For tridiagonal pairs, the raising and lowering maps defined by the eigenspaces of A* and by the split decomposition are intertwined by a single bijection, with explicit formulas and rank consequences.
The Tetrahedron algebra and its finite-dimensional irreducible modules
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Recently Terwilliger and the present author found a presentation for the three-point $\mathfrak{sl}_2$ loop algebra via generators and relations. To obtain this presentation we defined a Lie algebra $\boxtimes$ by generators and relations and displayed an isomorphism from $\boxtimes$ to the three-point $\mathfrak{sl}_2$ loop algebra. In this paper we classify the finite-dimensional irreducible $\boxtimes$-modules.
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Raising and lowering maps for tridiagonal pairs
For tridiagonal pairs, the raising and lowering maps defined by the eigenspaces of A* and by the split decomposition are intertwined by a single bijection, with explicit formulas and rank consequences.