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While I don't know the answer to the questions as I don't quite understand them, let me offer some remarks.

There's sort of a ''misunderstanding'' of what a deformation is. What you are discussing here is only i very special case of a formal deformation. In general a deformation should take place in a (dream) moduli space of the objects you're deforming. As such a ''global'' space is rarely exists, one is forced to study the problem locally and/or formally (meaning that the deformation is described with a formal power series ring).

Let us look at the formal case first as this is usually the first step in a deformation problem, the goal being to construct the completion of the local ring of the moduli space. To every deformation problem there is a cohomology theory attached. The first thing that one has to do is determine what this theory is. When this is done one computes is the tangent space to the moduli space in the point (object) one is interested in (in your case a matrix). This tangent space is described by the first cohomology group in the cohomology theory (usually some $\mathrm{Ext}^1$-group). Then one tries to ''lift'' this deformation to higher order deformations in order to obtain something formal. This is done by ''killing'' obstructions at each level in the lifting. These obstructions live in a second cohomology group (usually some $\mathrm{Ext}^2$). When all obstructions are zero the deformation problem is said to be ''unobstructed'' and the ring describing the deformation is a formal power series ring. This case is very rare but happens in many deformation problems at specific points.

(I want to point out that deformations are only defined modulo some equivalence relation and this is in fact a very important thing to remember.)

Ok, this formal ring (i.e., a ring that is a quotient of a formal power series ring) is now to be considered the completion of the local ring to the point in the moduli space. In your setup this corresponds to a deformation in one formal direction in the moduli space. However, you are implictly assuming that the deformation is unobstructed as you haven't factored out by any obstructions (i.e., there are no relations in the deformation ring).

Alas, this is only a formal construction and not "useful" in geometry (at least in algebraic geometry; I suppose that in complex or analytic geometry the formal case would be ok, but I'm not really qualified to answer that). In any case, one needs an algebraization (think from formal power series to polynomial algebras). There is a deep theorem due to Micheal Artin saying that in many situation such an algebraization exists, but it is far from true in general and can be very hard to decide in specific cases.

Let's say we have algebraizations around every point in ''moduli space''. One initial hope is to glue all these algebraizations to a global object. Unfortunately this is seldom possible in the category of schemes. A way out of this is to consider algebraic stacks but that's a whole other story. In any case, locally (formally) the moduli is described by the (completed) deformation rings.

The deformation you're considering comes from the ''Gerstenhaber school'' which is, as I said, only a formal deformation in one direction with the assumption that the deformation problem is unobstructed. However, these kinds of deformations often appear as ''quantizations'' of Poisson algebras, Lie algebras, quantum groups etc. But in a strict sense they are not deformations.

Ok, as for your problems, I don't quite understand how you're supposed to deform a non-commutative algebra with a commutative deformation ring. To me this seems to be an utterly impossible thing to do, at least if I understand your setup correctly. You could possibly make sense of this in some way but I don't know how. Therefore I would certainly say that the problem is rigid (or undefined even).

If you were to deform specific matrices that is a very different thing, and certainly possible, at least if you forget the equivalences. However, and this is important, taking into account these equivalence effectively destroys the moduli space although locally it may well exist. In other words, it is not possible to glue the deformation rings in a sensible way without going to the stack language.

As far as I can see, the Azumaya locus has no direct connection to deformation theory. It has to do with sheaves of algebras over a scheme such that locally it is a full matrix algebra. The Azumaya locus is the points on the schemes where this occurs.

The other questions seem to be related to Cherednik algebras. See papers by Ian Gordon and collaborators.

Edit: Oh, I forgot question 2, but I see that DamienC has already given a correct answer to that question. Not much more to say. Let me note however, that the only reasonable deformations in algebraic geometry are the flat ones, although in principle you could take more general morphisms.

1

While I don't know the answer to the questions as I don't quite understand them, let me offer some remarks.

There's sort of a ''misunderstanding'' of what a deformation is. What you are discussing here is only i very special case of a formal deformation. In general a deformation should take place in a (dream) moduli space of the objects you're deforming. As such a ''global'' space is rarely exists, one is forced to study the problem locally and/or formally (meaning that the deformation is described with a formal power series ring).

Let us look at the formal case first as this is usually the first step in a deformation problem, the goal being to construct the completion of the local ring of the moduli space. To every deformation problem there is a cohomology theory attached. The first thing that one has to do is determine what this theory is. When this is done one computes is the tangent space to the moduli space in the point (object) one is interested in (in your case a matrix). This tangent space is described by the first cohomology group in the cohomology theory (usually some $\mathrm{Ext}^1$-group). Then one tries to ''lift'' this deformation to higher order deformations in order to obtain something formal. This is done by ''killing'' obstructions at each level in the lifting. These obstructions live in a second cohomology group (usually some $\mathrm{Ext}^2$). When all obstructions are zero the deformation problem is said to be ''unobstructed'' and the ring describing the deformation is a formal power series ring. This case is very rare but happens in many deformation problems at specific points.

(I want to point out that deformations are only defined modulo some equivalence relation and this is in fact a very important thing to remember.)

Ok, this formal ring (i.e., a ring that is a quotient of a formal power series ring) is now to be considered the completion of the local ring to the point in the moduli space. In your setup this corresponds to a deformation in one formal direction in the moduli space. However, you are implictly assuming that the deformation is unobstructed as you haven't factored out by any obstructions (i.e., there are no relations in the deformation ring).

Alas, this is only a formal construction and not "useful" in geometry (at least in algebraic geometry; I suppose that in complex or analytic geometry the formal case would be ok, but I'm not really qualified to answer that). In any case, one needs an algebraization (think from formal power series to polynomial algebras). There is a deep theorem due to Micheal Artin saying that in many situation such an algebraization exists, but it is far from true in general and can be very hard to decide in specific cases.

Let's say we have algebraizations around every point in ''moduli space''. One initial hope is to glue all these algebraizations to a global object. Unfortunately this is seldom possible in the category of schemes. A way out of this is to consider algebraic stacks but that's a whole other story. In any case, locally (formally) the moduli is described by the (completed) deformation rings.

The deformation you're considering comes from the ''Gerstenhaber school'' which is, as I said, only a formal deformation in one direction with the assumption that the deformation problem is unobstructed. However, these kinds of deformations often appear as ''quantizations'' of Poisson algebras, Lie algebras, quantum groups etc. But in a strict sense they are not deformations.

Ok, as for your problems, I don't quite understand how you're supposed to deform a non-commutative algebra with a commutative deformation ring. To me this seems to be an utterly impossible thing to do, at least if I understand your setup correctly. You could possibly make sense of this in some way but I don't know how. Therefore I would certainly say that the problem is rigid (or undefined even).

If you were to deform specific matrices that is a very different thing, and certainly possible, at least if you forget the equivalences. However, and this is important, taking into account these equivalence effectively destroys the moduli space although locally it may well exist. In other words, it is not possible to glue the deformation rings in a sensible way without going to the stack language.

As far as I can see, the Azumaya locus has no direct connection to deformation theory. It has to do with sheaves of algebras over a scheme such that locally it is a full matrix algebra. The Azumaya locus is the points on the schemes where this occurs.

The other questions seem to be related to Cherednik algebras. See papers by Ian Gordon and collaborators.