The function holonomyLie(A), see holonomyLie, constructs the holonomy Lie algebra of a hyperplane arrangement or a matroid given by the set A of 2-flats. The input may be any set of subsets of a finite set A, such that all subsets have at most one element in common and are of length at least three (the 2-flats of size two are determined by those). Indeed, for any such set of subsets there is a unique simple matroid of rank at most three with the given set as the set of 2-flats of size at least three, and holonomyLie(A) gives the holonomy Lie algebra of this matroid.
i1 : L = holonomyLie({{a1,a2,a3},{a1,a4,a5},{a2,a4,a6}}) o1 = L o1 : LieAlgebra |
i2 : L.relsLie o2 = {(a2 a1) - (a3 a2), (a3 a1) + (a3 a2), (a4 a1) - (a5 a4), (a5 a1) + (a5 ------------------------------------------------------------------------ a4), (a4 a2) - (a6 a4), (a6 a2) + (a6 a4), (a4 a3), (a5 a2), (a5 a3), ------------------------------------------------------------------------ (a6 a1), (a6 a3), (a6 a5)} o2 : List |
The sum of the generators is a central element. Hence, by dividing out by this element and using minPresLie one obtains a presentation of a Lie algebra with one generator less, which is isomorphic to the holonomy Lie algebra in degrees >=2.
i3 : L0 = L/{a1+a2+a3+a4+a5+a6} o3 = L0 o3 : LieAlgebra |
i4 : L0 = minPresLie 3 o4 = L0 o4 : LieAlgebra |
i5 : peekLie L0 o5 = gensLie => {a2, a3, a4, a5, a6} genWeights => {{1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}} genSigns => {0, 0, 0, 0, 0} relsLie => {(a6 a5), (a6 a3), (a6 a2) + (a6 a4), (a5 a3), (a5 a2), (a4 a3), (a4 a2) - (a6 a4)} genDiffs => {0, 0, 0, 0, 0} field => QQ diffl => false compdeg => 1 |
It is possible to get this Lie algebra directly by choosing one of the variables, picking all 2-flats containing that variable, deleting the variable and putting A equal to the set of deleted 2-flats and B equal to the remaining 2-flats and finally applying holonomyLie(A,B).
i6 : L1 = holonomyLie({{a2,a3},{a4,a5}},{{a2,a4,a6}}) o6 = L1 o6 : LieAlgebra |
i7 : peekLie L1 o7 = gensLie => {a2, a3, a4, a5, a6} genWeights => {{1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}} genSigns => {0, 0, 0, 0, 0} relsLie => {(a4 a2) - (a6 a4), (a6 a2) + (a6 a4), (a4 a3), (a5 a2), (a5 a3), (a6 a3), (a6 a5)} genDiffs => {0, 0, 0, 0, 0} field => QQ diffl => false compdeg => 0 |
Choosing another generator to delete gives another presentation (which is still isomorhic to the holonomy Lie algebra in degrees >=2).
i8 : L6 = holonomyLie({{a2,a4}},{{a1,a2,a3},{a1,a4,a5}}) o8 = L6 o8 : LieAlgebra |
i9 : peekLie L6 o9 = gensLie => {a2, a4, a1, a3, a5} genWeights => {{1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}} genSigns => {0, 0, 0, 0, 0} relsLie => { - (a1 a2) - (a3 a2), (a3 a2) + (a3 a1), - (a1 a4) - (a5 a4), (a5 a4) + (a5 a1), (a3 a4), (a5 a2), (a5 a3)} genDiffs => {0, 0, 0, 0, 0} field => QQ diffl => false compdeg => 0 |
i10 : dimsLie 6 o10 = {5, 3, 6, 9, 18, 27} o10 : List |
i11 : useLie L1 o11 = L1 o11 : LieAlgebra |
i12 : dimsLie 6 o12 = {5, 3, 6, 9, 18, 27} o12 : List |
i13 : useLie L o13 = L o13 : LieAlgebra |
i14 : dimsLie 6 o14 = {6, 3, 6, 9, 18, 27} o14 : List |
The above corresponds to the deconing process of a central hyperplane arrangement, yielding an affine hyperplane arrangement. The first input set in holonomyLie should be all maximal sets of parallel hyperplanes of size at least two and the second input set should be all maximal sets of hyperplanes of size at least three, which intersect in an affine space of codimension 2.
A local Lie algebra of a holonomy Lie algebra, see localLie, is the Lie subalgebra generated by the generators in one of the sets defined in the input. If this 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 (observe that the numbering of the sets begins with 0).
i15 : useLie L1 o15 = L1 o15 : LieAlgebra |
i16 : peekLie localLie 1 o16 = gensLie => {a4, a5} genWeights => {{1, 0}, {1, 0}} genSigns => {0, 0} relsLie => {} genDiffs => {0, 0} field => QQ diffl => false compdeg => 0 |
i17 : peekLie localLie 2 o17 = gensLie => {a2, a4, a6} genWeights => {{1, 0}, {1, 0}, {1, 0}} genSigns => {0, 0, 0} relsLie => {(a4 a2) - (a6 a4), (a6 a2) + (a6 a4)} genDiffs => {0, 0, 0} field => QQ diffl => false compdeg => 0 |
The kernel of the map, in degrees >=2, from L to the direct sum of the local Lie algebras, see localLie, is obtained by decompidealLie. This ideal is generated by the basis elements in degree 3 of the form (a b c) where not all of a,b,c belong to the same local Lie algebra.
