v_0^{({\bf c})}$  ${\bf c}+3$            .
  --------------------------------------------------------- ------------- ---------------------

The following proposition is a generalization of Proposition \[reduction\]:

\[reduction\_n\] For generic $n$ we define ${\bf U}^{(j,N)}\subset {\bf W}^{(j,{\bf c}+N)}$ with $j\leq N$ as: $${\bf U}^{(j,N)}=\langle \; (\Delta^{(n)} \alpha^+)^{N-j}\,\opo_1^{j_1} \opo_2^{j_2}\cdots\opo_{n-1}^{j_{n-1}}\,v_0^{({\bf c})}\, ;\, j_1+\cdots+j_{n-1}=N\; \rangle\ ,$$ with $j_i\in \N_0$, where ${\bf c}=c_1+c_2+\cdots+c_n$ and $v_0^{({\bf c})}\in {\bf V}^{({\bf c})}$.

The vectors in ${\bf U}^{(j,N)}$ are eigenvectors of $\Delta^{(n)} C_q$ and $\Delta^{(n)} \epsilon$ with eigenvalues respectively ${\bf c}+j$ and ${\bf c}+N$. We have that: $$\label{eq_reduction_n}
{\bf W}^{(\gamma,{\bf c}+N)}=\bigoplus_{j=0}^N{\bf U}^{(j,N)}\ .$$

[*Proof*]{}: We have that: $$\label{recursion}
{\bf W}^{(j,{\bf c}+N)}=\langle \; (1^{\otimes i-1}\otimes \alpha^+\otimes1^{\otimes n-i})\ {\bf W}^{(\gamma,{\bf c}+N-1)}\,;i=1,\cdots,n\;\rangle\ .$$ Each of the $n$ vectors $(1^{\otimes i-1}\otimes \alpha^+\otimes1^{\otimes n-i})\, w^{({\bf c}+N-1)}$, where $w^{({\bf c}+N-1)}\in {\bf W}^{(\gamma,{\bf c}+N-1)}$ has the form (\[vector\_n\]) with $m_1+m_2+\cdots+m_n=N-1$, can be written as a linear combination of the vector $\Delta^{(n)} \alpha^+\, w^{({\bf c}+N-1)}$ and the $n-1$ vectors $\opo_k\,w^{({\bf c}+N-1)}$, note that the terms containing $\Gamma$ take a numerical value on vectors of the form (\[vector\_n\]). This implies that if the equation (\[eq\_reduction\_n\]) is valid for $N-1$ we have that: $$\label{induction_U}
 {\bf U}^{(j,N)}=\langle \;\Delta^{(n)}\alpha^+\, {\bf U}^{(j,N-1)}\ ,\ \opo_k\,{\bf U}^{(j-1,N-1)}\,;\, k=1,\cdots,n-1\;\rangle.$$ With an induction procedure making use of (\[induction\_U\]), we derive that ${\bf U}^{(j,N)}$ coincides with the definition in the proposition.

The following Lemma derives directly from the previous Proposition:

\[lwso\] The lowest weight spaces ${\bf V}^{({\bf c}+N)}$ are obtained as the vector spaces spanned by $\langle\;\opo_1^{j_1} \opo_2^{j_2}\cdots\opo_{n-1}^{j_{n-1}}\,{\bf V}^{({\bf c})}\,;\, j_1+\cdots+j_{n-1}=N\;\rangle$, with $j_i\in\N_0$.

Representations of the braid group $B_n$ {#repBn}
========================================

We are ready to build the representations of the braid group $B_n$. We use the presentation of the generators of the braid group given in (\[sigma\]). From the fact that $\sigma_i$ commute with $\Delta^{(n)} \epsilon$ we derive that ${\bf W}^{(\gamma,{\bf c}+N)}$ defined in Definition \[weight\_space\], where ${\bf c}=c_1+c_2+\cdots+c_n$ is the sum of the Casimir labels of the representations, is a representation space for $B_n$, moreover $\sigma_i$ commute with $\Delta^{(n)} C_q$ then this space is reducible. The lowest weight spaces ${\bf V}^{({\bf c}+N)}$ are again $B_n$ invariant.

We present here the main proposition:

\[representations\] We denote ${\bf V}^{({\bf c}+N)}$ the lowest weight spaces as in the Lemma \[lwso\].

For each choice of the representations ${\bf H}^{(\gamma_i,c_i)}$, we obtain a representation of the braid group $B_n$ given by an automorphism of ${\bf V}^{(c+N)}$.

