# Name for algebra and its tensor products

Is there a name for the algebra (and its tensor products) given by generators $U_{j}$, $j \in \mathbb{Z}_{n}$

under the conditions $U_{j} = (1 - U_{j-1})(1-U_{j+1})$? There is no restriction on the commutativity of $U_{j}$. I am interested in structures for all possible cases for $U_{j}$.

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How do I interpret your relation when $j=1$ or $n$? –  David Hill Apr 9 '13 at 14:28
For $j=1$, $j-1$ will be $n$ and for $j=n$, $j+1$ will be $1$. –  Turbo Apr 9 '13 at 14:37
In that case you can just renumber indices $i \to i-1$ and write $U_i$, $i\in \mathbb{Z}_n$. Adding some context would be helpful. Are the variables $U_i$ (anti)commutative? –  Vít Tuček Apr 10 '13 at 8:53
I don't know if this is helpful, but when $n=2$, $U_1$ can be written in terms of $U_0$, so we can regard the algebra as a quotient of a polynomial algebra: $k[x]/\langle x(x^3-4x^2+2x-1)\rangle$. Perhaps it would be easier for someone to recognize this algebra. –  David Hill Apr 11 '13 at 19:39