Jones-Ocneanu trace - 편집 역사

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imported>Pythagoras0: 새 문서: ==introduction== * linear functional on the Hecke algebra of type $A_n$ * Jones related a trace found by Ocneanu with HOMFLY polynomial ==construction== * let $W=S_n$, the symm...

2017-05-25T08:08:28Z

새 문서: ==introduction== * linear functional on the Hecke algebra of type $A_n$ * Jones related a trace found by Ocneanu with HOMFLY polynomial ==construction== * let $W=S_n$, the symm...

새 문서

==introduction==
* linear functional on the Hecke algebra of type $A_n$
* Jones related a trace found by Ocneanu with [[HOMFLY polynomial]]

==construction==
* let $W=S_n$, the symmetric group on $n$ letters and $S$ the set of transpositions $s_i: = (i,i+1)$
* Coxeter matrix : $m_{ii}=1$ and $m_{i,i+1}=3$, 2 otherwise

Let $A$ be a commutative ring with 1 and fix two invertible elements $u, v \in A$.

For $n\geq 1$, consider $H_A(S_n)$ associated with $S_n$ over the ring $A$ and with parameters $a_{s_i} = u$, $b_{s_i} = v$ for $1\leq i \leq n - 1$.

For simplicity, let $H_n: = H_A(S_n)$.

Regard $H_n$ as a subalgebra of $H_{n+1}$.

;thm (Jones, Ocneanu)
There is a unique family $\{\tau_n\}_{n\geq1}$ of $A$-linear maps $\tau_n : H_n \to A$ s.t. the following conditions hold :
$$
\begin{array}{ll}
(M1) & \tau_1(T_1)=1 \\
(M2) & \tau_{n+1}(hT_{s_n}^{\pm})=\tau_{n}(h) \quad& \text{for $n\geq1$ and $h\in H_n$} \\
(M3) & \tau_n(hh')=\tau_{n}(h'h) & \text{for $n\geq1$ and $h,h'\in H_n$}
\end{array}
$$
Moreover, $\tau_{n+1}(h)=v^{-1}(1-u)\tau_{n}(h)$ for all $n\geq 1$ and $h\in H_n$.

;proof
Let us define $\tau_n$ recursively as follows.

For $n=1$, set $\tau_1(T_1)=1$.

Let $n\geq 1$ and assume that $\tau_n$ has been defined. Then we set
\begin{equation}\label{star}
\tau_{n+1}(a+b T_{s_n}c):=\frac{1-u}{v}\tau_n(a)+\tau_n(bc), \, a,b,c\in H_n
\end{equation}

need to check that $\tau_{n+1}$ is well-defined and satisfies M2, M3.

we need the isomorphism of $A$-modules for $n\geq 2$:

$$
\psi_n: H_n\oplus (H_n\otimes_{H_{n-1}} H_n) \to H_{n+1},\, a\oplus (b\otimes c)\mapsto a+bT_{s_n}c
$$
■

* $v^{-1}$ is used when we define $\tau_n$
* note that to have $T_{s_n}^{-1}$, we need $u^{-1}$.

← 이전 판		2020년 11월 13일 (금) 11:10 판
1번째 줄:		1번째 줄:
−	~~==introduction==~~
−	* linear functional on the Hecke algebra of type $A_n$
−	* Jones related a trace found by Ocneanu with [[HOMFLY polynomial]]

