# HallBasisΒΆ

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`