"Rank 2 cluster algebra"의 두 판 사이의 차이

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1번째 줄: 1번째 줄:
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<h5>introduction</h5>
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==introduction==
  
 
* cluster algebra defined by a 2x2 matrix
 
* cluster algebra defined by a 2x2 matrix
6번째 줄: 6번째 줄:
 
* finite classification
 
* finite classification
  
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+
  
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+
  
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<h5>cluster variables and exchange relations</h5>
+
==cluster variables and exchange relations==
 +
* Fix two positive integers b and c.
 +
* Let <math>y_1</math> and <math>y_2</math> be variable in the field <math>F=\mathbb{Q}(y_1,y_2)</math>
 +
* Define a sequence <math>\{y_n\}</math> by
 +
:<math>
 +
y_{m-1}y_{m+1}=
 +
\begin{cases}
 +
y_m^b+1, & \text{if <math>m</math> is odd}\\
 +
y_m^c+1, & \text{if <math>m</math> is even} \\
 +
\end{cases}
 +
</math>
 +
* We call this ''''exchange relation''''
 +
* <math>y_m</math>'s are called ''''cluster variable''''
 +
* <math>\{y_i,y_{i+1}\}</math> "'''cluster'''"
 +
* <math>\{y_m^py_{m+1}^q\}</math> "'''cluster monomials'''" (supported on a given cluster)
 +
* 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)
  
−
Fix two positive integers b and c.
 
  
−
Let y_1 and y_2 be variable. Define a sequence {y_n}.
+
  
−
<math>y_{m-1}y_{m+1}=y_m^b+1</math> if m odd
+
===matrix formulation===
 +
:<math>B=\begin{bmatrix} 0 & -b\\ c  &\,0 \end{bmatrix}</math>
 +
:<math>\mu_{1}(B)=\begin{bmatrix} 0 & b\\ -c  &\,0 \end{bmatrix}</math>
 +
:<math>\mu_{2}(B)=\begin{bmatrix} 0 & b\\ -c  &\,0 \end{bmatrix}</math>
 +
* <math>x_1x_1'=x_2^c+1</math> call <math>x_1'=x_3</math>
 +
* <math>x_2x_2'=x_1^b+1</math> call <math>x_2'=x_4</math>
  
−
<math>y_{m-1}y_{m+1}=y_m^c+1</math> if m even
 
  
−
We call this ''''exchange relation''''
+
  
−
y_m's are called ''''cluster variable''''
+
==observations==
 +
;theorem (FZ) : For any <math>b,c</math>, <math>y_m</math> is a Laurent polynomial.
 +
* Positivity conjecture: coefficients of these Laurent polynomials are positive (numerator and denomonator always have )
 +
* In this example, <math>bc\leq 3</math> iff the recurrence is periodic
  
−
<math>\{y_i,y_{i+1}\}</math> "'''cluster'''"
+
  
−
Note : we can use the exchange relation any y_m in terms of arbitrary cluster <math>\{y_i,y_{i+1}\}</math> (rational expression)
+
  
−
 
+
==cluster algebra associated to Cartan matrices==
 +
* Finite type classification <math>A(b,c)</math> related to root systems of Cartan matrix
 +
:<math> \begin{bmatrix} 2 & -b \\ -c & 2 \end{bmatrix}</math>
 +
* 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>
 +
* 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]]
 +
* <math>bc=1, h=2</math>
 +
* <math>bc=2, h=4</math>
 +
* <math>bc=3, h=6</math>
 +
* <math>bc\geq 4, h=\infty</math>
 +
* If <math>bc\geq 4</math>, all <math>y_m</math> are distinct
 +
  
