M is a minimal model of the currently used Lie algebra L up to degree d, that is, f: M -> L is a differential homomorphism such that H(f) is an isomorphism and M is free as a Lie algebra and the differential on M has no linear part. The homomophism f is available as M.modelmap, and M is available as L.minmodel and L is obtained as M.modelmap.targetLie.
The generators of M yield a basis for the cohomology of L, i.e., ExtUL(k,k), where k is L.field. The dimensions of this cohomology algebra is obtained by extTableLie. Also, the polynomial ring with generators equal to the basis elements of the cohomology algebra is obtained by extRepRing as L.cache.extRepRing. The linear polynomials in this ring gives a representation of ExtUL(k,k) (similar to the representation of L by linear polynomials in L.cache.mbRing, see mbRing). Multiplication of elements in ExtUL(k,k) is obtained using SPACE, see RingElement RingElement
Observe that the homological weight in the cohomology algebra is one higher than the homological weight in the minimal model.
Since R in the following example is a Koszul algebra, the cohomology algebra of L is equal to R, which means that the minimal model of L has generators in each degree (d,d-1).
i1 : R=QQ[x] o1 = R o1 : PolynomialRing |
i2 : L=koszulDualLie R o2 = L o2 : LieAlgebra |
i3 : peekLie L o3 = gensLie => {ko } 0 genWeights => {{1, 0}} genSigns => {1} relsLie => { - (1/2)(ko_0 ko_0)} genDiffs => {0} field => QQ diffl => false compdeg => 1 |
i4 : extTableLie 5 o4 = | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 1 0 0 | | 0 0 0 1 0 | | 0 0 0 0 1 | 5 5 o4 : Matrix ZZ <--- ZZ |
i5 : peekLie minmodelLie 5 o5 = gensLie => {fr , fr , fr , fr , fr } 0 1 2 3 4 genWeights => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4}} genSigns => {1, 1, 1, 1, 1} relsLie => {} genDiffs => {0, (fr_0 fr_0), (fr_0 fr_1), (fr_1 fr_1) + 4 (fr_0 fr_2), 2 (fr_1 fr_2) + (fr_0 fr_3)} field => QQ diffl => true compdeg => 5 modelmap => MapLie{fr_0 => ko_0 } fr_1 => 0 fr_2 => 0 fr_3 => 0 fr_4 => 0 sourceLie => LieAlgebra{...14...} targetLie => L |
In the following example the enveloping algebra of L1 has global dimension two, which means that the computed minimal model is in fact the full minimal model of L1.
i6 : L1=lieAlgebra{a,b,c}/{a b,a b c} o6 = L1 o6 : LieAlgebra |
i7 : peekLie minmodelLie 4 o7 = gensLie => {fr , fr , fr , fr , fr } 0 1 2 3 4 genWeights => {{1, 0}, {1, 0}, {1, 0}, {2, 1}, {3, 1}} genSigns => {0, 0, 0, 1, 1} relsLie => {} genDiffs => {0, 0, 0, (fr_1 fr_0), (fr_1 fr_2 fr_0)} field => QQ diffl => true compdeg => 4 modelmap => MapLie{fr_0 => a } fr_1 => b fr_2 => c fr_3 => 0 fr_4 => 0 sourceLie => LieAlgebra{...14...} targetLie => L1 |
Below is a differential Lie algebra which is non-free and where the differential has a linear part.
i8 : L2=lieAlgebra({a,b,c,r3,r4,r42},genWeights=>{{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}}, genSigns=>{0,0,0,1,1,0},diffl=>true) o8 = L2 o8 : LieAlgebra |
i9 : L2=diffLieAlgebra{L2.zz,L2.zz,L2.zz,a c,a a c,r4-a r3}/{b c-a c,a b,b r4-a r4} o9 = L2 o9 : LieAlgebra |
i10 : peekLie minmodelLie 5 o10 = gensLie => {fr , fr , fr , fr , fr , fr , fr } 0 1 2 3 4 5 6 genWeights => {{1, 0}, {1, 0}, {2, 0}, {2, 1}, {3, 1}, {3, 1}, {5, 2}} genSigns => {0, 0, 0, 1, 1, 1, 0} relsLie => {} genDiffs => {0, 0, 0, (fr_1 fr_0), (fr_1 fr_2), (fr_0 fr_2), (fr_0 fr_3 fr_2) + (fr_0 fr_0 fr_4) - (fr_0 fr_1 fr_5)} field => QQ diffl => true compdeg => 5 modelmap => MapLie{fr_0 => a } fr_1 => b fr_2 => c fr_3 => 0 fr_4 => r3 fr_5 => r3 fr_6 => - (a r42) + (b r42) sourceLie => LieAlgebra{...14...} targetLie => L2 |