  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
  $\mathfrak{sl}_2$.

Quantum Algebra