holonomyLocal -- compute the Lie algebra for a local subalgebra of the holonomy Lie algebra

Synopsis

Usage:

L = holonomyLocal(i,H)
Inputs:
- i, an integer, referring to the $i$th set, where the sets in the first input of holonomy are counted before the sets in the second input
- H, an instance of the type LieAlgebra, the holonomy Lie algebra
Outputs:
- L, an instance of the type LieAlgebra, the Lie subalgebra

Description

The generators in the $i$th 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 $\ge 2$ 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

Ways to use holonomyLocal :

holonomyLocal(ZZ,LieAlgebra)

For the programmer

The object holonomyLocal is a method function.

holonomyLocal -- compute the Lie algebra for a local subalgebra of the holonomy Lie algebra

Synopsis

Description

See also

Ways to use holonomyLocal :

For the programmer