[*Proof:*]{} The building block is the computation of the conjugation of $\opo_k$ by $\sigma_i$ given by (\[sigma\]), for $k=i$ and $k=i\pm 1$, we recall that $\sigma_i^{-1}$ is obtained from $\sigma_i$ by the exchange $q\leftrightarrow q^{-1}$ : $$\label{sigmaopo}
\begin{split}
\sigma_i\; \opo_i&=-(1^{\otimes\,i-1}\otimes q^{-\Gamma}\otimes q^{-\Gamma}\otimes\, 1^{\otimes\,n-i-1})\;\opo_i\;\sigma_i\quad  ,\\
\sigma_i\;\opo_{i+1}&=(1^{\otimes\,i}\otimes q^{-\Gamma}[\Gamma]_q^{-1/2}\otimes\, [\Gamma]_q^{1/2}\otimes\, 
                      1^{\otimes\,n-i-2})\;\opo_i\;\sigma_i\,+ \\
                      & +(1^{\otimes\,i-1}\otimes [\Gamma]_q^{1/2}\otimes\, [\Gamma]_q^{-1/2}\otimes\, 1^{\otimes\,n-i-1})\;\opo_{i+1}\;\sigma_i\quad  ,\\
\sigma_i\;\opo_{i-1}&=(1^{\otimes\,i-1}\otimes [\Gamma]_q^{-1/2}\otimes\, [\Gamma]_q^{1/2}\otimes\, 
                      1^{\otimes\,n-i-1})\;\opo_{i-1}\;\sigma_i\,+ \\
                      & +(1^{\otimes\,i-2}\otimes\,[\Gamma]_q^{1/2}\otimes\, q^{-\Gamma}[\Gamma]_q^{-1/2}\otimes\, 
                      1^{\otimes\,n-i})\;\opo_i\;\sigma_i\quad .
\end{split}$$

It is clear that $\sigma_i\;\opo_j=\opo_j\,\sigma_i$ for $|i-j|>1$.

If we act with $\sigma_i$ on the lowest weight vector $v_0^{({\bf c})}=h_0^{(1)}\otimes\, \cdots \otimes\, h_0^{(n)}\in {\bf V}^{({\bf c})}$ from (\[p\_er\]) and (\[sigma\]) we get the permuted vector up a numerical factor: $$\label{sigma0}
\begin{split}
 \sigma_i\;&h_0^{(1)}\otimes\, \cdots\otimes\,h_0^{(i)}\otimes h_0^{(i+1)}\cdots \otimes\, h_0^{(n)}= \\
 &q^{-c_i\gamma_{i+1}-c_{i+1}\gamma_i}\, h_0^{(1)}\otimes\, \cdots\otimes\,h_0^{(i+1)}\otimes h_0^{(i)}\cdots \otimes\, h_0^{(n)}\ .
\end{split}$$

Given a vector $o^{(N)}= \opo_1^{j_1} \opo_2^{j_2}\cdots\opo_{n-1}^{j_{n-1}}\,v_0^{({\bf c})}\in {\bf V}^{({\bf c}+N)}$ we compute $\sigma_i\, o^{(N)}$ by using (\[sigmaopo\]) repeatedly on $\sigma_i\, \opo_{i-1}^{j_{i-1}}\, \opo_i^{j_i}\,\opo_{i+1}^{j_{i+1}}$ to write $\sigma_i$ on the right. With $\sigma_i$ fully on the right we use (\[sigma0\]) and we apply on the left side of equation (\[sigma0\]) the combintion of operators $\opo_1^{k_1} \opo_2^{k_2}\cdots\opo_{n-1}^{k_{n-1}}$ that have come out from the use of (\[sigmaopo\]). Note that the action of $\sigma_i$ does not change the sum: $j_1+j_2+\cdots j_{n-1}=k_1+k_2+\cdots + k_{n-1}$, then we remain with vectors in ${\bf V}^{({\bf c}+N)}$.

Examples
========

Here would like to present some examples of representations of $B_n$ obtained using the Proposition \[representations\].

N=1
---

Let us choose as a first example $N=1$ and all $\gamma_i$’s and $c_i$’s equal.

The representation has dimension ${\cal M}_{n,1}=n-1$.

If we define for $k=1,\cdots,n-1$ $$w_k=\opo_k\, v_0^{(n\,c)}\quad \text{with}\quad v_0^{(n\,c)}=\underbrace{h_0^{(1)}\otimes h_0^{(1)} \cdots \otimes\, h_0^{(1)}}_n\ ,$$ (where all the representations are labeled by $c$ and $\gamma$) we have from (\[sigma0\]) that $\sigma_i\;v_0^{(n\, c)}= q^{-2\,c\gamma} v_0^{(n\, c)}$.

