Lowering-Raising triples and $U_q({\mathfrak{sl}}_2)$

Abstract

We introduce the notion of a lowering-raising (or LR) triple of linear transformations on a nonzero finite-dimensional vector space. We show how to normalize an LR triple, and classify up to isomorphism the normalized LR triples. We describe the LR triples using various maps, such as the reflectors, the inverters, the unipotent maps, and the rotators. We relate the LR triples to the equitable presentation of the quantum algebra $U_q({\mathfrak{sl}}_2)$ and Lie algebra ${\mathfrak{sl}}_2$.

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Cited by in corpus (3)

• Linear transformations that are tridiagonal with respect to the three decompositions for an LR triple
• An LR pair that can be extended to an LR triple
• The Lusztig automorphism of $U_q({\mathfrak{sl}}_2)$ from the equitable point of view