Abstract
We construct a family of right coideal subalgebras of quantum groups, which have the property that all irreducible representations are one-dimensional, and which are maximal with this property. The obvious examples for this are the standard Borel subalgebras expected from Lie theory, but in a quantum group there are many more. Constructing and classifying them is interesting for structural reasons, and because they lead to unfamiliar induced (Verma-)modules for the quantum group. The explicit family we construct in this article consists of quantum Weyl algebras combined with parts of a standard Borel subalgebra, and they have a triangular decomposition. Our main result is proving their Borel subalgebra property. Conversely we prove under some restrictions a classification result, which characterizes our family. Moreover we list for Uq(4) all possible triangular Borel subalgebras, using our underlying results and additional by-hand arguments. This gives a good working example and puts our results into context.
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Acknowledgements
Both authors thank Istvan Heckenberger for answering questions and giving valuable impulses. Another valuable occasion was the Micro-Workshop on Quantum Symmetric Pairs with Stefan Kolb (Hamburg, February 2017), and we also thank the RTG 1670 of the University of Hamburg for support and hospitality. We thank Giovanna Carnovale for proving for us Lemma 3.26. We thank the referee for very valuable remarks and corrections.
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Presented by: Kenneth Goodearl
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Lentner, S., Vocke, K. A Family of New Borel Subalgebras of Quantum Groups. Algebr Represent Theor 24, 473–503 (2021). https://doi.org/10.1007/s10468-020-09956-y
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DOI: https://doi.org/10.1007/s10468-020-09956-y