Introduction: Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, let $G \leq \mathrm{GL}(V)$ be a finite subgroup and let $\kappa:V \times V \rightarrow \mathbb{C}G$ be an alternating bilinear map. Let $I$ be the ideal in $\mathrm{T}(V) \sharp \mathbb{C}G$ generated by the elements $vw-wv-\kappa(v,w)$ with $v,w \in V$. ($\sharp$ denotes the smash product). If we let elements of $V$ have degree $1$ and elements of $\mathbb{C}G$ have degree 0, we get a grading on $\mathrm{T}(V) \sharp \mathbb{C}G$ and so a filtration on $A := \mathrm{T}(V) \sharp \mathbb{C} G/I$. As $\mathrm{gr}(A)$ is commutative in degree 1, the quotient morphism $\mathrm{T}(V) \sharp \mathbb{C} G$ induces a surjective graded algebra morphism $\xi: \mathrm{S}(V) \sharp \mathbb{C} G \rightarrow \mathrm{gr}(A)$.

Now, $A$ is called a rational Cherednik algebra (or perhaps in this more general setting it's called a graded Hecke algebra or Drinfeld-Hecke algebra) if $\xi$ is an isomorphism.

If I did not make a mistake, the condition that $\xi$ is an isomorphism is equivalent to the following: for any linear section $s$ of the quotient map $\mathrm{T}(V) \rightarrow \mathrm{S}(V)$, the vector space morphism $\theta_s:\mathrm{S}(V) \otimes \mathbb{C} G \rightarrow A$, $x \otimes g \mapsto q(s(x) \otimes g)$, is an isomorphism, where $q:\mathrm{T}(V) \otimes \mathbb{C} G \rightarrow A$ is the quotient map.

Hence, a rational Cherednik algebra $A$ is $\mathrm{S}(V) \otimes \mathbb{C}G$ as a vector space, but only up to the choice of a linear section. Now, is there a canonical way to identify $A$ with $\mathrm{S}(V) \otimes \mathbb{C}G$?

This might be a little pedantic, but nobody ever mentions a choice of a section and I fear that I miss an important point here. Here is a further problem: is there a canonical way to identify $\mathrm{S}(V)$ as a subalgebra of $A$? If I choose a basis of $V$ and let $s$ be the linear section defined by this choice, then $\theta_s$ embeds $\mathrm{S}(V)$ as a subalgebra in $A$. Here I don't even now if this works for any linear section (probably it does?).

An additional (not totally unrelated) question: If the PBW morphism $\xi$ is an isomorphism, the algebra $A$ is already pretty nice because several properties of $\mathrm{S}(V) \sharp \mathbb{C}G$ (noetherian, prime, finite homolgical dimension, Cohen-Macaulay) are transported to $A$. But $\xi$ would in the same way as above already exist if $\kappa$ would be a map to $V \cdot \mathbb{C} G \subseteq \mathrm{T}(V) \sharp \mathbb{C}G$, i.e. if it would involve degree 1 elements. Why aren't those algebras interesting? (Perhaps because they don't have a triangular decomposition or they aren't deformations of $\mathrm{S}(V) \sharp \mathbb{C}G$?) I'm just looking for a natural reason to look at the $A_\kappa$ (I remember very well my question and the answers about reasons for studying rational Cherednik algebras but let's try it this way...)

Introduction: Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, let $G \leq \mathrm{GL}(V)$ be a finite subgroup and let $\kappa:V \times V \rightarrow \mathbb{C}G$ be an alternating bilinear map. Let $I$ be the ideal in $\mathrm{T}(V) \sharp \mathbb{C}G$ generated by the elements $vw-wv-\kappa(v,w)$ with $v,w \in V$. ($\sharp$ denotes the smash product). If we let elements of $V$ have degree $1$ and elements of $\mathbb{C}G$ have degree 0, we get a grading on $\mathrm{T}(V) \sharp \mathbb{C}G$ and so a filtration on $A := \mathrm{T}(V) \sharp \mathbb{C} G/I$. As $\mathrm{gr}(A)$ is commutative in degree 1, the quotient morphism $\mathrm{T}(V) \sharp \mathbb{C} G$ induces a surjective graded algebra morphism $\xi: \mathrm{S}(V) \sharp \mathbb{C} G \rightarrow \mathrm{gr}(A)$.

