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

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

2006-06-10

arxiv.org/abs/math/0606237v1

Mathematics

Representation Theory

19 pages

Scientific paper

As part of our study of the $q$-tetrahedron algebra $\boxtimes_q$ 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 $\boxtimes_q$-modules of type 1; (ii) the isomorphism classes of $q$-inverting pairs.

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