Pick $\dim V=2$, $G$ trivial, and $\kappa$ a symplectic form. Then $A$ is the first Weyl algebra, which does not contain any commutative algebra isomorphic to $S(V)$. The answer to your further problem is thus no.

The algebras you mention in your related problem are indeed worthy of study! They are deformations of $S(V)\\#\mathbb CG$, and some of them are even PBW deformations. You can write down the explicit condition for this to happen by unrolling the so called Jacobi condition; you'll find this discussed, for example, in Roland Berger and Victor Ginzburg's paper on non homogeneous PBW deformations of $N$-Koszul algebras (you case is quadratic, so simpler, but they discuss it it)

As for your pedantic point, I don't know. I guess you want someting like the symmetrization map that exists for enveloping algebras?

Pick $\dim V=2$, $G$ trivial, and $\kappa$ a symplectic form. Then $A$ is the first Weyl algebra, which does not contain any commutative algebra isomorphic to $S(V)$. The answer to your further problem is thus no. A reference for this result is [Dixmier, Jacques. Sur les algèbres de Weyl. (French) Bull. Soc. Math. France 96 1968 209--242. MR0242897 (39 #4224)], where Dixmier shows that whenever $x$ is a non-scalar element in $A_1$ the centralizer $C(x)$ of $x$ in $A_1$ is a module of finite type over $k[x]\subseteq A_1$.

The algebras you mention in your related problem are indeed worthy of study! They are deformations of $S(V)\\#\mathbb CG$, and some of them are even PBW deformations. You can write down the explicit condition for this to happen by unrolling the so called Jacobi condition; you'll find this discussed, for example, in Roland Berger and Victor Ginzburg's paper on non homogeneous PBW deformations of $N$-Koszul algebras (you case is quadratic, so simpler, but they discuss it it)

As for your pedantic point, I don't know. I guess you want someting like the symmetrization map that exists for enveloping algebras?

Pick $\dim V=2$, $G$ trivial, and $\kappa$ a symplectic form. Then $A$ is the first Weyl algebra, which does not contain any commutative algebra isomorphic to $S(V)$. The answer to your further problem is thus no.

Pick $\dim V=2$, $G$ trivial, and $\kappa$ a symplectic form. Then $A$ is the first Weyl algebra, which does not contain any commutative algebra isomorphic to $S(V)$. The answer to your further problem is thus no.

The algebras you mention in your related problem are indeed worthy of study! They are deformations of $S(V)\\#\mathbb CG$, and some of them are even PBW deformations. You can write down the explicit condition for this to happen by unrolling the so called Jacobi condition; you'll find this discussed, for example, in Roland Berger and Victor Ginzburg's paper on non homogeneous PBW deformations of $N$-Koszul algebras (you case is quadratic, so simpler, but they discuss it it)

As for your pedantic point, I don't know. I guess you want someting like the symmetrization map that exists for enveloping algebras?