HallBasis

fnla.spad line 70 [edit on github]

Generate a basis for the free Lie algebra on n generators over a ring R with identity up to basic commutators of length c using the algorithm of P. Hall as given in Serre's book Lie Groups -- Lie Algebras.

basis : (NonNegativeInteger, NonNegativeInteger) -> Vector(List(Integer))

basis(numberOfGens, maximalWeight) generates a vector of elements of the form [left, weight, right] which represents a P. Hall basis element for the free lie algebra on numberOfGens generators. We only generate those basis elements of weight less than or equal to maximalWeight

inHallBasis? : (Integer, Integer, Integer, Integer) -> Boolean

inHallBasis?(numberOfGens, leftCandidate, rightCandidate, left) tests to see if a new element should be added to the P. Hall basis being constructed. The list [leftCandidate, wt, rightCandidate] is included in the basis if in the unique factorization of rightCandidate, we have left factor leftOfRight, and leftOfRight <= leftCandidate

lfunc : (Integer, Integer) -> Integer

lfunc(d, n) computes the rank of the nth factor in the lower central series of the free d-generated free Lie algebra; This rank is d if n = 1 and binom(d, 2) if n = 2