FreeLieAlgebra(VarSet, R)¶

VarSet: OrderedSet
R: CommutativeRing

The category of free Lie algebras. It is used by domains of non-commutative algebra: LiePolynomial and XPBWPolynomial. Author: Michel Petitot (petitot@lifl.fr)

0: %: from AbelianMonoid

*: (%, R) -> %: from RightModule R
*: (Integer, %) -> %: from AbelianGroup
*: (NonNegativeInteger, %) -> %: from AbelianMonoid
*: (PositiveInteger, %) -> %: from AbelianSemiGroup
*: (R, %) -> %: from LeftModule R

+: (%, %) -> %: from AbelianSemiGroup

-: % -> %: from AbelianGroup
-: (%, %) -> %: from AbelianGroup

/: (%, R) -> % if R has Field: from LieAlgebra R

=: (%, %) -> Boolean: from BasicType

~=: (%, %) -> Boolean: from BasicType

coef: (XRecursivePolynomial(VarSet, R), %) -> R: coef(x, y) returns the scalar product of x by y, the set of words being regarded as an orthogonal basis.

coerce: % -> OutputForm: from CoercibleTo OutputForm

coerce: % -> XDistributedPolynomial(VarSet, R): coerce(x) returns x as distributed polynomial.

coerce: % -> XRecursivePolynomial(VarSet, R): coerce(x) returns x as a recursive polynomial.

coerce: VarSet -> %: coerce(x) returns x as a Lie polynomial.

construct: (%, %) -> %: from LieAlgebra R

degree: % -> NonNegativeInteger: degree(x) returns the greatest length of a word in the support of x.

eval: (%, List VarSet, List %) -> %: eval(p, [x1, ..., xn], [v1, ..., vn]) replaces xi by vi in p.

eval: (%, VarSet, %) -> %: eval(p, x, v) replaces x by v in p.

latex: % -> String: from SetCategory

LiePoly: LyndonWord VarSet -> %: LiePoly(l) returns the bracketed form of l as a Lie polynomial.

lquo: (XRecursivePolynomial(VarSet, R), %) -> XRecursivePolynomial(VarSet, R): lquo(x, y) returns the left simplification of x by y.

mirror: % -> %: mirror(x) returns Sum(r_i mirror(w_i)) if x is Sum(r_i w_i).

opposite?: (%, %) -> Boolean: from AbelianMonoid

rquo: (XRecursivePolynomial(VarSet, R), %) -> XRecursivePolynomial(VarSet, R): rquo(x, y) returns the right simplification of x by y.

sample: %: from AbelianMonoid

subtractIfCan: (%, %) -> Union(%, failed): from CancellationAbelianMonoid

trunc: (%, NonNegativeInteger) -> %: trunc(p, n) returns the polynomial p truncated at order n.

varList: % -> List VarSet: varList(x) returns the list of distinct entries of x.

zero?: % -> Boolean: from AbelianMonoid

BiModule(R, R)

CancellationAbelianMonoid

CoercibleTo OutputForm

Module R