# LieElement @ LieElement -- formal multiplication of Lie elements

## Synopsis

• Operator: @
• Usage:
u=x (symbol @) y
• Inputs:
• x, an instance of the type LieElement, $x$ is of type $L$, where $L$ is of type LieAlgebra
• y, an instance of the type LieElement, $y$ is of type $L$
• Outputs:
• u, an instance of the type LieElement, u is of type $L$, the formal Lie product of $x$ and $y$

## Description

The "at sign" $@$ is used as infix notation for a "formal" Lie multiplication (and also formal multiplication by scalars) where no simplifications are performed. (The formal addition is written as ++ and / is used as formal subtraction.) In this sense, it is different from the use of SPACE as multiplication operator, which always gives an object of normal form as output. The formal operations are useful when relations are introduced in a big free Lie algebra, since then it might be too hard to compute the normal form of the relations, which is not needed in order to define a quotient Lie algebra. For an example, see Minimal models, Ext-algebras and Koszul duals.

 i1 : L = lieAlgebra{a,b} o1 = L o1 : LieAlgebra i2 : (b@b)@a/3@b@a@b++2@a@b@b o2 = (b b a) - (b b a) - 3 (b a b) + 2 (a b b) o2 : L i3 : (b b) a - 3 b a b + 2 a b b o3 = 3 (b b a) o3 : L