Let $W$ be a Coxeter group. The [Iwahori-Hecke algebra](https://en.wikipedia.org/wiki/Iwahori%E2%80%93Hecke_algebra) $H\_q(W)$ is a deformation of $k W$. 



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>  **Question:**  is there some way to interpret the deformation $H\_q(W)$ as a cohomology class? It doesn't seem like $kW$ twisted by a class in $H^2(W,k^{\times})$...
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Do Iwahori-Hecke algebras come from cohomology classes?
Not in an interesting way. Cohomology classes describe formal deformations (i.e. deformations over an Artinian ring, or more generally complete local rings), and the Iwahori-Hecke algebra is trivial on a formal neighborhood of $q=1$. In fact, the only points where it is not trivial on a formal neighborhood is when $q$ is a root of unity of small order. 


