"Rank 2 cluster algebra"의 두 판 사이의 차이

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==introduction==
  
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* cluster algebra defined by a 2x2 matrix
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* Laurent phenomenon
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* Positivity conjecture
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* finite classification
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==cluster variables and exchange relations==
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* Fix two positive integers b and c.
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* Let <math>y_1</math> and <math>y_2</math> be variable in the field <math>F=\mathbb{Q}(y_1,y_2)</math>
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* Define a sequence <math>\{y_n\}</math> by
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:<math>
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y_{m-1}y_{m+1}=
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\begin{cases}
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y_m^b+1, & \text{if <math>m</math> is odd}\\
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y_m^c+1, & \text{if <math>m</math> is even} \\
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\end{cases}
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</math>
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* We call this ''''exchange relation''''
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* <math>y_m</math>'s are called ''''cluster variable''''
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* <math>\{y_i,y_{i+1}\}</math> "'''cluster'''"
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* <math>\{y_m^py_{m+1}^q\}</math> "'''cluster monomials'''" (supported on a given cluster)
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* Note : we can use the exchange relation any <math>y_m</math> in terms of arbitrary cluster <math>\{y_i,y_{i+1}\}</math> (rational expression)
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===matrix formulation===
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:<math>B=\begin{bmatrix} 0 & -b\\ c  &\,0 \end{bmatrix}</math>
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:<math>\mu_{1}(B)=\begin{bmatrix} 0 & b\\ -c  &\,0 \end{bmatrix}</math>
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:<math>\mu_{2}(B)=\begin{bmatrix} 0 & b\\ -c  &\,0 \end{bmatrix}</math>
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* <math>x_1x_1'=x_2^c+1</math> call <math>x_1'=x_3</math>
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* <math>x_2x_2'=x_1^b+1</math> call <math>x_2'=x_4</math>
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==observations==
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;theorem (FZ) : For any <math>b,c</math>, <math>y_m</math> is a Laurent polynomial.
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* Positivity conjecture: coefficients of these Laurent polynomials are positive (numerator and denomonator always have )
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* In this example, <math>bc\leq 3</math> iff the recurrence is periodic
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==cluster algebra associated to Cartan matrices==
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* Finite type classification <math>A(b,c)</math> related to root systems of Cartan matrix
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:<math> \begin{bmatrix} 2 & -b \\ -c & 2 \end{bmatrix}</math>
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* Say <math>A(b,c)</math> is of finite/affine/indefinite type if <math>bc\leq 3</math>, <math>bc=4</math>, <math>bc>4</math>
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* when <math>bc\leq 3</math>, <math>y_m=y_n</math> if and only if <math>m\equiv n \mod (h+2)</math> where h is [[Coxeter number and dual Coxeter number|Coxeter number]]
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* <math>bc=1, h=2</math>
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* <math>bc=2, h=4</math>
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* <math>bc=3, h=6</math>
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* <math>bc\geq 4, h=\infty</math>
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* If <math>bc\geq 4</math>, all <math>y_m</math> are distinct
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==algebraic structure==
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* By "Laurent phenomenon" each element in <math>A(b,c)</math> can be uniquely expressed as Laurent polynomial in <math>y_m</math> and <math>y_{m+1}</math> for any <math>m</math>
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;theorem (Berenstein, Fomin and Zelevinsky) :
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:<math>A(b,c)=\cap_{m\in\mathbb{Z}}\mathbb{Z}[y_m^{\pm 1},y_{m+1}^{\pm 1}] =\cap_{m=0}^{2}\mathbb{Z}[y_m^{\pm 1},y_{m+1}^{\pm 1}]</math>
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* standard monomial basis : the following set  is a <math>\mathbb{Z}</math>-basis of <math>A(b,c)</math>
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:<math>\{y_0^{a_0}y_1^{a_1}y_2^{a_2}y_3^{a_3} : a_{m}\in\mathbb{Z}_{\geq 0}, a_0a_2=a_1a_3=0\}</math>
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* Here support of any such monomial is
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:<math>\{y_0,y_1\},\{y_1,y_2\},\{y_2,y_3\},\{y_0,y_3\}</math>
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* <math>A(b,c)</math> is finitely generated. In fact,
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:<math>A(b,c)=\mathbb{Z}[y_0,y_1,y_2,y_3]/\langle y_0y_2-y_1^b-1,y_1y_3-y_2^c-1\rangle</math>
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==related items==
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* [[Rank 2 cluster algebra examples]]
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==articles==
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* '''[SZ2003]'''Sherman, Paul, and Andrei Zelevinsky. 2003. Positivity and canonical bases in rank 2 cluster algebras of finite and affine types. math/0307082 (July 7). http://arxiv.org/abs/math/0307082.
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[[분류:cluster algebra]]
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[[분류:math and physics]]
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[[분류:math]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q944095 Q944095]
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===Spacy 패턴 목록===
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* [{'LOWER': 'kaluza'}, {'OP': '*'}, {'LOWER': 'klein'}, {'LEMMA': 'theory'}]

