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

Journal of Physics: Conference Series. 2013. Vol. 474. No. 012026. P. 1-14.

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 Fréchet algebras O(D^n) and O(B^n) of holomorphic functions are not topologically isomorphic. Our goal is to prove a noncommutative version of the above result. Given a nonzero complex number q, we define two noncommutative power series algebras O_q(D^n) and O_q(B^n) which can be viewed as q-analogs of O(D^n) and O(B^n), respectively. Both O_q(D^n) and O_q(B^n) are the completions of the algebraic quantum affine space w.r.t. certain families of seminorms. In the case where 0 < q < 1, the algebra O_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 O_q(D^n) and O_q(B^n) can be interpreted as Fréchet algebra deformations (in a suitable sense) of O(D^n) and O(B^n), respectively. Our main result is that O_q(D^n) and O_q(B^n) are not isomorphic if n ≥ 2 and |q| = 1, but are isomorphic if |q| ≠ 1.