Let $E$ be a central $A$ -extension of $G$ . A $1$ -cochain $\tau\in C^{1}(E;A)$ is called a connection cochain of $E$ if $\tau$ satisfies

$$\tau(ga)=\tau(g)+a$$

for any $g\in G,a\in A$ . The coboundary $\delta\tau\in C^{2}(E;A)$ is called a curvature of $\tau$ .

We say that a function $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is multiplicatively scale invariant if it satisfies

$$f(cx)=u(c)f(x)$$

(2)

and $f:\mathbb{R}^{d}\setminus\{0\}\rightarrow\mathbb{R}$ is additively scale invariant if it satisfies

$$f(cx)=f(x)+v(c)$$

(3)

for some even functions $u:\mathbb{R}\rightarrow\mathbb{R}^{+}$ with $u(0)=0$ and $v:\mathbb{R}\setminus\{0\}\rightarrow\mathbb{R}$ with $v(1)=0$ .

Let $Y$ be a complex Banach space. For each $f\in Y$ denote by $\hat{f}$ the function on the dual $Y^{\prime}$ defined by

(14)

$$\hat{f}(\phi)=\phi(f).$$

The weak-star topology on $Y^{\prime}$ is the weak topology on $Y^{\prime}$ induced by the family $\{\hat{f}\colon f\in Y\}.$

Let $\mathcal{F},\mathcal{G}$ be two $\sigma$ -algebras on $X,Y$ respectively. Let $\sigma\colon\mathcal{F}\to\mathbb{R}^{m}$ and let $\theta\colon\mathcal{G}\to\mathbb{R}$ be two measures. An unique measure $\sigma\otimes\theta\colon\mathcal{F}\otimes\mathcal{G}\to\mathbb{R}^{n}$ such that

$$\langle\sigma\otimes\theta,v\rangle=\langle\sigma,v\rangle\otimes\theta$$

for all $v\in\mathbb{R}^{m}$ we shall call the product measure . Here $\langle\sigma,v\rangle\otimes\theta$ is the usual product measure of $\mathbb{R}$ -valued measures.

Let $n\geq 1$ , then we set

$\displaystyle\iota(n)=(-1)^{\pi(n)},$

Full extraction holds if, given $v:T\to{\bf R}$ , there exists a collection $\{c(t)\in C(S):t\in T\}$ such that for each $t\in T$ :

$$v(t)-\pi(t)\cdot c(t)=0$$

and

$$v(t)-\pi(t)\cdot c(s)\leq 0\ \ \ \forall s\not=t$$

Virtual extraction holds if, given $v:T\to{\bf R}$ , for each $\varepsilon>0$ there exists a collection $\{c_{\varepsilon}(t)\in C(S):t\in T\}$ such that for each $t\in T$ :

$$0\leq v(t)-\pi(t)\cdot c_{\varepsilon}(t)\leq\varepsilon$$

and

$$v(t)-\pi(t)\cdot c_{\varepsilon}(s)\leq\varepsilon\ \ \ \forall s\not=t$$

A type $t^{*}\in T$ is strongly detectable if there exists $z\in{\bf R}^{S}$ such that

$$\pi(t^{*})\cdot z=0$$

and

$$\inf_{t\not=t^{*}}\pi(t)\cdot z>0$$

A rack is a set $X$ with two binary operations $\triangleright$ and $\triangleright^{-1}$ satisfying for all $x,y,z\in X$

• (i)

  $(x\triangleright y)\triangleright^{-1}y=x=(x\triangleright^{-1}y)\triangleright y$ and

• (ii)

  $(x\triangleright y)\triangleright z=(x\triangleright z)\triangleright(y\triangleright z)$ .

A rack which further satisfies $x\triangleright x=x$ for all $x\in X$ is a quandle .

The shuffle product $\phantom{}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup\phantom{}:\mathcal{W}(\mathcal{A}_{d})\times\mathcal{W}(\mathcal{A}_{d})\to\mathcal{W}(\mathcal{A}_{d})$ is defined inductively by

$${\color[rgb]{0,0,1}\mathbf{ua}}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup{\color[rgb]{0,0,1}\mathbf{vb}}=({\color[rgb]{0,0,1}\mathbf{u}}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup{\color[rgb]{0,0,1}\mathbf{vb}}){\color[rgb]{0,0,1}\mathbf{a}}+({\color[rgb]{0,0,1}\mathbf{ua}}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup{\color[rgb]{0,0,1}\mathbf{v}}){\color[rgb]{0,0,1}\mathbf{b}},$$

