FreeLieAlgebra(VarSet, R)

xlpoly.spad line 325 [edit on github]

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

* : (%, R) -> %
from RightModule(R)
* : (R, %) -> %
from LeftModule(R)
* : (Integer, %) -> %
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
+ : (%, %) -> %
from AbelianSemiGroup
- : % -> %
from AbelianGroup
- : (%, %) -> %
from AbelianGroup
/ : (%, R) -> % if R has Field
from LieAlgebra(R)
0 : () -> %
from AbelianMonoid
= : (%, %) -> Boolean
from BasicType
LiePoly : LyndonWord(VarSet) -> %

LiePoly(l) returns the bracketed form of l as a Lie polynomial.

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 : VarSet -> %

coerce(x) returns x as a Lie polynomial.

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.

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

degree(x) returns the greatest length of a word in the support of x.

eval : (%, VarSet, %) -> %

eval(p, x, v) replaces x by v in p.

eval : (%, List(VarSet), List(%)) -> %

eval(p, [x1, ..., xn], [v1, ..., vn]) replaces xi by vi in p.

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

BiModule(R, R)

CancellationAbelianMonoid

LeftModule(R)

RightModule(R)

Module(R)

AbelianGroup

LieAlgebra(R)

AbelianSemiGroup

SetCategory

AbelianMonoid

BasicType

CoercibleTo(OutputForm)