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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:
- L, an instance of the type LieAlgebra
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

boundaries -- make the subalgebra of boundaries
cycles -- make the subalgebra of cycles

Ways to use lieHomology:

lieHomology(LieAlgebra)

For the programmer

The object lieHomology is a method function.

The source of this document is in GradedLieAlgebras/doc2.m2:1171:0.