HallBasis¶

Author : Larry Lambe Date Created : August 1988 Related Constructors: OrderedSetInts, Commutator, FreeNilpotentLie AMS Classification: Primary 17B05, 17B30; Secondary 17A50 Keywords: free Lie algebra, Hall basis, basic commutators Description : 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(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?(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(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