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1. Is there a name for the algebra (and its tensor products) given by generators $M_{j}$, $j \in \mathbb{Z}_{n}$ under the conditions $$M_{j} = (1 - M_{j-1})(1-M_{j+1})$$ where $j=1\implies j-1=n$ and $j=n\implies j+1=1$?

2. There is no restriction on the commutativity of $M_{j}$ and so is there a matrix representation of this algebra for all possible cases for $M_{j}$?

By tensor product I mean generalization of following extension to multiple indices:

For single index $$M_{j} = \prod_{\substack{i\in\{-1,0,1\}\\i\neq j}}(1 - M_{j+i})$$ which is same as above and for two indices $$M_{j,j'} = \prod_{\substack{i,i'\in\{-1,0,1\}\\i\neq j\wedge i'\neq j}}(1 - M_{j+i,j'+i'})$$ relation holds.

1. Is there a name for the algebra (and its tensor products) given by generators $M_{j}$, $j \in \mathbb{Z}_{n}$ under the conditions $$M_{j} = (1 - M_{j-1})(1-M_{j+1})$$ where $j=1\implies j-1=n$ and $j=n\implies j+1=1$?

2. There is no restriction on the commutativity of $M_{j}$ and so is there a matrix representation of this algebra for all possible cases for $M_{j}$?

By tensor product I mean generalization of following extension to multiple indices:

For single index $$M_{j} = \prod_{\substack{i\in\{-1,0,1\}\\i\neq j}}(1 - M_{j+i})$$ which is same as above and for two indices $$M_{j,j'} = \prod_{\substack{i,i'\in\{-1,0,1\}\\i\neq j\wedge i'\neq j}}(1 - M_{j+i,j'+i'})$$ relation holds.

Update Like Lena I too thought it is similar to a Hecke algebra however since I am not familiar I could not pin down details.

Is there a name for the algebra (and its tensor products) given by generators $M_{j}$, $j \in \mathbb{Z}_{n}$ under the conditions $$M_{j} = (1 - M_{j-1})(1-M_{j+1})$$ where $j=1\implies j-1=n$ and $j=n\implies j+1=1$?

There is no restriction on the commutativity of $M_{j}$. I am interested in structures (in particular matrix representations) for all possible cases for $M_{j}$.

By tensor product I mean generalization of following extension to multiple indices:

For single index $$M_{j} = \prod_{\substack{i\in\{-1,0,1\}\\i\neq j}}(1 - M_{j+i})$$ which is same as above and for two indices $$M_{j,j'} = \prod_{\substack{i,i'\in\{-1,0,1\}\\i\neq j\wedge i'\neq j}}(1 - M_{j+i,j'+i'})$$ relation holds.

1. Is there a name for the algebra (and its tensor products) given by generators $M_{j}$, $j \in \mathbb{Z}_{n}$ under the conditions $$M_{j} = (1 - M_{j-1})(1-M_{j+1})$$ where $j=1\implies j-1=n$ and $j=n\implies j+1=1$?

2. There is no restriction on the commutativity of $M_{j}$ and so is there a matrix representation of this algebra for all possible cases for $M_{j}$?

By tensor product I mean generalization of following extension to multiple indices:

For single index $$M_{j} = \prod_{\substack{i\in\{-1,0,1\}\\i\neq j}}(1 - M_{j+i})$$ which is same as above and for two indices $$M_{j,j'} = \prod_{\substack{i,i'\in\{-1,0,1\}\\i\neq j\wedge i'\neq j}}(1 - M_{j+i,j'+i'})$$ relation holds.

Is there a name for the algebra (and its tensor products) given by generators $M_{j}$, $j \in \mathbb{Z}_{n}$ under the conditions $$M_{j} = (1 - M_{j-1})(1-M_{j+1})$$ where $j=1\implies j-1=n$ and $j=n\implies j+1=1$?

There is no restriction on the commutativity of $M_{j}$. I am interested in structures including matrix representations for all possible cases for $M_{j}$.

By tensor product I mean generalization of following extension to multiple indices:

For single index $$M_{j} = \prod_{\substack{i\in\{-1,0,1\}\\i\neq j}}(1 - M_{j+i})$$ which is same as above and for two indices $$M_{j,j'} = \prod_{\substack{i,i'\in\{-1,0,1\}\\i\neq j\wedge i'\neq j}}(1 - M_{j+i,j'+i'})$$ relation holds.

Is there a name for the algebra (and its tensor products) given by generators $M_{j}$, $j \in \mathbb{Z}_{n}$ under the conditions $$M_{j} = (1 - M_{j-1})(1-M_{j+1})$$ where $j=1\implies j-1=n$ and $j=n\implies j+1=1$?

There is no restriction on the commutativity of $M_{j}$. I am interested in structures (in particular matrix representations) for all possible cases for $M_{j}$.

By tensor product I mean generalization of following extension to multiple indices:

For single index $$M_{j} = \prod_{\substack{i\in\{-1,0,1\}\\i\neq j}}(1 - M_{j+i})$$ which is same as above and for two indices $$M_{j,j'} = \prod_{\substack{i,i'\in\{-1,0,1\}\\i\neq j\wedge i'\neq j}}(1 - M_{j+i,j'+i'})$$ relation holds.