lieAlgebra(...,Weights=>...) -- optional argument for lieAlgebra

Synopsis

Usage:

L = lieAlgebra(gen,Weights => w)
Inputs:
- gen, a list,
- w, a list, of lists of weights or a list of positive integers
- w, an integer, 1, the same weight (1,0) is assigned to each generator
Outputs:
- L, an instance of the type LieAlgebra,

Description

This is an option to tell lieAlgebra to assign the given weights to the generators. A weight is a list of integers of a length that is the same for all generators, see degreeLength(LieAlgebra). The first component (also just called the degree) is positive, see firstDegree(LieElement). If the option LastWeightHomological is true, then the last component in a weight is the homological degree, which is non-negative and less than the first degree. If the option LastWeightHomological is false, then the program adds a last component 0 to the existing degrees. When the option is given as a list of integers $n1,n2,\ldots$, which is not possible when the option LastWeightHomological is true, then the program defines the weights for the generators to be $\{n1,0\},\ \{n2,0\},\ldots$. The default value is 1, which has the effect that all generators have weight $\{1,0\}$.

i1 : describe lieAlgebra({a,b},Weights=>{{1,2},{2,3}})

o1 = generators => {a, b}
     Weights => {{1, 2, 0}, {2, 3, 0}}
     Signs => {0, 0}
     ideal => {}
     ambient => LieAlgebra{...10...}
     diff => {}
     Field => QQ
     computedDegree => 0

i2 : describe lieAlgebra({a,b},Weights=>{{2,1},{3,2}},
         LastWeightHomological=>true)

o2 = generators => {a, b}
     Weights => {{2, 1}, {3, 2}}
     Signs => {0, 0}
     ideal => {}
     ambient => LieAlgebra{...10...}
     diff => {}
     Field => QQ
     computedDegree => 0

i3 : describe lieAlgebra{a,b}

o3 = generators => {a, b}
     Weights => {{1, 0}, {1, 0}}
     Signs => {0, 0}
     ideal => {}
     ambient => LieAlgebra{...10...}
     diff => {}
     Field => QQ
     computedDegree => 0

i4 : describe lieAlgebra({a,b},Weights=>{{1,0},{1,0}})

o4 = generators => {a, b}
     Weights => {{1, 0, 0}, {1, 0, 0}}
     Signs => {0, 0}
     ideal => {}
     ambient => LieAlgebra{...10...}
     diff => {}
     Field => QQ
     computedDegree => 0

i5 : describe lieAlgebra({a,b},Weights=>{{1,0},{1,0}},
         LastWeightHomological=>true)

o5 = generators => {a, b}
     Weights => {{1, 0}, {1, 0}}
     Signs => {0, 0}
     ideal => {}
     ambient => LieAlgebra{...10...}
     diff => {}
     Field => QQ
     computedDegree => 0

i6 : describe lieAlgebra({a,b},Weights=>{1,2})

o6 = generators => {a, b}
     Weights => {{1, 0}, {2, 0}}
     Signs => {0, 0}
     ideal => {}
     ambient => LieAlgebra{...10...}
     diff => {}
     Field => QQ
     computedDegree => 0

Further information

Default value: 1
Function: lieAlgebra -- make a free Lie algebra
Option key: Weights -- assigning weights to the variables

Functions with optional argument named Weights :

betti(...,Weights=>...) -- see betti(BettiTally) -- view and set the weight vector of a Betti diagram
lieAlgebra(...,Weights=>...) -- optional argument for lieAlgebra
minimalBetti(...,Weights=>...) -- see minimalBetti -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
monoid(...,Weights=>...) -- specify weights of the variables
newRing(...,Weights=>...) -- see newRing -- make a copy of a ring, with some features changed
regularity(...,Weights=>...) -- see regularity -- compute the Castelnuovo-Mumford regularity
symmetricAlgebra(...,Weights=>...) -- see symmetricAlgebra -- the symmetric algebra of a module
tensor(Monoid,Monoid,Weights=>...) -- see tensor(Monoid,Monoid) -- tensor product of monoids
tensor(Ring,Ring,Weights=>...) -- see tensor(Monoid,Monoid) -- tensor product of monoids

lieAlgebra(...,Weights=>...) -- optional argument for lieAlgebra

Synopsis

Description

Further information

See also

Functions with optional argument named Weights :