Summary
This paper is the second part of our paper “Cluster ensembles, quantization, and the dilogarithm” [FG2].1 Its main result is a construction, by means of the quantum dilogarithm, of certain intertwiner operators, which play a crucial role in the quantization of the cluster \(\mathcal X\)-varieties and construction of the corresponding canonical representation.
2000 Mathematics Subject Classifications: 17,337, 11655, 16W35 81R10, 33E30
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Notes
- 1.
Both papers were originally combined into a single ArXiv preprint math/0311245, version 1.
- 2.
The precise meaning we put into “intertwining” is clear from the computation carried out in the proof: we deal with the generators of the Heisenberg algebra only, and thus are not concerned with the nature of the topological completion, which was left unspecified in Section 3. Furthermore, we understand the intertwining property formally without paying attention to the domains of the definition of the generators of the Heisenberg algebra. For a different approach see [FG3].
- 3.
See also a more transparent and less computational definition given in Section 3 of [FG3].
References
Barnes E.W., The genesis of the double gamma function, Proc. London Math. Soc. 31 (1899) 358–381.
Baxter R., Exactly solved models in statistical mechanics, Academic Press (1982).
Berenstein A., Fomin S., Zelevinsky A., Cluster algebras III, Upper bounds and double Bruhat cells, math.RT/0305434. J. Amer. Math. Soc. 15 (2002), no. 2, 497–529.
Faddeev L.D., Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995), no. 3, 249–254.
Fock, V. V., Chekhov, L. O., Quantum Teichmüller spaces, Teoret. Mat. Fiz. 120 (1999), no. 3, 511–528.
Fock V.V., Goncharov A.B., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. IHES, n. 103 (2006) 1–212. math.AG/0311149.
Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm I, Ann Scient de Ec. Norm. Sup., t. 42, N6, (2009), 865–929. math.AG/0311245.
Fock V.V., Goncharov A.B., The quantum dilogarithm and representation of quantam cluster varieties, math.QA/0702397.
Fomin S., Zelevinsky A., Cluster algebras. I, Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529.
Kashaev R.M., Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998), no. 2, 105–115.
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Dedicated to Yu. I. Manin for his 70th Birthday
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Fock, V.V., Goncharov, A.B. (2009). Cluster Ensembles, Quantization and the Dilogarithm II: The Intertwiner. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_15
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