Y-system and functional dilogarithm identities - 편집 역사

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@@ 113번째 줄: / 113번째 줄: @@
 * [http://fuji.cec.yamanashi.ac.jp/%7Ering/icra14/lecture/conf/nakanishi.pdf Periodicities in cluster algebras and dilogarithm identities]
-−
-−
 ==articles==

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
 * {{수학노트|url=함수_다이로그_항등식(functional_dilogarithm_identity)}}
-* $\mathbb{Y}(X,X')$ the order of $X$ and $X'$ matters!
+* <math>\mathbb{Y}(X,X')</math> the order of <math>X</math> and <math>X'</math> matters!
 * Caracciolo, R., F. Gliozzi, and R. Tateo in 1999 proves the functional dilogarithm identities associated ADExADE Y-systems assuming the periodicity of restricted T-system
@@ 9번째 줄: / 9번째 줄: @@
 ===level restricted Y-system===
 * Suppose that a family of positive real numbers
-$$\{Y^{(a)}_m(u)\mid a\in I, 1\leq m \leq t_a\ell-1,
+:<math>\{Y^{(a)}_m(u)\mid a\in I, 1\leq m \leq t_a\ell-1,
-u\in \mathbb{Z}\}$$
+u\in \mathbb{Z}\}</math>
-satisfy the level $\ell$ restricted Y-system for $\mathfrak{g}$.
+satisfy the level <math>\ell</math> restricted Y-system for <math>\mathfrak{g}</math>.
 Then, the following identities hold:
 \begin{align}\label{t:eq:DI2}
@@ 22번째 줄: / 22번째 줄: @@
 t(\ell h - h^{\vee})\mathrm{rank}\,\mathfrak{g},
 \end{align}
-where $h$ is the Coxeter number of ${\mathfrak g}$.
+where <math>h</math> is the Coxeter number of <math>{\mathfrak g}</math>.
 * in simply-laced case, we get
-$$
+:<math>
 \begin{align}
 \frac{6}{\pi^2}\sum_{a\in I}\sum_{m=1}^{\ell-1}
@@ 34번째 줄: / 34번째 줄: @@
 (\ell h - h)r=2h(\ell-1)r
 \end{align}
-$$
+</math>
 ===Y-systems for a pair of Dynkin diagrams===
 * Bloch group element
-$$
+:<math>
 \sum_{(\mathbf{i},u)\in S_{+}} Y_{\mathbf{i}}(u)\wedge (1+Y_{\mathbf{i}}(u))=0\in \Lambda^2 \mathbb{Q}(y)^{\times}
-$$ where $S_{+}=\{(\mathbf{i},u) |0\leq u \leq 2(h+h')-1,(\mathbf{i},u)\in P_{+}\}$.
+</math> where <math>S_{+}=\{(\mathbf{i},u) |0\leq u \leq 2(h+h')-1,(\mathbf{i},u)\in P_{+}\}</math>.
 * functional dilogarithm identity
-$$
+:<math>
 \sum_{(\mathbf{i},u)\in S_{+}}L\left(\frac{Y_\mathbf{i}(u)}{1+Y_\mathbf{i}(u)}\right)=h r r' L(1)
-$$
+</math>
 ==bicoloring==
-* what's $P_{+}$?
+* what's <math>P_{+}</math>?
-* We give an alternate bicoloring on the pair of Dynkin diagrams. Let us fix bipartite decompositions of $I$ and $I'$.
+* We give an alternate bicoloring on the pair of Dynkin diagrams. Let us fix bipartite decompositions of <math>I</math> and <math>I'</math>.
-* Let $ \mathbf{I}= I\times I'$ and $\mathbf{I}=\mathbf{I}_{+}\sqcup \mathbf{I}_{-}$ where $\mathbf{I}_{+}=(I_{+}\times I'_{+}) \sqcup (I_{-}\times I'_{-})$ and $\mathbf{I}_{-}=(I_{+}\times I'_{-}) \sqcup (I_{-}\times I'_{+})$.
+* Let <math> \mathbf{I}= I\times I'</math> and <math>\mathbf{I}=\mathbf{I}_{+}\sqcup \mathbf{I}_{-}</math> where <math>\mathbf{I}_{+}=(I_{+}\times I'_{+}) \sqcup (I_{-}\times I'_{-})</math> and <math>\mathbf{I}_{-}=(I_{+}\times I'_{-}) \sqcup (I_{-}\times I'_{+})</math>.
-* Let $\epsilon : \mathbf{I}\to \{1,-1\}$ be the function defined by $\epsilon(\mathbf{i})=\pm 1$ for $\mathbf{i}\in \mathbf{I}_{\pm}$ and $P_{\pm} =\{(\mathbf{i},u)\in \mathbf{I}\times\mathbb{Z}| \epsilon(\mathbf{i})(-1)^u=\pm 1\}$.
+* Let <math>\epsilon : \mathbf{I}\to \{1,-1\}</math> be the function defined by <math>\epsilon(\mathbf{i})=\pm 1</math> for <math>\mathbf{i}\in \mathbf{I}_{\pm}</math> and <math>P_{\pm} =\{(\mathbf{i},u)\in \mathbf{I}\times\mathbb{Z}| \epsilon(\mathbf{i})(-1)^u=\pm 1\}</math>.
-* Roughly speaking, we want our alternate bicoloring interchanges their colors as $u\in \mathbb{Z}$ changes by 1.
+* Roughly speaking, we want our alternate bicoloring interchanges their colors as <math>u\in \mathbb{Z}</math> changes by 1.
 ==an example==
-* compute $\mathbb{Y}(A_2,A_1)$ explicitly
+* compute <math>\mathbb{Y}(A_2,A_1)</math> explicitly
 * <math>y_{m-1}y_{m+1}=y_m+1</math>
 * Start with two variables <math>y_1,y_2</math>.
@@ 67번째 줄: / 67번째 줄: @@
 ===observations===
-* we saw that $$S=\left\{x,y,\frac{y+1}{x},\frac{x+y+1}{x y},\frac{x+1}{y}\right\}$$ forms a half-period of $\mathbb{Y}(A_2,A_1)$.
+* we saw that :<math>S=\left\{x,y,\frac{y+1}{x},\frac{x+y+1}{x y},\frac{x+1}{y}\right\}</math> forms a half-period of <math>\mathbb{Y}(A_2,A_1)</math>.
-* So we have $r=2,h=3$ and $r'=1,h'=2$.
+* So we have <math>r=2,h=3</math> and <math>r'=1,h'=2</math>.
-* They are all Laurent polynomials in $x$ and $y$.
+* They are all Laurent polynomials in <math>x</math> and <math>y</math>.

