Truncated tensor algebra
= Truncated tensor algebra
{title2=$T^{(N)}(V)=\bigoplus_{k=0}^N V^{\otimes k}$}
The truncated <tensor algebra> discards products of degree greater than $N$. Its multiplication is $(a\otimes b)^{(k)}=\sum_{j=0}^k a^{(j)}\otimes b^{(k-j)}$. Every element with scalar part $1$ is invertible by a finite geometric series in its positive-degree part. Truncated exponential and logarithm are finite polynomials.