# holonomy -- compute the holonomy Lie algebra associated to an arrangement or matroid

## 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 2 and all sets in $y$ have length at least 3. In the case of a single argument $y$, there is a unique simple matroid of rank at most 3 such that $y$ is the set of all 2-flats of size at least 3. 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 this 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 2, and $y$ is the set of all maximal sets of hyperplanes of size at least 3 that intersect in an affine space of codimension 2. The output holonomy(x,y)is the holonomy Lie algebra of the affine arrangement, which is the same in degrees at least 2 as holonomy(z) where $z$ is obtained by choosing a new variable and adding it to all sets in $x$ and then taking the union with $y$.

## Synopsis

• Usage:
L=holonomy(y)
• Inputs:
• y, a list, a list of subsets of the finite set $X$
• Outputs:
• L, an instance of the type LieAlgebra, the holonomy Lie algebra with generators $X$

## Synopsis

• Usage:
L=holonomy(x,y)
• Inputs:
• x, a list, a list of subsets of the finite set $X$
• y, a list, a list of subsets of the finite set $X$
• Outputs:
• L, an instance of the type LieAlgebra, the holonomy Lie algebra with generators $X$
 i1 : L=holonomy({{a0,a1,a2,a3},{a0,a4,a5},{a1,a4,a6}}) o1 = L o1 : LieAlgebra i2 : describe L o2 = generators => {a0, a1, a2, a3, a4, a5, a6} Weights => {{1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}} Signs => {0, 0, 0, 0, 0, 0, 0} ideal => {(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)} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i3 : dims(1,4,L) o3 = {7, 5, 12, 24} o3 : List i4 : M=holonomy({{a1,a2,a3},{a4,a5}},{{a1,a4,a6}}) o4 = M o4 : LieAlgebra i5 : describe M o5 = generators => {a1, a2, a3, a4, a5, a6} Weights => {{1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}, {1, 0}} Signs => {0, 0, 0, 0, 0, 0} ideal => {(a4 a1) - (a6 a4), (a6 a1) + (a6 a4), (a4 a2), (a4 a3), (a5 a1), (a5 a2), (a5 a3), (a6 a2), (a6 a3), (a6 a5)} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i6 : dims(1,4,M) o6 = {6, 5, 12, 24} o6 : List