I'm studying certain games on graphs on $n$ vertices and it turns out that I get a matrix algebra with generators $T_i , i=1,\dotsc,n$ satisfying a certain type of relations. Here the $T_i$ are matrices attached to the vertices of the graph and $q$ is a real parameter .
A) In the case $q=1$ the relations are
$T_i^2 - 1 = 0$ for every vertex $i$.
$T_i T_j = T_j T_i$ if vertices $i$ and $j$ are not adjacent.
$T_iT_{j}T_i = T_{j}T_{i}T_{j}$
if vertices $i$ and $j$ are adjacent.
B) For $q \ne 1$ the relations are
$T_i^2 +(1-q)T_i -q = 0$ for every vertex $i$.
$T_i T_j = T_j T_i $ if vertices $i$ and $j$ are not adjacent.
$T_iT_jT_i - T_jT_iT_j + (1-q)\cdot(T_j-T_i) = 0\quad$ or $\ \ T_iT_jT_i - T_jT_iT_j -q\cdot (T_j-T_i) = 0 $ if vertices $i$ and $j$ are adjacent "skewed" braid relations.
I have managed the case $q=1$ and now I want to study the case $q\ne 1$. It turns out that for $R$ a commutative integral domain with 1 and $q \in R$ an arbitrary element, the associated so called Iwahori–Hecke algebra is a unital associative $R$-algebra with generators $T_1,\dotsc,T_n$ subject to the relations
$T_i^2 +(1-q)T_i -q = 0$ for $i=1,\dotsc,n-1$.
$T_i T_j = T_j T_i$ for $1 \le i < j-1 \le n-2$.
$T_iT_{i+1}T_i = T_{i+1}T_{i}T_{i+1}$ for $i=1,\dotsc,n-2$.
For the case $q=1$ these relations are exactly the same as mine if I consider the path graph on n vertices.
For the case $q\ne 1$ my generators fail to satisfy the "standard" braid relations but satisfy a slightly modified version.
My question is: I'm not at all a specialist and my question is maybe too broad but do we know something about algebras with generators satisfying relations like mine for the case $q \ne 1$ above?