2021년 2월 17일 (수) 03:01 기준 최신판

introduction

  • cluster algebra defined by a 2x2 matrix
  • Laurent phenomenon
  • Positivity conjecture
  • finite classification



cluster variables and exchange relations

  • Fix two positive integers b and c.
  • Let \(y_1\) and \(y_2\) be variable in the field \(F=\mathbb{Q}(y_1,y_2)\)
  • Define a sequence \(\{y_n\}\) by

\[ y_{m-1}y_{m+1}= \begin{cases} y_m^b+1, & \text{if \(m\] is odd}\\ 

y_m^c+1, & \text{if <math>m\) is even} \\

\end{cases} </math>

  • We call this 'exchange relation'
  • \(y_m\)'s are called 'cluster variable'
  • \(\{y_i,y_{i+1}\}\) "cluster"
  • \(\{y_m^py_{m+1}^q\}\) "cluster monomials" (supported on a given cluster)
  • Note : we can use the exchange relation any \(y_m\) in terms of arbitrary cluster \(\{y_i,y_{i+1}\}\) (rational expression)



matrix formulation

\[B=\begin{bmatrix} 0 & -b\\ c &\,0 \end{bmatrix}\] \[\mu_{1}(B)=\begin{bmatrix} 0 & b\\ -c &\,0 \end{bmatrix}\] \[\mu_{2}(B)=\begin{bmatrix} 0 & b\\ -c &\,0 \end{bmatrix}\]

  • \(x_1x_1'=x_2^c+1\) call \(x_1'=x_3\)
  • \(x_2x_2'=x_1^b+1\) call \(x_2'=x_4\)



observations

theorem (FZ)
For any \(b,c\), \(y_m\) is a Laurent polynomial.
  • Positivity conjecture: coefficients of these Laurent polynomials are positive (numerator and denomonator always have )
  • In this example, \(bc\leq 3\) iff the recurrence is periodic



cluster algebra associated to Cartan matrices

  • Finite type classification \(A(b,c)\) related to root systems of Cartan matrix

\[ \begin{bmatrix} 2 & -b \\ -c & 2 \end{bmatrix}\]

  • Say \(A(b,c)\) is of finite/affine/indefinite type if \(bc\leq 3\), \(bc=4\), \(bc>4\)
  • when \(bc\leq 3\), \(y_m=y_n\) if and only if \(m\equiv n \mod (h+2)\) where h is Coxeter number
  • \(bc=1, h=2\)
  • \(bc=2, h=4\)
  • \(bc=3, h=6\)
  • \(bc\geq 4, h=\infty\)
  • If \(bc\geq 4\), all \(y_m\) are distinct


algebraic structure

  • By "Laurent phenomenon" each element in \(A(b,c)\) can be uniquely expressed as Laurent polynomial in \(y_m\) and \(y_{m+1}\) for any \(m\)
theorem (Berenstein, Fomin and Zelevinsky)

\[A(b,c)=\cap_{m\in\mathbb{Z}}\mathbb{Z}[y_m^{\pm 1},y_{m+1}^{\pm 1}] =\cap_{m=0}^{2}\mathbb{Z}[y_m^{\pm 1},y_{m+1}^{\pm 1}]\]

  • standard monomial basis : the following set is a \(\mathbb{Z}\)-basis of \(A(b,c)\)

\[\{y_0^{a_0}y_1^{a_1}y_2^{a_2}y_3^{a_3} : a_{m}\in\mathbb{Z}_{\geq 0}, a_0a_2=a_1a_3=0\}\]

  • Here support of any such monomial is

\[\{y_0,y_1\},\{y_1,y_2\},\{y_2,y_3\},\{y_0,y_3\}\]

  • \(A(b,c)\) is finitely generated. In fact,

\[A(b,c)=\mathbb{Z}[y_0,y_1,y_2,y_3]/\langle y_0y_2-y_1^b-1,y_1y_3-y_2^c-1\rangle\]


related items


articles

  • [SZ2003]Sherman, Paul, and Andrei Zelevinsky. 2003. Positivity and canonical bases in rank 2 cluster algebras of finite and affine types. math/0307082 (July 7). http://arxiv.org/abs/math/0307082.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'kaluza'}, {'OP': '*'}, {'LOWER': 'klein'}, {'LEMMA': 'theory'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Rank_2_cluster_algebra&oldid=51879"