random(ZZ,LieAlgebra) -- get a random element of a Lie algebra

Synopsis

Function: random
Optional inputs:
- Density => ..., default value 1,
- Height => ..., default value 10,
- MaximalRank => ..., default value false,
- UpperTriangular => ..., default value false,

Description

The optional inputs given above are not relevant for Lie algebras. Below is an example of a periodic Lie algebra (a periodization of $sl_3$) with five generators and seven random quadratic relations.

Synopsis

Usage:

random(d,L)
Inputs:
- d, an integer, the degree
- L, an instance of the type LieAlgebra
Outputs:
- an instance of the type LieElement, a random Lie element of degree $d$

Synopsis

Usage:

random(m,L)
Inputs:
- m, a list, the multi-degree
- L, an instance of the type LieAlgebra
Outputs:
- an instance of the type LieElement, a random Lie element of multi-degree $m$

Synopsis

Usage:

random(m,n,L)
Inputs:
- m, an integer, the degree
- n, an integer, the homological degree
- L, an instance of the type LieAlgebra
Outputs:
- an instance of the type LieElement, a random Lie element of degree $m$ and homological degree $n$

i1 : L = lieAlgebra({a,b,c,d,e}, Field=>ZZ/7)

o1 = L

o1 : LieAlgebra

i2 : Q = L/apply(7,i->random(2,L))

o2 = Q

o2 : LieAlgebra

i3 : dims(1,8,Q)

o3 = {5, 3, 5, 3, 5, 3, 5, 3}

o3 : List

i4 : random({4,0},Q)

o4 = (-2)(d e e a) - (e e e a) + (-3)(e e e d)

o4 : Q

Ways to use this method: