next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
GradedLieAlgebras :: idealTableLie

idealTableLie -- a table of dimensions of an ideal of a Lie algebra

Synopsis

Description

The ideal is the least subspace containing the generators of the ideal and which is closed under Lie multiplication by the Lie generators and closed under application of the differential. The columns are referring to the degree, indexed from 1, and the rows are referring to the homological degree, indexed from 0. In the case of three arguments, if the second argument s is zero (one), the dimensions of the even (odd) elements are displayed.

i1 : L = lieAlgebra({a,b,c,r3,r4,r42},
          genWeights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}},
          genSigns=>{1,1,1,1,0,1},diffl=>true)

o1 = L

o1 : LieAlgebra
i2 : L = diffLieAlgebra{L.zz,L.zz,L.zz,a c,a a c,r4 + a r3}

o2 = L

o2 : LieAlgebra
i3 : idealTableLie(5,{a b,b r4 - a r4})
warning: new generators for the ideal have been added to get invariance of the differential

o3 = | 0 1 2 4 11 |
     | 0 0 0 0 2  |
     | 0 0 0 0 0  |
     | 0 0 0 0 0  |
     | 0 0 0 0 0  |

              5        5
o3 : Matrix ZZ  <--- ZZ
i4 : idealTableLie(5,{a b})

o4 = | 0 1 2 4 10 |
     | 0 0 0 0 1  |
     | 0 0 0 0 0  |
     | 0 0 0 0 0  |
     | 0 0 0 0 0  |

              5        5
o4 : Matrix ZZ  <--- ZZ

The ideal generated by (a b) and (b r4 - a r4) in degree (5,0) contains the differential of (b r4 - a r4), which is not in the ideal generated by (a b).

i5 : d = diffLie()

o5 = d

o5 : DerLie
i6 : d(b r4 - a r4)

o6 = (a a a c) - (b a a c)

o6 : L

See also

Ways to use idealTableLie :