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LieAlgebra -- a placeholder for the base Lie algebra

Description

Exporting this symbol gives the ring a name that the user can see when, for instance, matrices are constructed between modules over this ring

i1 : extensionAlg = exteriorExtension(3, 6, e, QQ);
 
i2 : gens extensionAlg.LieAlgebra

o2 = {h , h , h , h , h , E      , E      , E      , E      , E      , E   
       1   2   3   4   5   {0, 1}   {0, 2}   {0, 3}   {0, 4}   {0, 5}   {1,
     ------------------------------------------------------------------------
       , E      , E      , E      , E      , E      , E      , E      , E   
     2}   {1, 3}   {1, 4}   {1, 5}   {2, 3}   {2, 4}   {2, 5}   {3, 4}   {3,
     ------------------------------------------------------------------------
       , E      , E      , E      , E      , E      , E      , E      , E   
     5}   {4, 5}   {1, 0}   {2, 0}   {3, 0}   {4, 0}   {5, 0}   {2, 1}   {3,
     ------------------------------------------------------------------------
       , E      , E      , E      , E      , E      , E      , E      , E   
     1}   {4, 1}   {5, 1}   {3, 2}   {4, 2}   {5, 2}   {4, 3}   {5, 3}   {5,
     ------------------------------------------------------------------------
       }
     4}

o2 : List

This is the underlying matrix Lie algebra.

i3 : extensionAlg.bases#0

o3 = {| 1 0  0 0 0 0 |, | 0 0 0  0 0 0 |, | 0 0 0 0  0 0 |, | 0 0 0 0 0  0 |,
      | 0 -1 0 0 0 0 |  | 0 1 0  0 0 0 |  | 0 0 0 0  0 0 |  | 0 0 0 0 0  0 | 
      | 0 0  0 0 0 0 |  | 0 0 -1 0 0 0 |  | 0 0 1 0  0 0 |  | 0 0 0 0 0  0 | 
      | 0 0  0 0 0 0 |  | 0 0 0  0 0 0 |  | 0 0 0 -1 0 0 |  | 0 0 0 1 0  0 | 
      | 0 0  0 0 0 0 |  | 0 0 0  0 0 0 |  | 0 0 0 0  0 0 |  | 0 0 0 0 -1 0 | 
      | 0 0  0 0 0 0 |  | 0 0 0  0 0 0 |  | 0 0 0 0  0 0 |  | 0 0 0 0 0  0 | 
     ------------------------------------------------------------------------
     | 0 0 0 0 0 0  |, | 0 1 0 0 0 0 |, | 0 0 1 0 0 0 |, | 0 0 0 1 0 0 |, | 0
     | 0 0 0 0 0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0
     | 0 0 0 0 0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0
     | 0 0 0 0 0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0
     | 0 0 0 0 1 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0
     | 0 0 0 0 0 -1 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0
     ------------------------------------------------------------------------
     0 0 0 1 0 |, | 0 0 0 0 0 1 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0
     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 1 0 0 0 |  | 0 0 0 1 0 0 |  | 0 0 0
     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0
     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0
     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0
     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0
     ------------------------------------------------------------------------
     0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0
     0 1 0 |  | 0 0 0 0 0 1 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0
     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 1 0 0 |  | 0 0 0 0 1 0 |  | 0 0 0 0 0
     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0
     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0
     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0
     ------------------------------------------------------------------------
     0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |,
     0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 1 0 0 0 0 0 | 
     1 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 | 
     0 |  | 0 0 0 0 1 0 |  | 0 0 0 0 0 1 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 | 
     0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 1 |  | 0 0 0 0 0 0 | 
     0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 | 
     ------------------------------------------------------------------------
     | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0
     | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0
     | 1 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0
     | 0 0 0 0 0 0 |  | 1 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0
     | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 1 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0
     | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 1 0 0 0 0 0 |  | 0
     ------------------------------------------------------------------------
     0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0
     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0
     1 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0
     0 0 0 0 0 |  | 0 1 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 1
     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 1 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0
     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 1 0 0 0 0 |  | 0 0 0
     ------------------------------------------------------------------------
     0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0
     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0
     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0
     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0
     0 0 0 |  | 0 0 1 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 1 0 0 |  | 0 0 0 0 0
     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 1 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 1 0
     ------------------------------------------------------------------------
     0 |, | 0 0 0 0 0 0 |}
     0 |  | 0 0 0 0 0 0 |
     0 |  | 0 0 0 0 0 0 |
     0 |  | 0 0 0 0 0 0 |
     0 |  | 0 0 0 0 0 0 |
     0 |  | 0 0 0 0 1 0 |

o3 : List

This is another way to find the basis of the Lie algebra as it is the 0-graded piece of the Extension Algebra

See also

For the programmer

The object LieAlgebra is a symbol.

The source of this document is in ExteriorExtensions.m2:524:0.