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dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space

Description

Synopsis

  • Usage:
    dm=dims(n,L)
  • Inputs:
  • Outputs:
    • dm, a matrix, the matrix of dimensions of $L$ in first degree $i$ and homological degree $j$, where $i$ ranges from $1$ to $n$ and $j$ from 0 to $n-1$

Synopsis

  • Usage:
    dm=dims(n,E)
  • Inputs:
  • Outputs:
    • dm, a matrix, the matrix of dimensions of $E$ in first degree $i$ and homological degree $j$, where $i$ ranges from $1$ to $n$ and $j$ from 0 to $n-1$

Synopsis

  • Usage:
    dm=dims(n,V)
  • Inputs:
    • n, an integer, the maximal degree
    • V, an instance of the type VectorSpace, an instance of type VectorSpace
  • Outputs:
    • dm, a matrix, the matrix of dimensions of $V$ in first degree $i$ and homological degree $j$, where $i$ ranges from $1$ to $n$ and $j$ from 0 to $n-1$

Synopsis

  • Usage:
    dl=dims(n,m,L)
  • Inputs:
  • Outputs:
    • dl, a list, the dimensions of $L$ in first degree $i$, where $i$ ranges from $n$ to $m$

Synopsis

  • Usage:
    dl=dims(n,m,E)
  • Inputs:
  • Outputs:
    • dl, a list, the dimensions of $E$ in first degree $i$, where $i$ ranges from $n$ to $m$

Synopsis

  • Usage:
    dl=dims(n,m,V)
  • Inputs:
    • n, an integer, the starting degree
    • m, an integer, the ending degree
    • V, an instance of the type VectorSpace, an instance of type VectorSpace
  • Outputs:
    • dl, a list, the dimensions of $V$ in first degree $i$, where $i$ ranges from $n$ to $m$ 
i1 : L = lieAlgebra({a,b,c},Weights=>{{1,0},{2,1},{3,2}},
                  Signs=>{1,1,1},LastWeightHomological=>true)

o1 = L

o1 : LieAlgebra
 
i2 : D= differentialLieAlgebra({0_L,a a,a b})

o2 = D

o2 : LieAlgebra
 
i3 : J=lieIdeal({b b + 4 a c})

o3 = J

o3 : FGLieIdeal
 
i4 : Q=D/J

o4 = Q

o4 : LieAlgebra
 
i5 : dims(7,Q)

o5 = | 1 1 0 0 0 0 0 |
     | 0 1 1 1 1 1 1 |
     | 0 0 1 1 1 1 2 |
     | 0 0 0 0 1 1 2 |
     | 0 0 0 0 0 1 1 |
     | 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 |

              7       7
o5 : Matrix ZZ  <-- ZZ
 
i6 : Z=cycles Q

o6 = Z

o6 : LieSubAlgebra
 
i7 : dims(5,Z)

o7 = | 1 1 0 0 0 |
     | 0 0 1 1 1 |
     | 0 0 0 0 0 |
     | 0 0 0 0 1 |
     | 0 0 0 0 0 |

              5       5
o7 : Matrix ZZ  <-- ZZ
 
i8 : H=lieHomology Q

o8 = H

o8 : VectorSpace
 
i9 : dims(1,5,H)

o9 = {1, 0, 0, 0, 1}

o9 : List
 
i10 : E=extAlgebra(5,Q)

o10 = E

o10 : ExtAlgebra
 
i11 : dims(4,E)

o11 = | 1 0 0 0 |
      | 0 1 0 0 |
      | 0 0 1 0 |
      | 0 0 0 1 |

               4       4
o11 : Matrix ZZ  <-- ZZ

See also

Ways to use dims:

  • dims(ZZ,ExtAlgebra)
  • dims(ZZ,LieAlgebra)
  • dims(ZZ,VectorSpace)
  • dims(ZZ,ZZ,ExtAlgebra)
  • dims(ZZ,ZZ,LieAlgebra)
  • dims(ZZ,ZZ,VectorSpace)

For the programmer

The object dims is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc2.m2:984:0.