Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
LieART`
OrthogonalSimpleRoots |
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Examples
(2)
Basic Examples
(2)
Simple roots of all rank 4 classical Lie algebras in the orthogonal basis:
In[1]:=
OrthogonalSimpleRoots[A4]
Out[1]=
{(1, -1, 0, 0, 0),(0, 1, -1, 0, 0),(0, 0, 1, -1, 0),(0, 0, 0, 1, -1)}
In[2]:=
OrthogonalSimpleRoots[B4]
Out[2]=
{(1, -1, 0, 0),(0, 1, -1, 0),(0, 0, 1, -1),(0, 0, 0, 1)}
In[3]:=
OrthogonalSimpleRoots[C4]
Out[3]=
{(1, -1, 0, 0),(0, 1, -1, 0),(0, 0, 1, -1),(0, 0, 0, 2)}
In[4]:=
OrthogonalSimpleRoots[D4]
Out[4]=
{(1, -1, 0, 0),(0, 1, -1, 0),(0, 0, 1, -1),(0, 0, 1, 1)}
Simple roots of all exceptional Lie algebras in the orthogonal basis:
In[1]:=
OrthogonalSimpleRoots[E6]
Out[1]=
{(1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2),(-1, 1, 0, 0, 0, 0, 0, 0),(0, -1, 1, 0, 0, 0, 0, 0),(0, 0, -1, 1, 0, 0, 0, 0),(0, 0, 0, -1, 1, 0, 0, 0),(1, 1, 0, 0, 0, 0, 0, 0)}
In[2]:=
OrthogonalSimpleRoots[E7]
Out[2]=
{(1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2),(-1, 1, 0, 0, 0, 0, 0, 0),(0, -1, 1, 0, 0, 0, 0, 0),(0, 0, -1, 1, 0, 0, 0, 0),(0, 0, 0, -1, 1, 0, 0, 0),(0, 0, 0, 0, -1, 1, 0, 0),(1, 1, 0, 0, 0, 0, 0, 0)}
In[3]:=
OrthogonalSimpleRoots[E8]
Out[3]=
{(1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2),(-1, 1, 0, 0, 0, 0, 0, 0),(0, -1, 1, 0, 0, 0, 0, 0),(0, 0, -1, 1, 0, 0, 0, 0),(0, 0, 0, -1, 1, 0, 0, 0),(0, 0, 0, 0, -1, 1, 0, 0),(0, 0, 0, 0, 0, -1, 1, 0),(1, 1, 0, 0, 0, 0, 0, 0)}
In[4]:=
OrthogonalSimpleRoots[F4]
Out[4]=
{(1, -1, 0, 0),(0, 1, -1, 0),(0, 0, 1, 0),(-1/2, -1/2, -1/2, -1/2)}
In[5]:=
OrthogonalSimpleRoots[G2]
Out[5]=
{(1, -1, 0),(-2, 1, 1)}
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