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Here's one way of looking at it.

Basically what's going on here is a $3{+}\epsilon$-dimensional TQFT, which corresponds to a 3-category with the right sort of duality.

$Rep(U_q(\mathfrak g))$ is such a 3-category. This is your first level. We could also plug in a type A Hecke algebra or the BMW algebra -- these also can be thought of as 3-categories.

If we pair our 3-category with a circle, we get a 2-category. In the Hecke algebra case this is closely related to the affine Hecke algebra.

If we pair our 3-category with a torus, we get a 1-category. In the Hecke algebra case this is closely related to the double affine Hecke algebra.

We can go one step further and pair the 3-category with a closed 3-manifold, yielding a vector space.

If we take $q=1$ then $Rep(U_q(\mathfrak g)) = Rep(U(\mathfrak g))$ is a symmetric monoidal category. We are now in the stable range. We can pair $Rep(U(\mathfrak g))$ with a manifold of any dimension and get another symmetric monoidal category. So in the $q=1$ case there is no problem with doing an $n$-tuple affine construction. Just pair $Rep(U(\mathfrak g))$ with the $n$-torus $T^n$.

References for the above? There are my 2005 TQFT notes, which talk about the general theory but not the specific examples I mention above. There is a paper in progress with Monica Vazirani, which talks about the affine level above in great detail. David Ben-Zvi, Adrien Brochier, and David Jordan have a different take on these ideas, and I think they have preprint(s) available somewhere. Perhaps one of them will chime in with another answer.

Here's one way of looking at it.

Basically what's going on here is a $3{+}\epsilon$-dimensional TQFT, which corresponds to a 3-category with the right sort of duality.

$Rep(U_q(\mathfrak g))$ is such a 3-category. This is your first level. We could also plug in a type A Hecke algebra (by which I really mean the HOMFYPT completion thereof) or the BMW algebra -- these also can be thought of as 3-categories.

If we pair our 3-category with a circle, we get a 2-category. In the Hecke algebra case this is closely related to the affine Hecke algebra.

If we pair our 3-category with a torus, we get a 1-category. In the Hecke algebra case this is closely related to the double affine Hecke algebra.

We can go one step further and pair the 3-category with a closed 3-manifold, yielding a 0-category, i.e. a vector space.

If we take $q=1$ then $Rep(U_q(\mathfrak g)) = Rep(U(\mathfrak g))$ is a symmetric monoidal category. We are now in the stable range. We can pair $Rep(U(\mathfrak g))$ with a manifold of any dimension and get another symmetric monoidal category. So in the $q=1$ case there is no problem with doing an $n$-tuple affine construction. Just pair $Rep(U(\mathfrak g))$ with the $n$-torus $T^n$.

References for the above? There are my 2005 TQFT notes, which talk about the general theory but not the specific examples I mention above. There is a paper in progress with Monica Vazirani, which talks about the affine level above in great detail. David Ben-Zvi, Adrien Brochier, and David Jordan have a different take on these ideas, and I think they have preprint(s) available somewhere. Perhaps one of them will chime in with another answer.

Here's one way of looking at it.

Basically what's going on here is a $3{+}\epsilon$-dimensional TQFT, which corresponds to a 3-category with the right sort of duality.

$Rep(U_q(\mathfrak g))$ is such a 3-category. This is your first level. We could also plug in a type A Hecke algebra or the BMW algebra -- these also can be thought of as 3-categories.

If we pair our 3-category with a circle, we get a 2-category. In the Hecke algebra case this is closely related to the affine Hecke algebra.

If we pair our 3-category with a torus, we get a 1-category. In the Hecke algebra case this is closely related to the double affine Hecke algebra.

We can go one step further and pair the 3-category with a closed 3-manifold, yielding a vector space.

If we take $q=1$ then $Rep(U_q(\mathfrak g)) = Rep(U(\mathfrak g))$ is a symmetric monoidal category. We are now in the stable range. We can pair $Rep(U(\mathfrak g))$ with a manifold of any dimension and get another symmetric monoidal category. So in the $q=1$ case there is no problem with doing an $n$-tuple affine construction. Just pair $Rep(U(\mathfrak g))$ with the $n$-torus $T^n$.

References for the above? There are my 2005 TQFT notes, which talk about the general theory but not the specific examples I mention above. There is a paper in progress with Monica Vazirani, which talks about the affine level above in great detail. David Ben-Zvi and David Jordan have a different take on these ideas, and I think they have preprint(s) available somewhere. Perhaps one of them will chime in with another answer.

