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

SIGMA 15 (2019), 038, 33 pages      arXiv:1809.02951      https://doi.org/10.3842/SIGMA.2019.038

### The Laurent Extension of Quantum Plane: a Complete List of $U_q(\mathfrak{sl}_2)$-Symmetries

Sergey Sinel'shchikov
Mathematics Division, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., 61103 Kharkiv, Ukraine

Received September 11, 2018, in final form April 17, 2019; Published online May 09, 2019

Abstract
This work finishes a classification of $U_q(\mathfrak{sl}_2)$-symmetries on the Laurent extension $\mathbb{C}_q\big[x^{\pm 1},y^{\pm 1}\big]$ of the quantum plane. After reproducing the partial results of a previous paper of the author related to symmetries with non-trivial action of the Cartan generator(s) of $U_q(\mathfrak{sl}_2)$ and the generic symmetries, a complete collection of non-generic symmetries is presented. Together, these collections constitute a complete list of $U_q(\mathfrak{sl}_2)$-symmetries on $\mathbb{C}_q\big[x^{\pm 1},y^{\pm 1}\big]$.

Key words: quantum universal enveloping algebra; Hopf algebra; Laurent polynomial; quantum symmetry; weight.

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