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

Abstract

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 q ∈ C \ {0}, 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 O reg q (C n ) 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.

Noncommutative Kähler structures on quantum homogeneous spaces

“…One of the most exciting new trends in noncommutative geometry is the search for a theory of noncommutative complex geometry [17,1,31]. It is motivated by the appearance of noncommutative complex structures in a number of areas of noncommutative geometry, such as the construction of spectral triples for quantum groups [20,7,2], geometric representation theory for quantum groups [16,25,17,18], the interaction of noncommutative geometry and noncommutative projective algebraic geometry [17,18,1], the Baum-Connes conjecture for quantum groups [37,38], and the application of topological algebras to quantum group theory [34,33]. While there have been a number of occurrences of Kähler phenomena in the literature, the question of whether metrics have a role to play in noncommutative complex geometry remains largely unexplored.…”

The Arens-Michael envelopes of the Jordan plane and $U_q(\mathfrak{sl}(2))$

“…It is interesting to note that if we quantize the Weyl algebra by taking the commutation relation to be @x qx@ D 1 .q ¤ 0; 1/; the resulting Arens-Michael envelope would again be the algebra of "noncommutative" power series (see [6,Corollary 5.19]). In the case 0 < q < 1, this can also be explained by noting that the above q-Weyl algebra admits a different presentation as a quantum disk whose Arens-Michael envelope is an algebra of "noncommutative" power series; see [7,Sections 4 and 5]. The connection between quantum Weyl algebras and quantum balls and polydisks can be traced back to the work of Vaksman [11].…”