−
 
+
==algebraic structure==
 +
* 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>
 +
;theorem (Berenstein, Fomin and Zelevinsky) :
 +
:<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>
 +
* standard monomial basis : the following set  is a <math>\mathbb{Z}</math>-basis of <math>A(b,c)</math>
 +
:<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>
 +
* Here support of any such monomial is
 +
:<math>\{y_0,y_1\},\{y_1,y_2\},\{y_2,y_3\},\{y_0,y_3\}</math>
 +
* <math>A(b,c)</math> is finitely generated. In fact,
 +
:<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>
  
−
<h5>matrix formulation</h5>
+
  
−
<math>B=\begin{bmatrix} 0 & -b\\ c &\,0 \end{bmatrix}</math>
+
==related items==
 +
* [[Rank 2 cluster algebra examples]]
 +
   
  
−
<math>\mu_{1}(B)=\begin{bmatrix} 0 & b\\ -c  &\,0 \end{bmatrix}</math>
+
==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.
  
−
<math>\mu_{2}(B)=\begin{bmatrix} 0 & b\\ -c  &\,0 \end{bmatrix}</math>
+
[[분류:cluster algebra]]
 +
[[분류:math and physics]]
 +
[[분류:math]]
 +
[[분류:migrate]]
  
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For <math>k\in \{1,2,\cdots, n\}</math>,  <math>x_kx_k' = \prod_{b_{ik}>0} x_i^{b_{ik}}+\prod_{b_{ik}<0} x_i^{|b_{ik}|}</math>
+
==메타데이터==
−
 
+
===위키데이터===
−
<math>x_1x_1'=x_2^c+1</math> call x_1'=x_3
+
* ID : [https://www.wikidata.org/wiki/Q944095 Q944095]
−
 
+
===Spacy 패턴 목록===
−
<math>x_2x_2'=x_1^b+1</math> call x_2'=x_4
+
* [{'LOWER': 'kaluza'}, {'OP': '*'}, {'LOWER': 'klein'}, {'LEMMA': 'theory'}]
−
 
 
−
 
 
−
 
 
−
<math>\mu_k(B)</math>
 
−
 
 
−
<math>-b_{ij}</math> if k=i or j
 
−
 
 
−
<math>b_{ij}</math> if <math>b_{ik}b_{kj}\leq 0</math>
 
−
 
 
−
 
 
−
 
 
−
<math>b_{ij}+b_{ik}b_{kj}</math> if <math>b_{ik}, b_{kj}>0</math>
 
−
 
 
−
<math>b_{ij}-b_{ik}b_{kj}</math> if <math>b_{ik},{b_{kj}< 0</math>
 
−
 
 
−
<math>\mu_{1}(B)=\begin{bmatrix} 0 & b\\ -c  &\,0 \end{bmatrix}</math>
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
<h5>observations</h5>
 
−
 
 
−
(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, 
 
−
 
 
−
<math>bc\leq 3</math> iff the recurrence is periodic
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
<h5>canonical basis (1/25/2011)</h5>
 
−
 
 
−
y_m cluster variables<br> \{y_m,y_{m+1}\} clusters<br> \{y_m^py_{m+1}^q\} cluster monomials (supported on a given cluster)<br> Goal : define and construct 'canonical basis' B in A(b,c) for <math>bc\leq 4</math><br> By "Leurant phenomenen" each element in A(b,c) can be uniquely expressed as Laurent polynomial in y_m and y_{m+1} for any m<br> B.F.Zelevinsky 's result :
 
−
 
 
−
'''*cap인지 cup인지 확인 필요*''' A(b,c) =\cup_{m\in\mathbb{Z}\mathbb{Z}[y_n^{\pm 1,\y_{m+1}^{\pm 1]=\cup_{m=0}^{\alpha}\mathbb{Z}[y_n^{\pm 1,\y_{m+1}^{\pm 1]<br> standard monomial basis : the set <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> is a \mathbb{Z}-basis of A(b,c).<br> Here support of any such monomial is <math>\{y_0,y_1\},\{y_1,y_2\},\{y_2,y_3\},\{y_0,y_3\}</math>.<br> A(b,c) is finitely generated. In fact,<br><math>A(b,c)=\mathbb{Z}[y_0,y_1,y_2,y_3]/<y_0y_2-y_1^b-1,y_1y_3-y_2^c-1></math>
 
