Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 073, 14 pages      arXiv:2002.03439      https://doi.org/10.3842/SIGMA.2020.073
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

Nonstandard Quantum Complex Projective Line

Nicola Ciccoli ^a and Albert Jeu-Liang Sheu ^b
a) Dipartimento di Matematica e Informatica, University of Perugia, Italy
^b) Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA

Received March 06, 2020, in final form July 24, 2020; Published online August 03, 2020

Abstract
In our attempt to explore how the quantum nonstandard complex projective spaces $\mathbb{C}P_{q,c}^{n}$ studied by Korogodsky, Vaksman, Dijkhuizen, and Noumi are related to those arising from the geometrically constructed Bohr-Sommerfeld groupoids by Bonechi, Ciccoli, Qiu, Staffolani, and Tarlini, we were led to establish the known identification of $C\big(\mathbb{C}P_{q,c}^{1}\big) $ with the pull-back of two copies of the Toeplitz $C^*$-algebra along the symbol map in a more direct way via an operator theoretic analysis, which also provides some interesting non-obvious details, such as a prominent generator of $C\big( \mathbb{C}P_{q,c}^{1}\big) $ being a concrete weighted double shift.

Key words: quantum homogeneous space; Toeplitz algebra; weighted shift.

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