−
−	~~==construction==~~
−	* let $W=S_n$, the symmetric group on $n$ letters and $S$ the set of transpositions $s_i: = (i,i+1)$
−	* Coxeter matrix : $m_{ii}=1$ and $m_{i,i+1}=3$, 2 otherwise
−
−	~~Let $A$ be a commutative ring with 1 and fix two invertible elements $u, v \in A$.~~
−
−	~~For $n\geq 1$, consider $H_A(S_n)$ associated with $S_n$ over the ring $A$ and with parameters $a_{s_i} = u$, $b_{s_i} = v$ for $1\leq i \leq n - 1$.~~
−
−	~~For simplicity, let $H_n: = H_A(S_n)$.~~
−
−	~~Regard $H_n$ as a subalgebra of $H_{n+1}$.~~
−
−	~~;thm (Jones, Ocneanu)~~
−	~~There is a unique family $\{\tau_n\}_{n\geq1}$ of $A$-linear maps $\tau_n : H_n \to A$ s.t. the following conditions hold :~~
−	$$
−	~~\begin{array}{ll}~~
−	~~(M1) & \tau_1(T_e)=1 \\~~
−	~~(M2) & \tau_{n+1}(hT_{s_n}^{\pm})=\tau_{n}(h) \quad& \text{for $n\geq1$ and $h\in H_n$} \\~~
−	~~(M3) & \tau_n(hh')=\tau_{n}(h'h) & \text{for $n\geq1$ and $h,h'\in H_n$}~~
−	~~\end{array}~~
−	$$
−	~~Moreover, $\tau_{n+1}(h)=v^{-1}(1-u)\tau_{n}(h)$ for all $n\geq 1$ and $h\in H_n$.~~
−
−	~~;proof~~
−	~~Let us define $\tau_n$ recursively as follows.~~
−
−	~~For $n=1$, set $\tau_1(T_e)=1$.~~
−
−	~~Let $n\geq 1$ and assume that $\tau_n$ has been defined. Then we set~~
−	~~\begin{equation}\label{star}~~
−	~~\tau_{n+1}(a+b T_{s_n}c):=\frac{1-u}{v}\tau_n(a)+\tau_n(bc), \, a,b,c\in H_n~~
−	~~\end{equation}~~
−
−	~~need to check that $\tau_{n+1}$ is well-defined and satisfies M2, M3.~~
−
−	~~we need the isomorphism of $A$-modules for $n\geq 2$:~~
−
−	$$
−	~~\psi_n: H_n\oplus (H_n\otimes_{H_{n-1}} H_n) \to H_{n+1},\, a\oplus (b\otimes c)\mapsto a+bT_{s_n}c~~
−	$$
−	■
−
−	* $v^{-1}$ is used when we define $\tau_n$
−	* note that to have $T_{s_n}^{-1}$, we need $u^{-1}$.
−
−
−	~~[[분류:Knot theory]]~~