From (\[sigmaopo\]), after a renormalization of $\sigma_i$ by the constant factor $q^{-2\,c\gamma}$, we get: $$\label{Burau}
\begin{split}
 \sigma_k\; w_k&=-q^{-2\gamma}\,w_k\ ,\\
 \sigma_k\; w_{k+1}&= q^{-\gamma}\,w_k+ w_{k+1}\ ,\\
 \sigma_k\; w_{k-1}&=w_{k-1}+q^{-\gamma}\, w_k\ ,\\
 \sigma_k\; w_{k+i}&=w_{k+i} \quad \text{for}\ |i|>1\ . 
 \end{split}$$

This is the [*reduced Burau representation*]{} (see Section 3.3 of [@KT] with the rescaling on the vectors $q^{k\gamma} w_k\to b_{n-k}$ and the substitution $q^{-2\gamma}\to t$).

Next we consider again $N=1$ but we take one representation in the tensor product different with respect to the others.

We define $w_k^{(j)}=\opo_k (h_0^{(1)}\otimes\, \cdots\otimes\,\underbrace{h_0^{(2)}}_j\otimes\, h_0^{(1)})$ and we get the following representation for the $\sigma_i$:

$$\label{N=1c_1c_2}
\begin{split}
i>j+1&\quad\text{or}\quad i<j-2\\
 \sigma_i\; w_i^{(j)}&=-q^{-2\gamma_1}q^{-2c_1\gamma_1}\; w_i^{(j)}\ ,\\
 \sigma_i\; w_{i+1}^{(j)}&= q^{-\gamma_1}q^{-2c_1\gamma_1}\; w_i^{(j)}+ q^{-2c_1\gamma_1}\; w_{i+1}^{(j)}\ ,\\
 \sigma_i\; w_{i-1}^{(j)}&=q^{-2c_1\gamma_1}\; w_{i-1}^{(j)}+q^{-\gamma_1}q^{-2c_1\gamma_1}\; w_i^{(j)}\ ,\\
 \sigma_i\; w_{i+k}^{(j)}&=q^{-2c_1\gamma_1}\; w_{i+k}^{(j)}\qquad \text{for}\qquad |k|>1\ ,\\
 \\
i=j-2&\\
 \sigma_i\; w_i^{(j)}&=-q^{-2\gamma_1}q^{-2c_1\gamma_1}\; w_i^{(j)}\ ,\\
 \sigma_i\; w_{i+1}^{(j)}&= q^{-\gamma_1}q^{-2c_1\gamma_1}[\gamma_1]_q^{-1/2}[\gamma_2]_q^{1/2}\; w_i^{(j)}+ q^{-2c_1\gamma_1}\; w_{i+1}^{(j)}\ ,\\
 \sigma_i\; w_{i-1}^{(j)}&=q^{-2c_1\gamma_1}\; w_{i-1}^{(j)}+q^{-\gamma_1}q^{-2c_1\gamma_1}\; w_i^{(j)}\ ,\\
 \sigma_i\; w_{i+k}^{(j)}&=q^{-2c_1\gamma_1}\; w_{i+k}^{(j)}\qquad \text{for}\qquad |k|>1\ ,\\
 \\
i=j-1&\\
 \sigma_i\; w_i^{(j)}&=-q^{-\gamma_1-\gamma_2}q^{-c_2\gamma_1-c_1\gamma_2}\; w_i^{(j-1)}\ ,\\
 \sigma_i\; w_{i+1}^{(j)}&= q^{-\gamma_1}q^{-c_2\gamma_1-c_1\gamma_2}\; w_i^{(j-1)}+ q^{-c_2\gamma_1-c_1\gamma_2}[\gamma_1]_q^{-1/2}[\gamma_2]_q^{1/2}\; w_{i+1}^{(j-1)}\ ,\\
 \sigma_i\; w_{i-1}^{(j)}&=[\gamma_1]_q^{1/2}[\gamma_2]_q^{-1/2}\,(q^{-c_2\gamma_1-c_1\gamma_2}\;w_{i-1}^{(j-1)}+
 q^{-\gamma_2}q^{-c_2\gamma_1-c_1\gamma_2}\; w_i^{(j-1)}\,)\ ,\\
 \sigma_i\; w_{i+k}^{(j)}&=q^{-c_2\gamma_1-c_1\gamma_2}\; w_{i+k}^{(j-1)}\qquad \text{for}\qquad |k|>1\ ,\\
 \end{split}$$

