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GradedLieAlgebras :: localLie

localLie -- gives the Lie algebra for a local subalgebra of the holonomy Lie algebra

Synopsis

Description

The generators in the ith set (beginning with i=0) in the inputs of holonomyLie generate a subalgebra of the holonomy Lie algebra and the output of localLie(i) 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 degree >=2 if it belongs to the second input set.

i1 : L=holonomyLie({{a1,a2},{a3,a4}},{{a1,a3,a5},{a2,a4,a5}})

o1 = L

o1 : LieAlgebra
i2 : peekLie localLie(1)

o2 = gensLie => {a3, a4}
     genWeights => {{1, 0}, {1, 0}}
     genSigns => {0, 0}
     relsLie => {}
     genDiffs => {0, 0}
     field => QQ
     diffl => false
     compdeg => 0
i3 : peekLie localLie(2)

o3 = gensLie => {a1, a3, a5}
     genWeights => {{1, 0}, {1, 0}, {1, 0}}
     genSigns => {0, 0, 0}
     relsLie => {(a3 a1) - (a5 a3), (a5 a1) + (a5 a3)}
     genDiffs => {0, 0, 0}
     field => QQ
     diffl => false
     compdeg => 0

See also

Ways to use localLie :