Now, $A$ is called a rational Cherednik algebra (or perhaps in this more general setting it's called a graded Hecke algebra or Drinfeld-Hecke algebra) if $\xi$ is an isomorphism.

If I did not make a mistake, the condition that $\xi$ is an isomorphism is equivalent to the following: for any linear section $s$ of the quotient map $\mathrm{T}(V) \rightarrow \mathrm{S}(V)$, the vector space morphism $\theta_s:\mathrm{S}(V) \otimes \mathbb{C} G \rightarrow A$, $x \otimes g \mapsto q(s(x) \otimes g)$, is an isomorphism, where $q:\mathrm{T}(V) \otimes \mathbb{C} G \rightarrow A$ is the quotient map.

Hence, a rational Cherednik algebra $A$ is $\mathrm{S}(V) \otimes \mathbb{C}G$ as a vector space, but only up to the choice of a linear section. Now, is there a canonical way to identify $A$ with $\mathrm{S}(V) \otimes \mathbb{C}G$?

This might be a little pedantic, but nobody ever mentions a choice of a section and I fear that I miss an important point here.

In an earlier version of my question I also asked if there is also a canonical way to view $S(V)$ as a subalgebra of $A$ but I see that this can't be true in general.

An additional (not totally unrelated) question: If the PBW morphism $\xi$ is an isomorphism, the algebra $A$ is already pretty nice because several properties of $\mathrm{S}(V) \sharp \mathbb{C}G$ (noetherian, prime, finite homolgical dimension, Cohen-Macaulay) are transported to $A$. But $\xi$ would in the same way as above already exist if $\kappa$ would be a map to $V \cdot \mathbb{C} G \subseteq \mathrm{T}(V) \sharp \mathbb{C}G$, i.e. if it would involve degree 1 elements. Why aren't those algebras interesting? (Perhaps because they don't have a triangular decomposition or they aren't deformations of $\mathrm{S}(V) \sharp \mathbb{C}G$?) I'm just looking for a natural reason to look at the $A_\kappa$ (I remember very well my question and the answers about reasons for studying rational Cherednik algebras but let's try it this way...)

Introduction: Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, let $G \leq \mathrm{GL}(V)$ be a finite subgroup and let $\kappa:V \times V \rightarrow \mathbb{C}G$ be an alternating bilinear map. Let $I$ be the ideal in $\mathrm{T}(V) \sharp \mathbb{C}G$ generated by the elements $vw-wv-\kappa(v,w)$ with $v,w \in V$. ($\sharp$ denotes the smash product). If we let elements of $V$ have degree $1$ and elements of $\mathbb{C}G$ have degree 0, we get a grading on $\mathrm{T}(V) \sharp \mathbb{C}G$ and so a filtration on $A := \mathrm{T}(V) \sharp \mathbb{C} G/I$. As $\mathrm{gr}(A)$ is commutative in degree 1, the quotient morphism $\mathrm{T}(V) \sharp \mathbb{C} G$ induces a surjective graded algebra morphism $\xi: \mathrm{S}(V) \sharp \mathbb{C} G \rightarrow \mathrm{gr}(A)$.

Now, $A$ is called a rational Cherednik algebra (or perhaps in this more general setting it's called a graded Hecke algebra or Drinfeld-Hecke algebra) if $\xi$ is an isomorphism.

If I did not make a mistake, the condition that $\xi$ is an isomorphism is equivalent to the following: for any linear section $s$ of the quotient map $\mathrm{T}(V) \rightarrow \mathrm{S}(V)$, the vector space morphism $\theta_s:\mathrm{S}(V) \otimes \mathbb{C} G \rightarrow A$, $x \otimes g \mapsto q(s(x) \otimes g)$, is an isomorphism, where $q:\mathrm{T}(V) \otimes \mathbb{C} G \rightarrow A$ is the quotient map.

Question: Hence, a rational Cherednik algebra $A$ is $\mathrm{S}(V) \otimes \mathbb{C}G$ as a vector space, but only up to the choice of a linear section. Now, is there a canonical way to identify $A$ with $\mathrm{S}(V) \otimes \mathbb{C}G$?

This might be a little pedantic, but nobody ever mentions a choice of a section and I fear that I miss an important point here. Here is a further problem: is there a canonical way to identify $\mathrm{S}(V)$ as a subalgebra of $A$? If I choose a basis of $V$ and let $s$ be the linear section defined by this choice, then $\theta_s$ embeds $\mathrm{S}(V)$ as a subalgebra in $A$. Here I don't even now if this works for any linear section (probably it does?).

