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Deformation of relations

Answer to question 2 is the following: a deformation of an algebra $A_0$ parametrized by a pointed affine scheme $*\to X=Spec(B\to k)$ is the data of a $B$-algebra $A$ such that $A_0\cong A\otimes_B k$.

Observe that the kind of deformations your are looking at in question 1 are those that are called "flat".

Deforming structure constants of matrix algebras

In order to find the answer to question 1 (which is yes) I would suggest you to use Artin–Wedderburn theorem.

EDIT 1: more precisely, over $\mathbb{C}$ any finite-dimensional (semi-)simple algebra is isomorphic to (a Cartesian product of) matrix algebras.

EDIT 2: Anyway, it seems that there is a more general phenomenon (see the fondational paper of Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1), (1964), 59–104): $$ \textrm{absolute rigidity}\Rightarrow \textrm{analytic rigidity}\Rightarrow \textrm{geometric rigidity} $$ where absolute rigidity means that $HH^2(A)=0$, analytic rigidity means that there are no non-trivial formal deformations of $A$, and geometric rigidity means that the $GL(V)$-orbit of the point defined by $A=(V,c_{ij}^k)$ in the variety of structure constants on $V$ is open (this tells you in particular that sufficiently small deformations are trivial). You can conclude by using that $$ \textrm{semi-simplicity}\Rightarrow\textrm{absolute rigidity} $$

Azumaya algebras

Finally, if I understand well your question 3, your guess seems correct: the right notion to look at is precisely the one of Azumaya algebra (if you are interested by the relation to Cherednick algebras mentionned in the answer of Daniel Larsson, I would suggest you to take a look at the last two or three chapters of this book by Pavel Etingof).

Final remark

Let me also mention that there is a lot of work about deformation theory of quadratic algebras by sub-quadratic ones (see references given in this MO answer - EDIT: sorry, you already know this since you're the one who actually asked the question!)

Deformation of relations

Answer to question 2 is the following: a deformation of an algebra $A_0$ parametrized by a pointed affine scheme $*\to X=Spec(B\to k)$ is the data of a $B$-algebra $A$ such that $A_0\cong A\otimes_B k$.

Observe that the kind of deformations your are looking at in question 1 are those that are called "flat".

Deforming structure constants of matrix algebras

In order to find the answer to question 1 (which is yes) I would suggest you to use Artin–Wedderburn theorem.

EDIT 1: more precisely, over $\mathbb{C}$ any finite-dimensional (semi-)simple algebra is isomorphic to (a Cartesian product of) matrix algebras.

EDIT 2: Anyway, it seems that there is a more general phenomenon (see the fondational paper of Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1), (1964), 59–104): $$ \textrm{absolute rigidity}\Rightarrow \textrm{analytic rigidity}\Rightarrow \textrm{geometric rigidity} $$ where absolute rigidity means that $HH^2(A)=0$, analytic rigidity means that there are no non-trivial formal deformations of $A$, and geometric rigidity means that the $GL(V)$-orbit of the point defined by $A=(V,c_{ij}^k)$ in the variety of structure constants on $V$ is open (this tells you in particular that sufficiently small deformations are trivial). You can conclude by using that $$ \textrm{semi-simplicity}\Rightarrow\textrm{absolute rigidity} $$

Azumaya algebras

Finally, if I understand well your question 3, your guess seems correct: the right notion to look at is precisely the one of Azumaya algebra (if you are interested by the relation to Cherednick algebras mentionned in the answer of Daniel Larsson, I would suggest you to take a look at the last two or three chapters of this book by Pavel Etingof).

Final remark

Let me also mention that there is a lot of work about deformation theory of quadratic algebras by sub-quadratic ones (see references given in this MO answer - EDIT: sorry, you already know this since you're the one who actually asked the question!)

Deformation of relations

Answer to question 2 is the following: a deformation of an algebra $A_0$ parametrized by a pointed affine scheme $*\to X=Spec(B\to k)$ is the data of a $B$-algebra $A$ such that $A_0\cong A\otimes_B k$.

Observe that the kind of deformations your are looking at in question 1 are those that are called "flat".

Deforming structure constants of matrix algebras

In order to find the answer to question 1 (which is yes) I would suggest you to use Artin–Wedderburn theorem.

EDIT 1: more precisely, over $\mathbb{C}$ any finite-dimensional (semi-)simple algebra is isomorphic to (a Cartesian product of) matrix algebras.

EDIT 2: Anyway, it seems that there is a more general phenomenon (see the fondational paper of Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1), (1964), 59–104): $$ \textrm{absolute rigidity}\Rightarrow \textrm{analytic rigidity}\Rightarrow \textrm{geometric rigidity} $$ where absolute rigidity means that $HH^2(A)=0$, analytic rigidity means that there are no non-trivial formal deformations of $A$, and geometric rigidity means that the $GL(V)$-orbit of the point defined by $A=(V,c_{ij}^k)$ in the variety of structure constants on $V$ is open. You can conclude by using that $$ \textrm{semi-simplicity}\Rightarrow\textrm{absolute rigidity} $$

Azumaya algebras

Finally, if I understand well your question 3, your guess seems correct: the right notion to look at is precisely the one of Azumaya algebra (if you are interested by the relation to Cherednick algebras mentionned in the answer of Daniel Larsson, I would suggest you to take a look at the last two or three chapters of this book by Pavel Etingof).

