Root systems and Dynkin diagrams(mathematica)

• Root Systems and Dynkin diagrams
• http://en.wikipedia.org/wiki/root_systems
• [1][2][3][4]http://en.wikipedia.org/wiki/Dynkin_diagram

 

 

A_n root systems

(* A_n type Cartan matrix *)
r := 2
rt[i_] := UnitVector[r + 1, i] - UnitVector[r + 1, i + 1]
a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]
A := Table[a[i, j], {i, 1, r}, {j, 1, r}]
A // MatrixForm

 

 

B_n root systems

(*B_r type Cartan matrix*)
r := 2
rt[i_] := UnitVector[r, i] - UnitVector[r, i + 1]
rt[r] := UnitVector[r, r]
a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]
A := Table[a[i, j], {i, 1, r}, {j, 1, r}]
A // MatrixForm

 

 

C_n root systems

(* A_n type Cartan matrix *)
r := 2
rt[i_] := UnitVector[r + 1, i] - UnitVector[r + 1, i + 1]
a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]
A := Table[a[i, j], {i, 1, r}, {j, 1, r}]
A // MatrixForm

 

 

D_n root systems

(* A_n type Cartan matrix *)
r := 2
rt[i_] := UnitVector[r + 1, i] - UnitVector[r + 1, i + 1]
a[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]
A := Table[a[i, j], {i, 1, r}, {j, 1, r}]
A // MatrixForm

 

 

 

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• dilogarithm and Nahm's conjecture (mathematica)