Suppose I have an algebra $ A_K $ set on $ K = mathbb {C} (x, y) $. For example, let's say $ A_K $ is generated by $ E, F $ with relation $$ (E, F) = frac {x ^ a-x ^ {a}} {y ^ b-y ^ {- b}} quad (*) $$

or $ a, b in mathbb {Z} $, depend on $ E $ and $ F $. I'm trying to define a specialization of $ A_K $ to an algebra $ A _ { mathbb {C}} $ more than $ mathbb {C} $, in which I force the relationship $$ (E, F) = frac {a} {b} $$

hold, instead of $ (*) $.

What I did, I considered algebra $ mathbb {C} (x ^ { pm}, y ^ { pm1}), $ generated by $ E, F $ with relation $ (*) $. Then I specialize $ x = y $, to get an algebra $ A_x $ more than $ mathbb {C} (x ^ { pm1}) $ with relation $ (*) $ to become $$ (E, F) = frac {x ^ a-x ^ {a}} {x ^ b-x ^ {- b}}. $$

Now i get my algebra $ A_ mathbb {C} $ by specializing $ x = $ 1.

My question is. Is this specialization mathematically correct? Is there a way to formalize it?