−
 
 
−
 
 
−
 
 
−
Finite type classification :
 
−
 
 
−
A(b,c) related to root systems of Cartan matrix
 
−
 
 
−
<math> \begin{bmatrix} 2 & -b \\ -c & 2 \end{bmatrix}</math>
 
−
 
 
−
Say A(b,c) is of finite/affine/indefinite type if <math>bc\leq 3</math>, <math>bc=4</math>, <math>bc>4</math>
 
−
 
 
−
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|coxeter number]]
 
−
 
 
−
bc=1, h=2
 
−
 
 
−
bc=2, h=4
 
−
 
 
−
bc=3, h=6
 
−
 
 
−
bc\geq 4, h=\infity
 
−
 
 
−
If bc\geq 4, all y_m distinct
 
−
 
 
−
 
 
−
 
 
−
 
 
−
 
 
−
\def
 
−
 
 
−
A nonzero element <math>y\in A(b,c)</math> is positive if for any <math>m\in \mathbb{Z}</math>, all coefficients in the expansion of y as Laurent polynomial in y_{m} and y_{m+1} are positive.
 
−
 
 
−
<math>A^{+}(b,c) </math>= set of positive elements in A(b,c) semiring
 
−
 
 
−
\thm ('''[SZ2003]''')
 
−
 
 
−
Suppose <math>bc\leq 4</math>. Then there exists a unique basis B of A(b,c) such that <math>A^{+}(b,c) </math> = set of positive integer linear combinations of elements of B.
 
−
 
 
−
 
 
−
 
 
−
\remark
 
−
 
 
−
If exists, then the uniqueness follows : B must consist of all indecomposable elements of <math>A^{+}(b,c)</math> i.e. those one cannot be written as positive sum of 2 elements in <math>A^{+}(b,c) </math>.
 
−
 
 
−
Let Q=\mathbb{Z}^2 be rank2 lattice with basis \{\alpha_1, \alpha_2\}. \alpha=a_1\alpha_1+a_2\alpha_2 <-> (a_1,a_2)
 
−
 
 
−
 
 
−
 
 
−
\Theorem *
 
−
 
 
−
When bc\leq 4, for each \alpha=(a_1,a_2)\in Q, there exists unique basis element \chi[\alpha]\in B of form \chi[\alpha]=\frac{N_{\alpha}(y_1,y_2)}{y_1^{a_1}y_2^{a_2}} where N_{\alpha} is a polynomial with constant term 1.
 
−
 
 
−
Map \alpha \to \chi[\alpha] is bijection \mathbb{Z}^2\to B
 
−
 
 
−
when bc\leq 3, B=\{cluster monomials\}
 
−
 
 
−
 
 
−
 
 
−
examples : b=c=1 case and b=c=2 case
 
−
 
 
−
 
 
−
 
 
−
From definition, B is invariant under any automorphism of A(b,c) preserving <math>A^{+}(b,c)</math> (Call such a map positive)
 
−
 
 
−
For all <math>p\in \mathbb{Z}</math>, there exists a positive automorphism <math>\sigma_{p}</math>  of A(b,c) defined by <math>\sigma_p(y_m)=y_{2p-m}</math>.
 
−
 
 
−
<math><\sigma_i : i\in \mathbb{Z}> = <\sigma_p,\sigma_{p+1}></math> for any p<br> \thm
 
−
 
 
−
The bijection B<-> Q translates the action of each <math>\sigma_p</math> on B into piecewise linear transformation of Q ;
 
−
 
 
−
\sigma_1(a_1,a_2) = (a_1 , c \max (a_1,0) -a_2)
 
−
 
 
−
\sigma_2(a_1,a_2) = (b \max (a_2,0)-a_1, a_2)
 
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\thm 1 (finite case)<br> If bc\leq 3, then B is the set of all cluster monomials.
 
