HallBasisΒΆ

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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: (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