next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
GradedLieAlgebras :: weightLie

weightLie -- gives the multi-degree of a graded element in a Lie algebra

Synopsis

Description

If the Lie algebra has no differential, the program adds an extra last homological degree zero to each generator. The weight of [] is defined to be a list of zeroes of the same length as the weight of the generators. However, the weight of [] should be thought of as arbitrary since, in the example below, the element [b,b] has weight 2,2,0 and is equal to [] in L.

i1 : L=lieAlgebra({a,b,c},{},genWeights => {{1,1},{1,1},{2,2}})

o1 = L

o1 : LieAlgebra
i2 : weightLie(a)

o2 = {1, 1, 0}

o2 : List
i3 : weightLie([a,a,b,a])

o3 = {4, 4, 0}

o3 : List
i4 : g={{1,-1},{[a,c],[b,c]}}

o4 = {{1, -1}, {[a, c], [b, c]}}

o4 : List
i5 : weightLie g

o5 = {3, 3, 0}

o5 : List
i6 : m=indexFormLie g

o6 = mb       - mb
       {3, 2}     {3, 3}

o6 : QQ[mb      , mb      , mb      , mb      , mb      , mb      , mb      , mb      ]
          {1, 0}    {1, 1}    {2, 0}    {2, 1}    {3, 0}    {3, 1}    {3, 2}    {3, 3}
i7 : degree m

o7 = {3, 3, 0}

o7 : List
i8 : weightLie{[],[b,b],{{1,2},{[c],[a,b]}}}

o8 = {{0, 0, 0}, {2, 2, 0}, {2, 2, 0}}

o8 : List

It is possible to use weightLie also in the case when the generators are indexedVariables or integers.

i9 : L2=lieAlgebra({a_2,b,1},{}, genWeights => {1,2,3})

o9 = L2

o9 : LieAlgebra
i10 : weightLie(1)

o10 = {3, 0}

o10 : List

See also

Ways to use weightLie :