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In affine case, introduce z an element of A(b,c) by
 
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−
<math>z=y_0y_3-y_1y_2</math> if (b,c)=(2,2) or
 
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−
<math>z=y_0^2y_3-(y_1+2)y_2^2</math> if (b,c)=(1,4)
 
−
 
 
−
Let T_0, T_1,\cdots, be Chebyshev polynomials defined by <math>T_0=1</math>, <math>T_n(t+t^{-1}) = t^n+t^{-n}</math> for n >0
 
−
 
 
−
Then set <math>z_n = T_n(z)</math>
 
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\thm 2 (affine case)
 
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If bc=4, B = the set of all cluster monomials union <math>\{z_n : n\geq 1\}</math>
 
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'''open problem''' : Give combinatorial formula for Laurent expansion of cluster variable's y_m when bc>4.
 
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For any b,c identify Q with root lattice such that \alpha_1,\alpha_2 correspond to simple roots
 
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Initial cluster variables y_1,y_2 correspond to negative simple roots
 
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Each cluster variable y_m\neq y_1, y_2 has form
 
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<math>\frac{N_{\alpha}(y_1,y_2)}{y_1^{a_1}y_2^{a_2}}</math> for a positive real root <math>a_1\alpha_1+a_2\alpha_2</math>.
 
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In affine case bc=4, positive imaginary root are all positive integer multiples of root \delta given by
 
−
 
 
−
\delta =\alpha_1+\alpha_2 if (b,c)=(2,2)
 
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\delta = \alpha_1+2\alpha_2 if (b,c)=(1,4)
 
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−
\proposition
 
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bc=4
 
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Cluster monimials <-> root lattice - {imaginary roots}
 
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(proof) FZ
 
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−
Each cluster variable y_m = \chi[\alpha(m)] where m\neq 1,2 has denominator \alpha(m) which is positive root.
 
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−
Set of all cluster variable's \{y_m, m=1,2 \} <-> \{positive real roots \}
 
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−
To finish prop, it's enought to show
 
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−
(1) For each m\in \mathbb{Z}, \alpha(m) and \alpha(m+1) form \mathbb{Z}-basis of Q
 
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−
(2) For each m\in \mathbb{Z}, \alpha(m) and \alpha(m+1) are only positive real roots in additive semi group \mathbb{Z}_{\geq 0} \alpha(m)+\mathbb{Z}_{\geq 0} \alpha(m+1)
 
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−
(3) The union \cup_{m\in\mathbb{Z}} [ \mathbb{Z}_{\geq 0} \alpha(m)+\mathbb{Z}_{\geq 0} \alpha(m+1)] = Q-\phi_{+}^{im}
 
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<h5>history</h5>
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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<h5>related items</h5>
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
 
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* http://en.wikipedia.org/wiki/
 
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* http://www.scholarpedia.org/
 
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* http://www.proofwiki.org/wiki/
 
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
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<h5>books</h5>
 
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* [[2011년 books and articles]]
 
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* http://library.nu/search?q=
 
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* http://library.nu/search?q=
 
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<h5>expositions</h5>
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
 
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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. <br>  <br>
 
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://arxiv.org/
 
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* http://www.pdf-search.org/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://dx.doi.org/
 
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<h5>question and answers(Math Overflow)</h5>
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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<h5>blogs</h5>
 
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*  구글 블로그 검색<br>
 
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**  http://blogsearch.google.com/blogsearch?q=<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* http://ncatlab.org/nlab/show/HomePage
 
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<h5>experts on the field</h5>
 
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* http://arxiv.org/
 
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<h5>links</h5>
 
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
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* http://functions.wolfram.com/
 

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.

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Spacy 패턴 목록

  • [{'LOWER': 'kaluza'}, {'OP': '*'}, {'LOWER': 'klein'}, {'LEMMA': 'theory'}]
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