i18 : decompidealLie 3 o18 = {} o18 : List |
Hence L1, in degrees >=2, is the direct sum of its local Lie algebras, L1 is "decomposable". This is not true for the "quadrangel", i.e., the graphical arrangement of the complete graph on four vertices, which is also the braid arrangement in dimension 4.
i19 : Q = holonomyLie({{a1,a2,a3},{a1,a4,a5},{a2,a4,a6},{a3,a5,a6}}) o19 = Q o19 : LieAlgebra |
i20 : decompidealLie 3 o20 = {(a6 a5 a4), (a5 a6 a4)} o20 : List |
i21 : Q1 = holonomyLie({{a2,a3},{a4,a5}},{{a2,a4,a6},{a3,a5,a6}}) o21 = Q1 o21 : LieAlgebra |
i22 : decompidealLie 3 o22 = {(a6 a5 a4), (a5 a6 a4)} o22 : List |
Here is a way to obtain decompidealLie (which is not used in the program). The direct sum of the local Lie algebras of Q1 may be obtained as follows
i23 : L0 = localLie 0 o23 = L0 o23 : LieAlgebra |
i24 : L1 = localLie 1 o24 = L1 o24 : LieAlgebra |
i25 : L2 = localLie 2 o25 = L2 o25 : LieAlgebra |
i26 : L3 = localLie 3 o26 = L3 o26 : LieAlgebra |
i27 : M = L0**L1**L2**L3 o27 = M o27 : LieAlgebra |
i28 : M.gensLie o28 = {pr_0, pr_1, pr_2, pr_3, pr_4, pr_5, pr_6, pr_7, pr_8, pr_9} o28 : List |
and the map from Q1 to M is given as
i29 : f = mapLie(M,Q1,{pr_0+pr_4,pr_1+pr_7,pr_2+pr_5,pr_3+pr_8,pr_6+pr_9}) o29 = f o29 : MapLie |
i30 : peekLie f o30 = MapLie{a2 => pr_0 + pr_4} a3 => pr_1 + pr_7 a4 => pr_2 + pr_5 a5 => pr_3 + pr_8 a6 => pr_6 + pr_9 sourceLie => Q1 targetLie => M |
and hence the ideal may be obtained as the kernel of f
i31 : kernelBasisLie(3,f) o31 = {(a6 a5 a4), (a5 a6 a4)} o31 : List |
The symmetric group S4 operates on the vertices of K4 and this induces an action of S4 on the six edges, which in turn induces an action of S4 on Q as automorphisms. One such permutation of the edges is (123)(465) but not (123)(456). It is possible to check, using symmetryLie, if a permutation of the generators, written as a product of cycles, or as a rearrangement of the generators, defines an automorphism of the Lie algebra. If this is true, then the map is given as output, else an error message is written.
i32 : useLie Q o32 = Q o32 : LieAlgebra |
i33 : symmetryLie({{a1,a2,a3},{a4,a5,a6}}) the map is not welldefined |
i34 : f=symmetryLie({{a1,a2,a3},{a4,a6,a5}}) o34 = f o34 : MapLie |
i35 : peekLie f o35 = MapLie{a1 => a2 } a2 => a3 a3 => a1 a4 => a6 a5 => a4 a6 => a5 sourceLie => Q targetLie => Q |
i36 : g=symmetryLie({a2,a3,a1,a6,a4,a5}) o36 = g o36 : MapLie |
i37 : peekLie g o37 = MapLie{a1 => a2 } a2 => a3 a3 => a1 a4 => a6 a5 => a4 a6 => a5 sourceLie => Q targetLie => Q |
The ideal decompidealLie is invariant under all automorphisms of L. We may use characterLie and a character table for S4 to determine its irreducible representation constituents. There are four conjugacy classes (except id). Representatives for them as permutation of the six generators are (23)(45), (123)(465), (16)(2354) and (16)(25) corresponding to one 2-cycle, one 3-cycle, one 4-cycle and a product of two 2-cycles.
i38 : dec4 = decompidealLie 4 o38 = {(a6 a6 a5 a4), (a6 a5 a6 a4), (a6 a5 a5 a4), (a6 a4 a5 a4), (a5 a6 a6 ----------------------------------------------------------------------- a4), (a5 a6 a5 a4), (a5 a5 a6 a4), (a5 a4 a6 a4), (a4 a6 a5 a4)} o38 : List |
i39 : characterLie({{a2,a3},{a4,a5}},dec4) o39 = -1 o39 : QQ |
i40 : characterLie({{a1,a2,a3},{a4,a6,a5}},dec4) o40 = 0 o40 : QQ |
i41 : characterLie({{a1,a6},{a2,a3,a5,a4}},dec4) o41 = -1 o41 : QQ |
i42 : characterLie({{a1,a6},{a2,a5}},dec4) o42 = 1 o42 : QQ |
Making calculations with the character table for S4, we see that decompidealLie 4 is the sum of the irreducible representations except the trivial representation.