← 이전 판		2020년 11월 13일 (금) 15:38 판
135번째 줄:		135번째 줄:
	[[분류:dilogarithm]]		[[분류:dilogarithm]]
	[[분류:Y-system]]		[[분류:Y-system]]
		+	[[분류:migrate]]

@@ 2번째 줄: / 2번째 줄: @@
 * {{수학노트|url=함수_다이로그_항등식(functional_dilogarithm_identity)}}
 * $\mathbb{Y}(X,X')$ the order of $X$ and $X'$ matters!
@@ 121번째 줄: / 123번째 줄: @@
 * Inoue, Rei, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, and Tomoki Nakanishi. 2013. “Periodicities of T and Y-Systems, Dilogarithm Identities, and Cluster Algebras I: Type B_r.” Publications of the Research Institute for Mathematical Sciences 49 (1): 1–42. doi:10.4171/PRIMS/95.
 * Nakanishi, Tomoki. “Dilogarithm Identities for Conformal Field Theories and Cluster Algebras: Simply Laced Case.” arXiv:0909.5480 [math], September 30, 2009. http://arxiv.org/abs/0909.5480.
-*  Chapoton, Frédéric. 2005. Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems. Bulletin of the London Mathematical Society 37, no. 5 (October 1): 755 -760. doi:[http://dx.doi.org/10.1112/S0024609305004510 10.1112/S0024609305004510].
+* Chapoton, Frédéric. 2005. Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems. Bulletin of the London Mathematical Society 37, no. 5 (October 1): 755 -760. doi:[http://dx.doi.org/10.1112/S0024609305004510 10.1112/S0024609305004510].
 * Caracciolo, R., F. Gliozzi, and R. Tateo. “A Topological Invariant of RG Flows in 2D Integrable Quantum Field Theories.” arXiv:hep-th/9902094, February 12, 1999. http://arxiv.org/abs/hep-th/9902094.
 * Frenkel, Edward, and Andras Szenes. “Thermodynamic Bethe Ansatz and Dilogarithm Identities I.” arXiv:hep-th/9506215, July 2, 1995. http://arxiv.org/abs/hep-th/9506215.
-* Gliozzi, F., and R. Tateo. 1995. “ADE Functional Dilogarithm Identities and Integrable Models.” Physics Letters B 348 (1–2) (March 30): 84–88. http://dx.doi.org/10.1016/0370-2693(95)00125-5.
+* Gliozzi, F., and R. Tateo. “ADE Functional Dilogarithm Identities and Integrable Models.” Physics Letters B 348, no. 1–2 (March 30, 1995): 84–88. doi:[http://dx.doi.org/10.1016/0370-2693(95)00125-5 10.1016/0370-2693(95)00125-5].
 * Kuniba, A., and T. Nakanishi. “Rogers Dilogarithm in Integrable Systems.” arXiv:hep-th/9210025, October 5, 1992. http://arxiv.org/abs/hep-th/9210025.
 * Kuniba, Atsuo, and Tomoki Nakanishi. “Spectra in Conformal Field Theories from the Rogers Dilogarithm.” arXiv:hep-th/9206034, June 9, 1992. http://arxiv.org/abs/hep-th/9206034.

