Free associative algebra on a vector space

olivierd

Aug 20, 2023 1 min read

Free associative algebra on a vector space

Vector spaces and associative algebras will be implicitely all on the same field $\K = \R$ or $\C$ .

Let $U$ be a vector space. We wish to build a free associative algebra $V$ on $U$ .

For any natural number $n \ge 1$ , we will note $U^{\otimes n} = U \otimes U \otimes \ldots \otimes U$ (tensor product $n$ times).

Each $U^{\otimes n}$ is a vector space. We may form:

$V = U \oplus U^{\otimes 2} \oplus U^{\otimes 3} \ldots = \bigoplus_{n \in \N_1} {U^{\otimes n}}$

An element $\vec v$ of $V$ can be written $(\vec v_1, \vec v_2, \ldots \vec v_i, \ldots)$ with each $\vec v_i$ element of $U^{\otimes i}$ , and $\vec v_i$ being non null for only a finite number of indices $i$ .