Quantum polydisk, quantum ball, and a q-analog of Poincare's theorem

摘要

The classical Poincaré theorem (1907) asserts that the polydisk D~n and the ball B~n in C~n are not biholomorphically equivalent for n ≥ 2. Equivalently, this means that the Frechet algebras (D~n) and (B~n) of holomorphic functions are not topologically isomorphic. Our goal is to prove a noncommutative version of the above result. Given q C {0}, we define two noncommutative power series algebras q(D~n) and _q(B~n), which can be viewed as q-analogs of (D~n) and (B~n), respectively. Both _q(D_n) and _q (B~n) are the completions of the algebraic quantum affine space (C~n) w.r.t. certain families of seminorms. In the case where 0 < q < 1, the algebra _q(B~n) admits an equivalent definition related to L. L. Vaksman's algebra C_q(B~n) of continuous functions on the closed quantum ball. We show that both _q(D~n) and _q(B~n) can be interpreted as Frechet algebra deformations (in a suitable sense) of (D~n) and (Bn) respectively. Our main result is that _(q(Dn)) and _q(B~n) are not isomorphic if n ≥ 2 and |q| = 1, but are isomorphic if |q| ≠ 1.