Here's one way of looking at it.

Basically what's going on here is a $3{+}\epsilon$-dimensional TQFT, which corresponds to a 3-category with the right sort of duality.

$Rep(U_q(\mathfrak g))$ is such a 3-category. This is your first level. We could also plug in a type A Hecke algebra or the BMW algebra -- these also can be thought of as 3-categories.

If we pair our 3-category with a circle, we get a 2-category. In the Hecke algebra case this is closely related to the affine Hecke algebra.

If we pair our 3-category with a torus, we get a 1-category. In the Hecke algebra case this is closely related to the double affine Hecke algebra.

We can go one step further and pair the 3-category with a closed 3-manifold, yielding a vector space.

If we take $q=1$ then $Rep(U_q(\mathfrak g)) = Rep(U(\mathfrak g))$ is a symmetric monoidal category. We are now in the stable range. We can pair $Rep(U(\mathfrak g))$ with a manifold of any dimension and get another symmetric monoidal category. So in the $q=1$ case there is no problem with doing an $n$-tuple affine construction. Just pair $Rep(U(\mathfrak g))$ with the $n$-torus $T^n$.

References for the above? There are my 2005 TQFT notes, which talk about the general theory but not the specific examples I mention above. There is a paper in progress with Monica Vazirani, which talks about the affine level above in great detail. David Ben-Zvi, Adrien Brochier, and David Jordan have a different take on these ideas, and I think they have preprint(s) available somewhere. Perhaps one of them will chime in with another answer.

Here's one way of looking at it.

Basically what's going on here is a $3{+}\epsilon$-dimensional TQFT, which corresponds to a 3-category with the right sort of duality.

$Rep(U_q(\mathfrak g))$ is such a 3-category. This is your first level. We could also plug in a type A Hecke algebra or the BMW algebra -- these also can be thought of as 3-categories.

If we pair our 3-category with a circle, we get a 2-category. In the Hecke algebra case this is closely related to the affine Hecke algebra.

If we pair our 3-category with a torus, we get a 1-category. In the Hecke algebra case this is closely related to the double affine Hecke algebra.

We can go one step further and pair the 3-category with a closed 3-manifold, yielding a vector space.

If we take $q=1$ then $Rep(U_q(\mathfrak g)) = Rep(U(\mathfrak g))$ is a symmetric monoidal category. We are now in the stable range. We can pair $Rep(U(\mathfrak g))$ with a manifold of any dimension and get another symmetric monoidal category. So in the $q=1$ case there is no problem with doing an $n$-tuple affine construction. Just pair $Rep(U(\mathfrak g))$ with the $n$-torus $T^n$.

References for the above? There are my 2005 TQFT notes. There is a paper in progress with Monica Vazirani. David Ben-Zvi and David Jordan have a slightly different take on these ideas, and I think they have preprint(s) available somewhere. Perhaps one of them will chime in with another answer.

Here's one way of looking at it.

Basically what's going on here is a $3{+}\epsilon$-dimensional TQFT, which corresponds to a 3-category with the right sort of duality.

$Rep(U_q(\mathfrak g))$ is such a 3-category. This is your first level. We could also plug in a type A Hecke algebra or the BMW algebra -- these also can be thought of as 3-categories.

If we pair our 3-category with a circle, we get a 2-category. In the Hecke algebra case this is closely related to the affine Hecke algebra.

If we pair our 3-category with a torus, we get a 1-category. In the Hecke algebra case this is closely related to the double affine Hecke algebra.

We can go one step further and pair the 3-category with a closed 3-manifold, yielding a vector space.

If we take $q=1$ then $Rep(U_q(\mathfrak g)) = Rep(U(\mathfrak g))$ is a symmetric monoidal category. We are now in the stable range. We can pair $Rep(U(\mathfrak g))$ with a manifold of any dimension and get another symmetric monoidal category. So in the $q=1$ case there is no problem with doing an $n$-tuple affine construction. Just pair $Rep(U(\mathfrak g))$ with the $n$-torus $T^n$.

References for the above? There are my 2005 TQFT notes, which talk about the general theory but not the specific examples I mention above. There is a paper in progress with Monica Vazirani, which talks about the affine level above in great detail. David Ben-Zvi and David Jordan have a different take on these ideas, and I think they have preprint(s) available somewhere. Perhaps one of them will chime in with another answer.