Final remark

Let me also mention that there is a lot of work about deformation theory of quadratic algebras by sub-quadratic ones (see references given in this MO answer - EDIT: sorry, you already know this since you're the one who actually asked the question!)

Deformation of relations

Answer to question 2 is the following: a deformation of an algebra $A_0$ parametrized by a pointed affine scheme $*\to X=Spec(B\to k)$ is the data of a $B$-algebra $A$ such that $A_0\cong A\otimes_B k$.

Observe that the kind of deformations your are looking at in question 1 are those that are called "flat".

Deforming structure constants of matrix algebras

In order to find the answer to question 1 (which is yes) I would suggest you to use Artin–Wedderburn theorem.

EDIT 1: more precisely, over $\mathbb{C}$ any finite-dimensional (semi-)simple algebra is isomorphic to (a Cartesian product of) matrix algebras.

EDIT 2: Anyway, it seems that there is a more general phenomenon (see the fondational paper of Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1), (1964), 59–104): $$ \textrm{absolute rigidity}\Rightarrow \textrm{analytic rigidity}\Rightarrow \textrm{geometric rigidity} $$ where absolute rigidity means that $HH^2(A)=0$, analytic rigidity means that there are no non-trivial formal deformations of $A$, and geometric rigidity means that the $GL(V)$-orbit of the point defined by $A=(V,c_{ij}^k)$ in the variety of structure constants on $V$ is open (this tells you in particular that sufficiently small deformations are trivial). You can conclude by using that $$ \textrm{semi-simplicity}\Rightarrow\textrm{absolute rigidity} $$

Azumaya algebras

Finally, if I understand well your question 3, your guess seems correct: the right notion to look at is precisely the one of Azumaya algebra (if you are interested by the relation to Cherednick algebras mentionned in the answer of Daniel Larsson, I would suggest you to take a look at the last two or three chapters of this book by Pavel Etingof).

Final remark

Let me also mention that there is a lot of work about deformation theory of quadratic algebras by sub-quadratic ones (see references given in this MO answer - EDIT: sorry, you already know this since you're the one who actually asked the question!)

Answer to question 2 is the following: a deformation of an algebra $A_0$ parametrized by a pointed affine scheme $*\to X=Spec(B\to k)$ is the data of a $B$-algebra $A$ such that $A_0\cong A\otimes_B k$.

Observe that the kind of deformations your are looking at in question 1 are those that are called "flat".

In order to find the answer to question 1 I would suggest you to use Artin–Wedderburn theorem.

Finally, if I understand well your question 3, your guess seems correct: the right notion to look at is precisely the one of Azumaya algebra (if you are interested by the relation to Cherednick algebras mentionned in the answer of Daniel Larsson, I would suggest you to take a look at the last two or three chapters of this book by Pavel Etingof).

Let me also mention that there is a lot of work about deformation theory of quadratic algebras by sub-quadratic ones (see references given in this MO answer - EDIT: sorry, you already know this since you're the one who actually asked the question!)

Deformation of relations

Answer to question 2 is the following: a deformation of an algebra $A_0$ parametrized by a pointed affine scheme $*\to X=Spec(B\to k)$ is the data of a $B$-algebra $A$ such that $A_0\cong A\otimes_B k$.

Observe that the kind of deformations your are looking at in question 1 are those that are called "flat".

Deforming structure constants of matrix algebras

In order to find the answer to question 1 (which is yes) I would suggest you to use Artin–Wedderburn theorem.

EDIT 1: more precisely, over $\mathbb{C}$ any finite-dimensional (semi-)simple algebra is isomorphic to (a Cartesian product of) matrix algebras.

EDIT 2: Anyway, it seems that there is a more general phenomenon (see the fondational paper of Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1), (1964), 59–104): $$ \textrm{absolute rigidity}\Rightarrow \textrm{analytic rigidity}\Rightarrow \textrm{geometric rigidity} $$ where absolute rigidity means that $HH^2(A)=0$, analytic rigidity means that there are no non-trivial formal deformations of $A$, and geometric rigidity means that the $GL(V)$-orbit of the point defined by $A=(V,c_{ij}^k)$ in the variety of structure constants on $V$ is open. You can conclude by using that $$ \textrm{semi-simplicity}\Rightarrow\textrm{absolute rigidity} $$

Azumaya algebras

Finally, if I understand well your question 3, your guess seems correct: the right notion to look at is precisely the one of Azumaya algebra (if you are interested by the relation to Cherednick algebras mentionned in the answer of Daniel Larsson, I would suggest you to take a look at the last two or three chapters of this book by Pavel Etingof).

Final remark

Let me also mention that there is a lot of work about deformation theory of quadratic algebras by sub-quadratic ones (see references given in this MO answer - EDIT: sorry, you already know this since you're the one who actually asked the question!)