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GradedLieAlgebras :: holonomyLie

holonomyLie -- gives the holonomy Lie algebra associated to an arrangement or matroid

Synopsis

Description

Two lists in the union of x and y have at most one element in common and the sets in x are disjoint. All sets in x have length at least two and all sets in y have length at least three. In the case of one argument y, there is a unique simple matroid of rank at most three such that y is the set of all 2-flats of size at least three. In some cases this matroid may be realized as the matroid of a central arrangement of hyperplanes.

The output holonomy(y) is the holonomy Lie algebra of the matroid (or arrangement).

In the geometric language the case with two arguments x and y, corresponds to the deconing process of a central hyperplane arrangement, see Holonomy Lie algebras and Symmetries, yielding an affine hyperplane arrangement. The set x consists of the maximal sets of parallel hyperplanes of size at least two, and y is the set of all maximal sets of hyperplanes of size at least three, which intersect in an affine space of codimension 2.

The output holonomyx(x,y) is the holonomy Lie algebra of the affine arrangement, which is the same in degrees at least two as holonomy(z), where z is obtained by choosing a new variable and add it two all sets in x and then take the union with y.

i1 : L=holonomyLie({{a0,a1,a2,a3},{a0,a4,a5},{a1,a4,a6}})

o1 = L

o1 : LieAlgebra
i2 : peekLie L

o2 = gensLie => {a0, a1, a2, a3, a4, a5, a6}
     genWeights => {{1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}}
     genSigns => {0, 0, 0, 0, 0, 0, 0}
     relsLie => {(a1 a0) - (a2 a1) - (a3 a1), (a2 a0) + (a2 a1) - (a3 a2), (a3 a0) + (a3 a1) + (a3 a2), (a4 a0) - (a5 a4), (a5 a0) + (a5 a4), (a4 a1) - (a6 a4), (a6 a1) + (a6 a4), (a4 a2), (a4 a3), (a5 a1), (a5 a2), (a5 a3), (a6 a0), (a6 a2), (a6 a3), (a6 a5)}
     genDiffs => {0, 0, 0, 0, 0, 0, 0}
     field => QQ
     diffl => false
     compdeg => 0
i3 : dimsLie 4

o3 = {7, 5, 12, 24}

o3 : List
i4 : M=holonomyLie({{a1,a2,a3},{a4,a5}},{{a1,a4,a6}})

o4 = M

o4 : LieAlgebra
i5 : peekLie M

o5 = gensLie => {a1, a2, a3, a4, a5, a6}
     genWeights => {{1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}}
     genSigns => {0, 0, 0, 0, 0, 0}
     relsLie => {(a4 a1) - (a6 a4), (a6 a1) + (a6 a4), (a4 a2), (a4 a3), (a5 a1), (a5 a2), (a5 a3), (a6 a2), (a6 a3), (a6 a5)}
     genDiffs => {0, 0, 0, 0, 0, 0}
     field => QQ
     diffl => false
     compdeg => 0
i6 : dimsLie 4

o6 = {6, 5, 12, 24}

o6 : List

See also

Ways to use holonomyLie :