Plantilla:Dynkin/testcases

Testing sandbox version

{{[[Template:Plantilla:TEMPLATENAMEE/sandbox|Plantilla:TEMPLATENAMEE/sandbox]]}}

Rank 2 Dynkin diagrams

Group name	Dynkin diagram		Cartan matrix		Symmetry order	Related simply-laced automorphic group3
	(Standard) multi-edged graph1	Valued graph2	$\left[{\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}}\right]$	Determinant (4-a21*a12)
Finite (Determinant>0)
A1xA1			$\left[{\begin{smallmatrix}2&0\\0&2\end{smallmatrix}}\right]$	4	2
A2			$\left[{\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}}\right]$	3	3
B2			$\left[{\begin{smallmatrix}2&-2\\-1&2\end{smallmatrix}}\right]$	2	4	${A}_{3}$
C2			$\left[{\begin{smallmatrix}2&-1\\-2&2\end{smallmatrix}}\right]$	2	4	${A}_{3}$
G2			$\left[{\begin{smallmatrix}2&-1\\-3&2\end{smallmatrix}}\right]$	1	6	${D}_{4}$
Affine (Determinant=0)
A1(1)			$\left[{\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}}\right]$	0	∞	${\tilde {A}}_{3}$
A2(2)			$\left[{\begin{smallmatrix}2&-1\\-4&2\end{smallmatrix}}\right]$	0	∞	${\tilde {D}}_{4}$
Hyperbolic (Determinant<0)
			$\left[{\begin{smallmatrix}2&-1\\-5&2\end{smallmatrix}}\right]$	-1	∞	H5(6)
			$\left[{\begin{smallmatrix}2&-b\\-a&2\end{smallmatrix}}\right]$	4-ab	∞
Note1: The multi-edged diagram corresponds to the nondiagonal Cartan matrix elements a21, a12, with the number of edges drawn equal to max(a21, a12), and an arrow pointing towards nonunity element(s). Note2: For hyperbolic groups, (a12*a21>4), the multiedge style is abandoned in favor of an explicit labeling (a21, a12) on the edge. These are usually not applied to finite and affine graphs. Note3: Many multi-edged groups are automorphic via a folding operation with a higher ranked simply-laced group.

Testing main template

{{[[Template:Plantilla:TEMPLATENAMEE|Plantilla:TEMPLATENAMEE]]}}

Rank 2 Dynkin diagrams

Group name	Dynkin diagram		Cartan matrix		Symmetry order	Related simply-laced automorphic group3
	(Standard) multi-edged graph1	Valued graph2	$\left[{\begin{matrix}2&a_{12}\\a_{21}&2\end{matrix}}\right]$	Determinant (4-a21*a12)
Finite (Determinant>0)
A1xA1			$\left[{\begin{smallmatrix}2&0\\0&2\end{smallmatrix}}\right]$	4	2
A2			$\left[{\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}}\right]$	3	3
B2			$\left[{\begin{smallmatrix}2&-2\\-1&2\end{smallmatrix}}\right]$	2	4	${A}_{3}$
C2			$\left[{\begin{smallmatrix}2&-1\\-2&2\end{smallmatrix}}\right]$	2	4	${A}_{3}$
G2			$\left[{\begin{smallmatrix}2&-1\\-3&2\end{smallmatrix}}\right]$	1	6	${D}_{4}$
Affine (Determinant=0)
A1(1)			$\left[{\begin{smallmatrix}2&-2\\-2&2\end{smallmatrix}}\right]$	0	∞	${\tilde {A}}_{3}$
A2(2)			$\left[{\begin{smallmatrix}2&-1\\-4&2\end{smallmatrix}}\right]$	0	∞	${\tilde {D}}_{4}$
Hyperbolic (Determinant<0)
			$\left[{\begin{smallmatrix}2&-1\\-5&2\end{smallmatrix}}\right]$	-1	∞	H5(6)
			$\left[{\begin{smallmatrix}2&-b\\-a&2\end{smallmatrix}}\right]$	4-ab	∞
Note1: The multi-edged diagram corresponds to the nondiagonal Cartan matrix elements a21, a12, with the number of edges drawn equal to max(a21, a12), and an arrow pointing towards nonunity element(s). Note2: For hyperbolic groups, (a12*a21>4), the multiedge style is abandoned in favor of an explicit labeling (a21, a12) on the edge. These are usually not applied to finite and affine graphs. Note3: Many multi-edged groups are automorphic via a folding operation with a higher ranked simply-laced group.