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

## Synopsis

• Usage:
dm=dims(n,L)
• Inputs:
• Outputs:
• dm, , 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, , 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, , 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