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

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