Finite dimensional modules for the \(q\)-tetrahedron algebra
Let \(B\) the complex Laurent polynomial algebra in one variable \(t\) and \(A\) the localisation of \(B\) by \(t-1\). The thetrahedron algebra \(\boxtimes\) is the complex Lie algebra \(sl_2\otimes A\) with the Lie bracketing \[ [u\otimes a,v\otimes b]= [u,v]\otimes ab. \] The Lie algebra \(L(sl_2)\) is the Lie subalgebra \(sl_2\otimes B\) in \(sl_2\otimes A\). In 2007 \textit{T. Ito} and \textit{P. Terwilliger} [Commun. Algebra 35, No. 11, 3415--3439 (2007; Zbl 1133.17011)] introduced \(q\)-analog \(\boxtimes_q\) of the algebra \(\boxtimes\). The aim of the paper is establish relations between finite dimensional modules over \(\boxtimes_q\) and finite dimensional modules over \(U_q(L(sl_2)).\) There relations modify similar relations between modules over \(\boxtimes\) and over the Onsager algebra.
scientific article; zbMATH DE number 5770027
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