periodicity of cluster Y systems

F. Chapoton

Description

Rogers dilogarithm and periodicity of cluster Y -systems F. Chapoton July 30, 2010 F. Chapoton Rogers dilogarithm and periodicity of cluster Y -systems

• identities can


• cluster algebras


• chapoton rogers


• rogers dilogarithm


• functional equations


• famous spence-abel

Sujets

• Rogers
• Functional equation

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 15
• Langue: English

Extrait

Rogers

periodicity

of

F.

dilogarithm

and cluster

Chapoton

July

30,

F. Chapoton

2010

Y-systems

Rogers dilogarithm and periodicity of clusterY-systems

Plan

1

2 3

ial

Some dilogarithm identities can be written in a spec manner. They are related to Y-systems and cluster algebras. The cluster category lurks in the background.

F. Chapoton

Rogers dilogarithm and periodicity of clusterY-systems

Dilogarithms

First, the classical dilogarithm (Leonhard Euler) x Li 2 (x) = n X ≥1 x n 2 n =− Z log(1−y) y dy . 0

F. Chapoton

Rogers dilogarithm and periodicity of clusterY-systems

Dilogarithms

First, the classical dilogarithm (Leonhard Euler) x Li 2 (x) = X n x n 2 =− Z 0 log(1−y

n≥1

) dy y .

Then, the Rogers dilogarithm (Leonard James Rogers)

L(x) = Li 2 (x+)ol 2 1 g(x) log(1−x).

see reference book by Leonard Lewin.

F. Chapoton

Rogers dilogarithm and periodicity of clusterY-systems

Dilogarithm values and identities

Values of L:

L(0) = 0

and

F. Chapoton

L(1) =ζ(2) =π 2 /6.

Rogers dilogarithm and periodicity of clusterY-systems

Dilogarithm values and identities

Values of L:

L(0) = 0

and

Functional equations for L:

L(1) =ζ(2) =π 2 /6.

L(x) + L(1−x) = L(1).

F. Chapoton

Rogers dilogarithm and periodicity of clusterY-systems

Dilogarithm values and identities

Values of L:

L(0) = 0

and

Functional equations for L:

and

L(1) =ζ(2) =π 2 /6.

L(x) + L(1−x) = L(1).

L(1/(1 +x)) + L(1/(1 +y)) + L(y/(1 +x+y)) + L(x/(1 +x+y)) + L(xy/(1 +x)(1 +y)) = 2 L(1).

F. Chapoton

Rogers dilogarithm and periodicity of clusterY-systems

Dilogarithm values and identities

Values of L:

L(0) = 0

and

Functional equations for L:

and

L(1) =ζ(2) =π 2 /6.

L(x) + L(1−x) = L(1).

L(1/(1 +x)) + L(1/(1 +y)) + L(y/(1 +x+y)) + L(x/(1 +x+y)) + L(xy/(1 +x)(1 +y)) = 2 L(1).

FamousSpence-Abel identity with hidden cyclic symmetry of order 5.

F. Chapoton

Rogers dilogarithm and periodicity of clusterY-systems