$${\color[rgb]{0,0,1}\mathbf{w}}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup{\color[rgb]{0,0,1}\mathbf{\varnothing}}={\color[rgb]{0,0,1}\mathbf{\varnothing}}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup{\color[rgb]{0,0,1}\mathbf{w}}={\color[rgb]{0,0,1}\mathbf{w}}$$

for all words ${\color[rgb]{0,0,1}\mathbf{u}},{\color[rgb]{0,0,1}\mathbf{v}}$ and letters ${\color[rgb]{0,0,1}\mathbf{a}},{\color[rgb]{0,0,1}\mathbf{b}}\in\mathcal{A}_{d}$ , which is then extended by bilinearity to $\mathcal{W}(\mathcal{A}_{d})$ . With some abuse of notation, the shuffle product on $T((\mathbb{R}^{d})^{\ast})$ induced by the shuffle product on words will also be denoted by $\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup\phantom{}$ .

Let $a=y+z$ , $b=x+z$ , and $c=x+y$ , that is, $x=\frac{(b+c-a)}{2}$ , $y=\frac{(a+c-b)}{2}$ , and $z=\frac{(a+b-c)}{2}$ , so that

$$\vbox{\hbox{ \centering\overpic[unit=1mm, scale = .5]{3vertex} \put(16.0,25.0){$c$} \put(22.0,0.0){$b$} \put(-3.0,0.0){$a$} \@add@centering}}=\vbox{\hbox{ \centering\overpic[unit=1mm, scale = .5]{3vertexfn} \par\put(17.0,26.0){$c$} \put(25.0,0.0){$b$} \put(-1.0,0.0){$a$} \put(21.5,13.0){$x$} \put(9.0,13.0){$y$} \put(15.0,6.0){$z$} \@add@centering}}.$$

Notice that $(a+b-c)$ , $(b+c-a)$ , and $(a+c-b)$ are all even integers. The Temperley - Lieb category $\boldsymbol{TL}$ consists of objects $n\in\mathbb{N},$ morphism spaces $TL_{m,n},$ and multiplication $TL_{m,n}\times TL_{l,m}\longrightarrow TL_{l,n}$ (see, for example, [ QW ] ). Then, the following figure depicts the Markov trace of the multiplication of $TL_{c,a+b}\times TL_{a+b,c}\longrightarrow TL_{a+b,a+b}.$ ^{2 2} 2 Recall that trace in the disk is called the Markov trace.

An arithmetical function $\psi$ on $\mathcal{M}_{q}$ is said to be multiplicative if

$\displaystyle\psi(fg)=\psi(f)\psi(g),\quad\text{ if }f\text{ and }g\text{ are coprime polynomials }.$

The shuffle product $\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup\phantom{}:\mathcal{W}(\mathcal{A}_{d})\times\mathcal{W}(\mathcal{A}_{d})\to\mathcal{W}(\mathcal{A}_{d})$ is defined inductively by

$${\color[rgb]{0,0,1}\mathbf{ua}}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup{\color[rgb]{0,0,1}\mathbf{vb}}=({\color[rgb]{0,0,1}\mathbf{u}}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup{\color[rgb]{0,0,1}\mathbf{vb}}){\color[rgb]{0,0,1}\mathbf{a}}+({\color[rgb]{0,0,1}\mathbf{ua}}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup{\color[rgb]{0,0,1}\mathbf{v}}){\color[rgb]{0,0,1}\mathbf{b}},$$

$${\color[rgb]{0,0,1}\mathbf{w}}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup{\color[rgb]{0,0,1}\mathbf{\varnothing}}={\color[rgb]{0,0,1}\mathbf{\varnothing}}\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup{\color[rgb]{0,0,1}\mathbf{w}}={\color[rgb]{0,0,1}\mathbf{w}}$$

for all words ${\color[rgb]{0,0,1}\mathbf{u}},{\color[rgb]{0,0,1}\mathbf{v}}$ and letters ${\color[rgb]{0,0,1}\mathbf{a}},{\color[rgb]{0,0,1}\mathbf{b}}\in\mathcal{A}_{d}$ , which is then extended by bilinearity to $\mathcal{W}(\mathcal{A}_{d})$ . With some abuse of notation, the shuffle product on $T((\mathbb{R}^{d})^{\ast})$ induced by the shuffle product on words will also be denoted by $\sqcup\mathchoice{\mskip-7.0mu }{\mskip-7.0mu }{\mskip-3.2mu }{\mskip-3.8mu }\sqcup\phantom{}$ .

