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sign(LieElement) -- get the sign of a homogeneous element

Description

The sign of a homogeneous Lie (or Ext) element $x$ is obtained as sign(x). The zero element has sign equal to 0; however, its sign should be thought of as arbitrary. The sign of a derivation $d$ is the sign of $d$ as a graded map and may also be obtained as d#sign.

Synopsis

  • Usage:
    s=sign(x)
  • Inputs:
    • x, an instance of the type LieElement, an element of type $L$ where $L$ is of type LieAlgebra
  • Outputs: 
i1 : L = lieAlgebra({a,b,c},Weights=>{{1,0},{2,1},{3,2}},
     	 LastWeightHomological=>true, Signs => 1)

o1 = L

o1 : LieAlgebra
 
i2 : D=differentialLieAlgebra{0_L,a a,a b}/{a a b, a a c, b a b}

o2 = D

o2 : LieAlgebra
 
i3 : x=a b c+2 c b a

o3 =  - 2 (b a c) - (a b c)

o3 : D
 
i4 : sign x

o4 = 1

Synopsis

  • Usage:
    s=sign(x)
  • Inputs:
    • x, an instance of the type ExtElement, an element of type $E$ where $E$ is of type ExtAlgebra
  • Outputs: 
i5 : E=extAlgebra(5,D)

o5 = E

o5 : ExtAlgebra
 
i6 : b=basis(5,E)

o6 = {ext_4, ext_5}

o6 : List
 
i7 : apply(b,sign)

o7 = {1, 1}

o7 : List

Synopsis

  • Usage:
    s=sign(d)
  • Inputs:
  • Outputs: 
i8 : sign differential D

o8 = 1

See also

Ways to use this method:

  • sign(ExtElement)
  • sign(LieDerivation)
  • sign(LieElement) -- get the sign of a homogeneous element

The source of this document is in GradedLieAlgebras/doc2.m2:1048:0.