# indexForm -- get a Lie element in the polynomial ring representation

## Synopsis

• Usage:
r=indexForm(a)
• Inputs:
• Outputs:
• r, , the element in L#cache.mbRing corresponding to the input.

## Description

The ring L#cache.mbRing, see mbRing, is used to get an output of Lie elements with indexed basis elements, which sometimes is better to use than the iterated Lie products of generators, especially in high degrees. Use indexForm to get the output in L#cache.mbRing and standardForm(RingElement,LieAlgebra) to get back the standard output. The ring mbRing is very large: it has as many generators as the total dimension of the computed Lie algebra. For this reason, you should give the ring a name to avoid a large output. When $x$ is a linear polynomial in L#cache.mbRing, the composition indexForm(standardForm(x) gives back $x$ . When $x$ is a Lie element in $L$, the composition standardForm(indexForm x) is equal to $x$ modulo the relations in $L$.

 i1 : L = lieAlgebra{a,b,c} o1 = L o1 : LieAlgebra i2 : x = (basis(2,L))_0 (basis(3,L))_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 : numgens R o4 = 80 i5 : indexForm x o5 = mb - mb - mb + mb {5, 5} {5, 7} {5, 12} {5, 15} o5 : R i6 : standardForm(oo,L) o6 = - (a c b b a) + (b a c b a) - (b c a b a) + (c b a b a) o6 : L i7 : indexForm a a b c o7 = - mb + 2mb - mb {4, 2} {4, 5} {4, 9} o7 : R i8 : standardForm(oo,L) o8 = 2 (a c b a) - (b a c a) - (c a b a) o8 : L i9 : a a b c===oo o9 = true