$$\label{N=1c_1c_2-bis}
 \begin{split}
i=j\qquad&\\
 \sigma_i\; w_i^{(j)}&=-q^{-\gamma_1-\gamma_2}q^{-c_2\gamma_1-c_1\gamma_2}\; w_i^{(j+1)}\ ,\\
 \sigma_i\; w_{i+1}^{(j)}&= [\gamma_1]_q^{1/2}[\gamma_2]_q^{-1/2}\,(q^{-\gamma_2}q^{-c_2\gamma_1-c_1\gamma_2}\; w_i^{(j+1)}+ q^{-c_2\gamma_1-c_1\gamma_2}\; w_{i+1}^{(j+1)}\,)\ ,\\
 \sigma_i\; w_{i-1}^{(j)}&=q^{-c_2\gamma_1-c_1\gamma_2}[\gamma_1]_q^{-1/2}[\gamma_2]_q^{1/2}\; w_{i-1}^{(j+1)}+ q^{-\gamma_1}q^{-c_2\gamma_1-c_1\gamma_2}\; w_i^{(j+1)}\ ,\\
 \sigma_i\; w_{i+k}^{(j)}&=q^{-c_2\gamma_1-c_1\gamma_2}\; w_{i+k}^{(j+1)}\qquad \text{for}\qquad |k|>1\ ,\\
 \\
 i=j+1&\\
 \sigma_i\; w_i^{(j)}&=-q^{-2\gamma_1}q^{-2c_1\gamma_1}\; w_i^{(j)}\ ,\\
 \sigma_i\; w_{i+1}^{(j)}&=q^{-\gamma_1}q^{-2c_1\gamma_1}\; w_i^{(j)}+q^{-2c_1\gamma_1}\; w_{i+1}^{(j)}\ ,\\
 \sigma_i\; w_{i-1}^{(j)}&= q^{-2c_1\gamma_1}\; w_{i-1}^{(j)}+q^{-\gamma_1}q^{-2c_1\gamma_1}[\gamma_1]_q^{-1/2}[\gamma_2]_q^{1/2}\; w_i^{(j)}\ ,\\
 \sigma_i\; w_{i+k}^{(j)}&=q^{-2c_1\gamma_1}\; w_{i+k}^{(j)}\qquad \text{for}\qquad |k|>1\ .
\end{split}$$

In the specific case with $n=3$ we have the following matrix expressions: $$\sigma_1=
\left(
\begin{array}{cccccc}
 -d_1\, q^{-2 \gamma_1} & 0 & 0 & d_1 d_3^{-1}\, q^{-\gamma_1} & 0 & 0 \\
 0 & 0 & -d_2\, q^{-\gamma_1-\gamma_2} & 0 & 0 & d_2 d_3\, q^{-\gamma_2} \\
 0 & -d_2\, q^{-\gamma_1-\gamma_2} & 0 & 0 & d_2\, q^{-\gamma_1} & 0 \\
 0 & 0 & 0 & d_1 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & d_2 d_3   \\
 0 & 0 & 0 & 0 & d_2 d_3^{-1} & 0   \\
\end{array}
\right)\ ,$$ $$\sigma_2=
\left(
\begin{array}{cccccc}
 0 & d_2d_3^{-1} & 0 & 0 & 0 & 0 \\
 d_2d_3 & 0 & 0 & 0 & 0 & 0   \\
 0 & 0 & d_1 & 0 & 0 & 0 \\
 0 & d_2\, q^{-\gamma_1} & 0 & 0 & -d_2\, q^{-\gamma_1-\gamma_2} & 0 \\
 d_2 d_3\, q^{-\gamma_2} & 0 & 0 & -d_2\, q^{-\gamma_1-\gamma_2} & 0 & 0 \\
 0 & 0 & d_1 d_3^{-1}\, q^{-\gamma_1} & 0 & 0 & -d_1 q^{-2\gamma_1} \\
\end{array}
\right)\ ,$$

with $\quad d_1=q^{-2c_1 \gamma_1}\ $, $\quad d_2=q^{-c_2\gamma_1-c_1\gamma_2}\quad$ and $\quad d_3=[\gamma_1]_q^{1/2}[\gamma_2]_q^{-1/2}$.

N=2
---

We would like to describe the simplest example with $N=2$.

