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GradedLieAlgebras :: lieHomology

lieHomology -- make the homology as a vector space

Description

The basis elements for the homology of a Lie algebra (which is of type VectorSpace) are represented as cycles in the Lie algebra.

Synopsis

  • Usage:
    H=lieHomology(L)
  • Inputs:
  • Outputs:
    • H, an instance of the type VectorSpace, the homology of the Lie algebra 
i1 : L=lieAlgebra({a,b},Signs=>{1,0},Weights=>{{2,0},{2,1}},
             Field=>ZZ/3,LastWeightHomological=>true)

o1 = L

o1 : LieAlgebra
 
i2 : D=differentialLieAlgebra({0_L,a})

o2 = D

o2 : LieAlgebra
 
i3 : H=lieHomology D

o3 = H

o3 : VectorSpace
 
i4 : dims(10,H)

o4 = | 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 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 |

              10        10
o4 : Matrix ZZ   <--- ZZ
 
i5 : basis(6,H)

o5 = {(b a a), (b b a)}

o5 : List

See also

Ways to use lieHomology :

For the programmer

The object lieHomology is a method function.