antipode(NCRingElement) -- The antipode of a tensor

Function: antipode

Description

Consider a free associative algebra $\mathtt{R}$ on the letters $\mathtt{1}, \dots, \mathtt{d}$. We define the antipode map $\mathtt{a}:\mathtt{R}\rightarrow \mathtt{R}$ first on a word $w:=\mathtt{i_1}\cdot\dots \cdot\mathtt{i_k}$ to be $$ \mathtt{a}(w) := (-1)^k \mathtt{i_k}\cdot\dots\cdot \mathtt{i_1}$$ and then extending it by linearity to the whole algebra.

i1 : R = wordAlgebra(3)

o1 = R

o1 : NCPolynomialRing

i2 : f = [1,2,3]_R + 2* [3,2]_R; f//wordFormat

o3 = 2 [3, 2] + [1, 2, 3]

i4 : antipode(f) //wordFormat

o4 = - [3, 2, 1] + 2 [2, 3]

Ways to use this method:

antipode(NCRingElement) -- The antipode of a tensor

The source of this document is in PathSignatures/documentation.m2:918:0.