@@ 122번째 줄: / 122번째 줄: @@
 * Nakanishi, Tomoki. “Dilogarithm Identities for Conformal Field Theories and Cluster Algebras: Simply Laced Case.” arXiv:0909.5480 [math], September 30, 2009. http://arxiv.org/abs/0909.5480.
 *  Chapoton, Frédéric. 2005. Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems. Bulletin of the London Mathematical Society 37, no. 5 (October 1): 755 -760. doi:[http://dx.doi.org/10.1112/S0024609305004510 10.1112/S0024609305004510].
 * Frenkel, Edward, and Andras Szenes. “Thermodynamic Bethe Ansatz and Dilogarithm Identities I.” arXiv:hep-th/9506215, July 2, 1995. http://arxiv.org/abs/hep-th/9506215.
 * Gliozzi, F., and R. Tateo. 1995. “ADE Functional Dilogarithm Identities and Integrable Models.” Physics Letters B 348 (1–2) (March 30): 84–88. http://dx.doi.org/10.1016/0370-2693(95)00125-5.

← 이전 판		2015년 2월 15일 (일) 16:53 판
93번째 줄:		93번째 줄:

	==history==		==history==
−	* The Y-system for general g including ADE was introduced in by Kuniba and Nakanishi in 1992 before RTV93
−	* http://www.google.com/search?hl=en&tbs=tl:1&q=
−
−

	==related items==		==related items==

@@ 126번째 줄: / 126번째 줄: @@
 * [http://arxiv.org/abs/0909.5480 Dilogarithm identities for conformal field theories and cluster algebras: simply laced case] Tomoki Nakanishi, 2009
 *  Chapoton, Frédéric. 2005. Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems. Bulletin of the London Mathematical Society 37, no. 5 (October 1): 755 -760. doi:[http://dx.doi.org/10.1112/S0024609305004510 10.1112/S0024609305004510].
-* [http://arxiv.org/abs/hep-th/9506215 Thermodynamic Bethe Ansatz and Dilogarithm Identities I] Edward Frenkel, Andras Szenes, 1995
+* Frenkel, Edward, and Andras Szenes. “Thermodynamic Bethe Ansatz and Dilogarithm Identities I.” arXiv:hep-th/9506215, July 2, 1995. http://arxiv.org/abs/hep-th/9506215.
 * Gliozzi, F., and R. Tateo. 1995. “ADE Functional Dilogarithm Identities and Integrable Models.” Physics Letters B 348 (1–2) (March 30): 84–88. http://dx.doi.org/10.1016/0370-2693(95)00125-5.
 * [http://arxiv.org/abs/hep-th/9210025 Rogers Dilogarithm in Integrable Systems] A. Kuniba, T. Nakanishi, 1992

@@ 102번째 줄: / 102번째 줄: @@
 * [[cluster algebra]]
 * [[Nahm's equation]]
-* [[Central charge, L-values and dilogarithm]]
+* [[Central charge and dilogarithm]]
 * [[dilogarithm and dilogarithm identities]]
 * [[Bloch group]]
 ==computational resource==