Two representations, $G\curvearrowright^{\pi}\mathcal{H}$ and $G\curvearrowright^{\pi^{\prime}}\mathcal{H}^{\prime}$ , are additive conjugates if there exists a bijection $\xi\colon\mathcal{H}\to\mathcal{H}^{\prime}$ such that for all $v,w\in\mathcal{H}$ and $g\in G$ ,

$\displaystyle\xi(v+w)=\xi(v)+\xi(w)$

and

$\displaystyle\xi(\pi_{g}v)=\pi^{\prime}_{g}\xi(v).$

Let $\mathcal{X}$ be a vector space over $\mathbb{C}$ equipped with product $\cdot:\mathcal{X}\times\mathcal{X}\mapsto\mathcal{X}$ . Denote the vector space addition by $+$ , we call $\mathcal{X}$ an algebra if for all $a,b,c\in\mathcal{X}$ and $\alpha\in\mathbb{C}$ ,

1. 1.

   $a(bc)=(ab)c$ ,

2. 2.

   $a(b+c)=ab+ac$ ,

3. 3.

   $\alpha(ab)=(\alpha a)b=a(\alpha b)$ .

We call $\mathcal{X}$ a unital algebra if there is a unital element $1_{\mathcal{X}}$ such that, for all $a\in\mathcal{X}$

$$a=a1_{\mathcal{X}}=1_{\mathcal{X}}a.$$

(54)

An algebra $\mathcal{X}$ is called a $*$ -algebra if it is also endowed with an antilinear $*$ -operation $\mathcal{X}\ni a\mapsto a^{*}\in\mathcal{X}$ , such that $(\alpha a)^{*}=\bar{\alpha}a^{*}$ , $(a^{*})^{*}=a$ and $(ab)^{*}=b^{*}a^{*}$ for all $\alpha\in\mathbb{C}$ , $a,b\in\mathcal{X}$ .

It is called exterior form of degree two or 2-form at point $m$ on the manifold $M^{2n}$ to the bilinear and antisymmetric application $\omega^{2}:T_{m}M^{2n}\times T_{m}M^{2n}\rightarrow\Re$ , i.e.

• •

  $\omega^{2}(\alpha x+\beta y,z)=\alpha\omega^{2}(x,z)+\beta\omega^{2}(y,z)$ ,

• •

  $\omega^{2}(x,y)=-\omega^{2}(y,x)$ ,

for all $x,y,z\in T_{m}M^{2n}$ and $\alpha,\beta\in\Re$ (see [ 2 ] ).

For a matrix $F\in MF(r,p,1)$ we define a $p$ admissible balanced $s$ -partition $\tau\in Partc(F,p,s)$ to be any ordered tupel

$$\tau=(\tau(1),\ldots,\tau(s))$$

(2.54)

of matrices $\tau(\alpha)\in MBaln(r+p)$ with

$$\sum_{k=r+1}^{r+p}dias(\tau(\alpha))_{k}\leq 1$$

(2.55)

for $\alpha=1,\ldots,s$ such that

$$F=\sum_{1\leq i\leq s}\tau(\alpha)$$

(2.56)

We define $Part(F,p,s)$ to be the set of equivalence classes of $Partc(F,p,s)$ under permutation in the $s$ -tupel and identify any representative of the class with it in the sequel. Because of equation ( 2.55 ) it is trivial to note that a $p$ -admissible partition must contain at least $p$ elements.

Let $A$ be an algebra and $X$ be an $A$ -bimodule. A linear map $d:A\longrightarrow X$ is called an $A$ - derivation if

$$d(ab)=d(a)\circ b+a\circ d(b)$$

for every $a,b\in A$ .

For a $k$ -algebra $A$ we define $\Omega(A)$ to the $A$ module generated by $df$ for $f\in A$ , satisfying the relations:

$$d(f+g)=df+dg$$

$$dfg=fdg+gdf$$

$$d\alpha f=\alpha df$$

For $f,g\in A$ , $\alpha\in k$ . The co-tangent bundle of a smooth variety $X$ is the sheaf satisfying $\Omega_{X}(O)=\Omega(\mathcal{O}_{X}(O))$ for any affine open subset $O$ . To construct this explicitly we look at the diagonal embedding $\Delta:X\rightarrow X\times_{k}X$ and if $\mathcal{I}$ is the ideal sheaf of the diagonal, then:

$$\Omega_{X}=\Delta^{*}(\mathcal{I}/\mathcal{I}^{2})$$