An additional (not totally unrelated) question: If the PBW morphism $\xi$ is an isomorphism, the algebra $A$ is already pretty nice because several properties of $\mathrm{S}(V) \sharp \mathbb{C}G$ (noetherian, prime, finite homolgical dimension, Cohen-Macaulay) are transported to $A$. But $\xi$ would in the same way as above already exist if $\kappa$ would be a map to $V \cdot \mathbb{C} G \subseteq \mathrm{T}(V) \sharp \mathbb{C}G$, i.e. if it would involve degree 1 elements. Why aren't those algebras interesting? (Perhaps because they don't have a triangular decomposition or they are't be deformations of $\mathrm{S}(V) \sharp \mathbb{C}G$?) I'm just looking for a natural reason to look at the $A_\kappa$ (I remember very well my question and the answers about reasons for studying rational Cherednik algebras but let's try it this way...)

Introduction: Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, let $G \leq \mathrm{GL}(V)$ be a finite subgroup and let $\kappa:V \times V \rightarrow \mathbb{C}G$ be an alternating bilinear map. Let $I$ be the ideal in $\mathrm{T}(V) \sharp \mathbb{C}G$ generated by the elements $vw-wv-\kappa(v,w)$ with $v,w \in V$. ($\sharp$ denotes the smash product). If we let elements of $V$ have degree $1$ and elements of $\mathbb{C}G$ have degree 0, we get a grading on $\mathrm{T}(V) \sharp \mathbb{C}G$ and so a filtration on $A := \mathrm{T}(V) \sharp \mathbb{C} G/I$. As $\mathrm{gr}(A)$ is commutative in degree 1, the quotient morphism $\mathrm{T}(V) \sharp \mathbb{C} G$ induces a surjective graded algebra morphism $\xi: \mathrm{S}(V) \sharp \mathbb{C} G \rightarrow \mathrm{gr}(A)$.

Now, $A$ is called a rational Cherednik algebra (or perhaps in this more general setting it's called a graded Hecke algebra or Drinfeld-Hecke algebra) if $\xi$ is an isomorphism.

If I did not make a mistake, the condition that $\xi$ is an isomorphism is equivalent to the following: for any linear section $s$ of the quotient map $\mathrm{T}(V) \rightarrow \mathrm{S}(V)$, the vector space morphism $\theta_s:\mathrm{S}(V) \otimes \mathbb{C} G \rightarrow A$, $x \otimes g \mapsto q(s(x) \otimes g)$, is an isomorphism, where $q:\mathrm{T}(V) \otimes \mathbb{C} G \rightarrow A$ is the quotient map.

Hence, a rational Cherednik algebra $A$ is $\mathrm{S}(V) \otimes \mathbb{C}G$ as a vector space, but only up to the choice of a linear section. Now, is there a canonical way to identify $A$ with $\mathrm{S}(V) \otimes \mathbb{C}G$?

This might be a little pedantic, but nobody ever mentions a choice of a section and I fear that I miss an important point here. Here is a further problem: is there a canonical way to identify $\mathrm{S}(V)$ as a subalgebra of $A$? If I choose a basis of $V$ and let $s$ be the linear section defined by this choice, then $\theta_s$ embeds $\mathrm{S}(V)$ as a subalgebra in $A$. Here I don't even now if this works for any linear section (probably it does?).

An additional (not totally unrelated) question: If the PBW morphism $\xi$ is an isomorphism, the algebra $A$ is already pretty nice because several properties of $\mathrm{S}(V) \sharp \mathbb{C}G$ (noetherian, prime, finite homolgical dimension, Cohen-Macaulay) are transported to $A$. But $\xi$ would in the same way as above already exist if $\kappa$ would be a map to $V \cdot \mathbb{C} G \subseteq \mathrm{T}(V) \sharp \mathbb{C}G$, i.e. if it would involve degree 1 elements. Why aren't those algebras interesting? (Perhaps because they don't have a triangular decomposition or they aren't deformations of $\mathrm{S}(V) \sharp \mathbb{C}G$?) I'm just looking for a natural reason to look at the $A_\kappa$ (I remember very well my question and the answers about reasons for studying rational Cherednik algebras but let's try it this way...)