A representation of $B_n$ is obtained by the set of elements $w_{i,j}=\opo_i\,\opo_j\, (h_0^{(1)}\otimes\, \cdots\otimes\, h_0^{(1)})$ with $1\leq i\leq j \leq n-1$; there are ${\cal M}_{n,2}=n(n-1)/2$ vectors.

The resulting formulas for the generators of $B_n$ are (where $\{i-1,i,i+1\}\cap\{j,k\}=0$) $$\begin{aligned}
\label{LKB}
\sigma_i\; &w_{j,k} =w_{j,k}                                        &\sigma_i\; &w_{i,i} =q^{-4\gamma}w_{i,i}\nonumber\\
\sigma_i\; &w_{i,k} =-q^{-2\gamma} w_{i,k}                          &\sigma_i\; &w_{k,i}=-q^{-2\gamma} w_{k,i}\nonumber\\
\sigma_i\; &w_{i+1,k} =q^{-\gamma} w_{i,k}+w_{i+1,k}                &\sigma_i\; &w_{k,i+1}=q^{-\gamma} w_{k,i}+w_{k,i+1}\nonumber\\
\sigma_i\; &w_{i-1,k} =q^{-\gamma} w_{i,k}+w_{i-1,k}                &\sigma_i\; &w_{k,i-1}=q^{-\gamma} w_{k,i}+w_{k,i-1}\nonumber\\
\sigma_i\; &w_{i-1,i} =-q^{-2\gamma}(w_{i-1,i}+ q^{-\gamma} w_{i,i}) &\sigma_i\; &w_{i,i+1} =-q^{-2\gamma}(w_{i,i+1}+ q^{-\gamma} w_{i,i}) \nonumber\\
\sigma_i\; &w_{i-1,i-1} =q^{-2\gamma}w_{i,i}+ 2q^{-\gamma} w_{i-1,i}+w_{i-1,i-1} &&\nonumber\\
\sigma_i\; &w_{i+1,i+1} =q^{-2\gamma}w_{i,i}+ 2q^{-\gamma} w_{i,i+1}+w_{i+1,i+1}  &&\nonumber\\
\sigma_i\; &w_{i-1,i+1} =q^{-2\gamma}w_{i,i}+ q^{-\gamma} (w_{i-1,i}+w_{i,i+1})+w_{i-1,i+1}&& \end{aligned}$$

This is the representation given in formula (44) of [@JK] using the following correspondence (with their elements $w$ rewritten as $W$ and their $q$ equal to $1$): $$\begin{aligned}
\label{corrJK}
&w_{i,i}=-2\, W_{i,i+1}\quad, \nonumber\\
&w_{i,i+1}=s^{-1}\,W_{i,i+1}-W_{i,i+2}+s\, W_{i+1,i+2}\quad,\\
&w_{i,r}=-\,W_{i,r+1}+s\,W_{i+1,r+1}+s^{-1}\, W_{i,r}-\,W_{i+1,r}\quad,\quad r\geq i+2\ .\nonumber\end{aligned}$$

In [@JK] it is shown the isomorphism between the representation ${\bf W}_{n,2}$ given in their formula (44) and the Lawrence-Krammer-Bigelow representation.

In the specific $n=3$ case we get the following matrices: $$\begin{aligned}
\label{LKB-3}
\sigma_1=\left(
\begin{array}{ccc}
 q^{-4 \gamma} & 0 & 0 \\
 -q^{-3 \gamma} & -q^{-2 \gamma} & 0 \\
 q^{-2 \gamma} & 2 q^{-\gamma} & 1 \\
\end{array}
\right)\ , &&
\sigma_2=\left(
\begin{array}{ccc}
 1 & 2 q^{-\gamma} & q^{-2 \gamma} \\
 0 & -q^{-2 \gamma} & -q^{-3 \gamma} \\
 0 & 0 & q^{-4 \gamma} \\
\end{array}
\right)\ .\end{aligned}$$

Conclusions
===========

In this paper we show how is possible to obtain representations of the braid groups starting from a quantum enveloping algebra whose classical limit is the Lie algebra of the harmonic oscillator.

The fact that the representations of this algebra are infinite dimensional obliges us to work with the lowest weights vectors that form a representation of the braid group due to the commutativity of the operator ${\cal P}\Re$ with the coproduct. The formulas (\[sigmaopo\]) permit to construct a braided version of a classical representation of the braid group. It would be interesting to study the possible relations with the link invariants following the quantum groups techniques.

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[^1]: INFN Sezione di Firenze, Dipartimento di Fisica e Astronomia, Università degli Studi Firenze, email: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>