A Family of New Borel Subalgebras of Quantum Groups

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 quantu...

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Bibliographic Details
Published in:Algebras and representation theory 2021-04, Vol.24 (2), p.473-503
Main Authors: Lentner, Simon, Vocke, Karolina
Format: Article
Language:English
Subjects:
Associative Rings and Algebras
Classification
Commutative Rings and Algebras
Mathematics
Mathematics and Statistics
Non-associative Rings and Algebras
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Summary: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 U q ( 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.
ISSN:1386-923X
1572-9079
DOI:10.1007/s10468-020-09956-y