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

indexFormLie -- returns an element in the ring representation corresponding to the input

Synopsis

Description

The ring L.cache.mbRing, see mbRing, is used to get an output of LieElements with indexed basis elements, which sometimes is better to use than the iterated Lie products of generators, especially in high degrees. Use indexFormLie to get the output in L.cache.mbRing and defLie to get back the standard output. The ring L.cache.mbRing is very large, it has as many generators as the total dimension of the computed Lie algebra, so to avoid a large output you should give the ring a name. The composition indexFormLie(defLie x) gives back x, when x is a linear polynomial in L.cache.mbRing. When x is a LieElement in L, the composition defLie(indexFormLie x) is equal to x modulo the relations in L.

i1 : L = lieAlgebra{a,b,c}

o1 = L

o1 : LieAlgebra
i2 : x = (basisLie 2)_0 (basisLie 3)_4

o2 =  - (a c b b a) + (b a c b a) - (b c a b a) + (c b a b a)

o2 : L
i3 : R = L.cache.mbRing

o3 = R

o3 : PolynomialRing
i4 : length gens R

o4 = 80
i5 : indexFormLie x

o5 = mb       - mb       - mb        + mb
       {5, 5}     {5, 7}     {5, 12}     {5, 15}

o5 : R
i6 : defLie oo

o6 =  - (a c b b a) + (b a c b a) - (b c a b a) + (c b a b a)

o6 : L
i7 : indexFormLie{c,b c,a b c,a a b c}

o7 = {mb      , -mb      , mb       - mb      , - mb       + 2mb       -
        {1, 2}     {2, 2}    {3, 2}     {3, 4}      {4, 2}      {4, 5}  
     ------------------------------------------------------------------------
     mb      }
       {4, 9}

o7 : List
i8 : defLie oo

o8 = {c,  - (c b),  - (b c a) + (c b a), 2 (a c b a) - (b a c a) - (c a b a)}

o8 : List
i9 : {c,b c,a b c,a a b c}

o9 = {c,  - (c b),  - (b c a) + (c b a), 2 (a c b a) - (b a c a) - (c a b a)}

o9 : List

See also

Ways to use indexFormLie :