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Finite-dimensional simple Poisson modules. (English) Zbl 1362.17042
Summary: We classify the finite-dimensional simple Poisson modules for two Poisson algebras. The first is related to the invariants for an automorphism of the torus and to the cyclically \(q\)-deformed algebra \(U'_q(\mathfrak{so}_3)\) of [M. Havliček et al., J. Math. Phys. 40, No. 4, 2135–2161 (1999; Zbl 0959.17015); 42, No. 1, 472–500 (2001; Zbl 1032.17022)]. We find that there are five \(d\)-dimensional simple Poisson modules for each \(d\geq 1\). The second is the Poisson algebra arising from the quantized enveloping algebra \(U_q(\mathfrak{sl}_2)\) using a presentation discovered by Ito, Terwilliger and Weng [T. Ito et al., J. Algebra 298, No. 1, 284–301 (2006; Zbl 1090.17004)] and we find that there are two \(d\)-dimensional simple Poisson modules for each \(d\geq 1\).
17B63 Poisson algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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