Abstract
The coideal subalgebra of the quantum \(\mathfrak {sl}_2\) is a polynomial algebra in a generator t which depends on a parameter κ. The existence of the ı-canonical basis (also known as the ı-divided powers) for the coideal subalgebra of the quantum \(\mathfrak {sl}_2\) were established by Bao and Wang. We establish closed formulae for the ı-divided powers as polynomials in t and also in terms of Chevalley generators of the quantum \(\mathfrak {sl}_2\) when the parameter κ is an arbitrary q-integer. The formulae were known earlier when κ = 0, 1.
Dedicated to Vyjayanthi Chari for her 60th birthday with admiration
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Acknowledgements
WW thanks Huanchen Bao for his insightful collaboration. The formula in the Appendix for the second ı-divided power with arbitrary parameter κ (which was obtained with help from Huanchen) was crucial to this project, and to a large extent this paper grows by exploring for what values for the parameter κ reasonable formulae for higher divided powers can be obtained. The research of WW and the undergraduate research of CB are partially supported by a grant DMS-1702254 from National Science Foundation. Mathematica was used intensively in this work.
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Wang, W., Berman, C. (2021). Formulae of ı-Divided Powers in \({\mathbf {U}}_q(\mathfrak {sl}_2)\), II. In: Greenstein, J., Hernandez, D., Misra, K.C., Senesi, P. (eds) Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification. Progress in Mathematics, vol 337. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-63849-8_7
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DOI: https://doi.org/10.1007/978-3-030-63849-8_7
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