In RobinHankin/freealg: The Free Algebra

knitr::opts_chunk$set(echo = TRUE)

Given that $R$ is the free associative algebra over some set of generators over a field $k$, Cohn conjectured (and Bergman proved) that any pair of commuting elements of $R$ can written in the form $P(z)$, $Q(z)$ for some $z\in R$. This is easy to illustrate in the package.

library("freealg")
(z <- as.freealg("2+a-aab+4*z"))

We then construct $P=P(z)$ and $Q=Q(z)$:

P <- 3+5*z^3
Q <- z - z^4

Objects P and Q are quite complicated, but they commute:

.[P,Q]

References

• G. M. Bergman 1969. "Centralizers in free associative algebras". Transactions of the American Mathematical Society 137:327-344
• P. M. Cohn 1963. "Rings with weak algorithm". _Transactions of the Americal Mathematical Society 109:332-356.

RobinHankin/freealg documentation built on Dec. 24, 2024, 3:16 a.m.