Cartan subalgebras for quantum symmetric pair coideals

Letzter, Gail

doi:10.1090/ert/523

Cartan subalgebras for quantum symmetric pair coideals

Author: Gail Letzter
Journal: Represent. Theory 23 (2019), 88-153
MSC (2010): Primary 17B37; Secondary 17B10, 17B22
DOI: https://doi.org/10.1090/ert/523
Published electronically: January 31, 2019
MathSciNet review: 3904162
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Abstract: There is renewed interest in the coideal subalgebras used to form quantum symmetric pairs because of recent discoveries showing that they play a fundamental role in the representation theory of quantized enveloping algebras. However, there is still no general theory of finite-dimensional modules for these coideals. In this paper, we establish an important step in this direction: we show that every quantum symmetric pair coideal subalgebra admits a quantum Cartan subalgebra which is a polynomial ring that specializes to its classical counterpart. The construction builds on Kostant and Sugiura’s classification of Cartan subalgebras for real semisimple Lie algebras via strongly orthogonal systems of positive roots. We show that these quantum Cartan subalgebras act semisimply on finite-dimensional unitary modules and identify particularly nice generators of the quantum Cartan subalgebra for a family of examples.

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