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

21st International Conference on Integrable Systems and Quantum Symmetries

Pirkovskii, A.Yu.^1

Faculty of Mathematics, National Research University Higher School of Economics, 7 Vavilova, 117312 Moscow, Russia^1

关键词: Affine space;    Continuous functions;    Holomorphic functions;    Non-commutative;    Poincare;    Power series;    Q-analogs;    Seminorms;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/474/1/012026/pdf
DOI  :  10.1088/1742-6596/474/1/012026

来源: IOP

【 摘 要 】

The classical Poincare´ theorem (1907) asserts that the polydisk Dnand the ball Bnin Cnare not biholomorphically equivalent for n ≥ 2. Equivalently, this means that the Fre´chet algebras O(Dn) and O(Bn) of holomorphic functions are not topologically isomorphic. Our goal is to prove a noncommutative version of the above result. Given q Ε C \ {0}, we define two noncommutative power series algebras Oq(Dn) and Oq(Bn), which can be viewed as q-analogs of O(Dn) and O(Bn), respectively. Both Oq(Dn) and Oq(Bn) are the completions of the algebraic quantum affine space Oregq(Cn) w.r.t. certain families of seminorms. In the case where 0 q(Bn) admits an equivalent definition related to L. L. Vaksmans algebra Cq(B¯n) of continuous functions on the closed quantum ball. We show that both Oq(Dn) and Oq(Bn) can be interpreted as Fre´chet algebra deformations (in a suitable sense) of O(Dn) and O(Bn), respectively. Our main result is that Oq(Dn) and Oq(Bn) are not isomorphic if n ≥ 2 and |q| = 1, but are isomorphic if |q| ≠ 1.