The generators in the ith set (beginning with i=0) in the inputs of holonomy generate a subalgebra of the holonomy Lie algebra H, and the output of holonomyLocal(i,H) is this Lie subalgebra. If the set is of size k, then the local Lie algebra is free on k generators if the set belongs to the first input set, and it is free on k-1 generators in degrees ≥2 if it belongs to the second input set.
i1 : H=holonomy({{a1,a2},{a3,a4}},{{a1,a3,a5},{a2,a4,a5}}) o1 = H o1 : LieAlgebra |
i2 : describe holonomyLocal(1,H) o2 = generators => {a3, a4} Weights => {{1, 0}, {1, 0}} Signs => {0, 0} ideal => {} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 |
i3 : describe holonomyLocal(2,H) o3 = generators => {a1, a3, a5} Weights => {{1, 0}, {1, 0}, {1, 0}} Signs => {0, 0, 0} ideal => {(a3 a1) - (a5 a3), (a5 a1) + (a5 a3)} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 |