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# coefficients(LieElement) -- get the coefficients and monomials of a Lie element

## Synopsis

• Function: coefficients
• Usage:
c = coefficients(x)
• Inputs:
• x, an instance of the type LieElement, $x$ is of type $L$, where $L$ is of type LieAlgebra
• Optional inputs:
• Outputs:
• c, a list, the lists of coefficients and monomials in $x$

## Description

The optional inputs given above are not relevant for Lie algebras. A Lie element $x$ has a normal form in a Lie algebra $L$, which is a linear combination of basis elements in a certain order; coefficients(x) gives the lists of coefficients and monomials in this representation. If the Lie element has been obtained using the "formal" operators, then coefficients(x) gives the coefficients and monomials for all the iterated Lie products used in the expression.

 i1 : L = lieAlgebra{a,b,c} o1 = L o1 : LieAlgebra i2 : x = a b c - 3 c b a +(1/3) b a c o2 = - (4/3)(b c a) - 2 (c b a) o2 : L i3 : coefficients x 4 o3 = {{- -, -2}, {(b c a), (c b a)}} 3 o3 : List i4 : y = a@b@c/3@c@b@a++(1/3)@b@a@c o4 = (a b c) - 3 (c b a) + (1/3)(b a c) o4 : L i5 : coefficients y 1 o5 = {{1, -3, -}, {(a b c), (c b a), (b a c)}} 3 o5 : List