$q$-Inverting pairs of linear transformations and the $q$-tetrahedron algebra

As part of our study of the $q$-tetrahedron algebra ${⊠}_q$ we introduce the notion of a $q$-inverting pair. Roughly speaking, this is a pair of invertible semisimple linear transformations on a finite-dimensional vector space, each of which acts on the eigenspaces of the other according to a certain rule. Our main result is a bijection between the following two sets: (i) the isomorphism classes of finite-dimensional irreducible ${⊠}_q$-modules of type 1; (ii) the isomorphism classes of $q$-inverting pairs.