← 이전 판		2017년 5월 25일 (목) 08:09 판
48번째 줄:		48번째 줄:
	* $v^{-1}$ is used when we define $\tau_n$		* $v^{-1}$ is used when we define $\tau_n$
	* note that to have $T_{s_n}^{-1}$, we need $u^{-1}$.		* note that to have $T_{s_n}^{-1}$, we need $u^{-1}$.
		+
		+
		+	[[분류:Knot theory]]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* linear functional on the Hecke algebra of type $A_n$
+* linear functional on the Hecke algebra of type <math>A_n</math>
 * Jones related a trace found by Ocneanu with [[HOMFLY polynomial]]
 ==construction==
-* let  $W=S_n$, the symmetric group on $n$ letters and $S$ the set of transpositions $s_i: = (i,i+1)$
+* let  <math>W=S_n</math>, the symmetric group on <math>n</math> letters and <math>S</math> the set of transpositions <math>s_i: = (i,i+1)</math>
-* Coxeter matrix  : $m_{ii}=1$  and $m_{i,i+1}=3$, 2 otherwise
+* Coxeter matrix  : <math>m_{ii}=1</math>  and <math>m_{i,i+1}=3</math>, 2 otherwise
-Let $A$ be a commutative ring with 1 and fix two invertible elements $u, v \in A$.
+Let <math>A</math> be a commutative ring with 1 and fix two invertible elements <math>u, v \in A</math>.
-For $n\geq 1$, consider $H_A(S_n)$ associated with $S_n$ over the ring $A$ and with parameters $a_{s_i} = u$, $b_{s_i} = v$ for $1\leq i \leq n - 1$.
+For <math>n\geq 1</math>, consider <math>H_A(S_n)</math> associated with <math>S_n</math> over the ring <math>A</math> and with parameters <math>a_{s_i} = u</math>, <math>b_{s_i} = v</math> for <math>1\leq i \leq n - 1</math>.
-For simplicity, let $H_n: = H_A(S_n)$.
+For simplicity, let <math>H_n: = H_A(S_n)</math>.
-Regard $H_n$ as a subalgebra of $H_{n+1}$.
+Regard <math>H_n</math> as a subalgebra of <math>H_{n+1}</math>.
 ;thm (Jones, Ocneanu)
-There is a unique family $\{\tau_n\}_{n\geq1}$ of $A$-linear maps $\tau_n : H_n \to A$  s.t. the following conditions hold :
+There is a unique family <math>\{\tau_n\}_{n\geq1}</math> of <math>A</math>-linear maps <math>\tau_n : H_n \to A</math>  s.t. the following conditions hold :
-$$
+:<math>
 \begin{array}{ll}
 (M1) & \tau_1(T_e)=1 \\
-(M2) & \tau_{n+1}(hT_{s_n}^{\pm})=\tau_{n}(h) \quad& \text{for $n\geq1$ and $h\in H_n$} \\
+(M2) & \tau_{n+1}(hT_{s_n}^{\pm})=\tau_{n}(h) \quad& \text{for </math>n\geq1<math> and </math>h\in H_n<math>} \\
-(M3) & \tau_n(hh')=\tau_{n}(h'h) & \text{for $n\geq1$ and $h,h'\in H_n$}
+(M3) & \tau_n(hh')=\tau_{n}(h'h) & \text{for </math>n\geq1<math> and </math>h,h'\in H_n<math>}
 \end{array}
-$$
+</math>
-Moreover, $\tau_{n+1}(h)=v^{-1}(1-u)\tau_{n}(h)$ for all $n\geq 1$ and $h\in H_n$.
+Moreover, <math>\tau_{n+1}(h)=v^{-1}(1-u)\tau_{n}(h)</math> for all <math>n\geq 1</math> and <math>h\in H_n</math>.
 ;proof
-Let us define $\tau_n$ recursively as follows.
+Let us define <math>\tau_n</math> recursively as follows.
-For $n=1$, set $\tau_1(T_e)=1$.
+For <math>n=1</math>, set <math>\tau_1(T_e)=1</math>.
-Let $n\geq 1$ and assume that $\tau_n$ has been defined. Then we set
+Let <math>n\geq 1</math> and assume that <math>\tau_n</math> has been defined. Then we set
 \begin{equation}\label{star}
 \tau_{n+1}(a+b T_{s_n}c):=\frac{1-u}{v}\tau_n(a)+\tau_n(bc), \, a,b,c\in H_n
 \end{equation}
-need to check that $\tau_{n+1}$ is well-defined and satisfies M2, M3.
+need to check that <math>\tau_{n+1}</math> is well-defined and satisfies M2, M3.
-we need the isomorphism of $A$-modules for $n\geq 2$:
+we need the isomorphism of <math>A</math>-modules for <math>n\geq 2</math>:
-$$
+:<math>
 \psi_n: H_n\oplus (H_n\otimes_{H_{n-1}} H_n) \to H_{n+1},\, a\oplus (b\otimes c)\mapsto a+bT_{s_n}c
-$$
+</math>
 ■
-* $v^{-1}$ is used when we define $\tau_n$
+* <math>v^{-1}</math> is used when we define <math>\tau_n</math>
-* note that to have $T_{s_n}^{-1}$, we need $u^{-1}$.
+* note that to have <math>T_{s_n}^{-1}</math>, we need <math>u^{-1}</math>.
 [[분류:Knot theory]]
 [[분류:migrate]]

← 이전 판	2020년 11월 16일 (월) 17:50 판
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