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Characteristic classes of $$SL(N, \mathbb C)$$-bundles and quantum dynamical elliptic R-matrices. (English) Zbl 1273.81117
Dynamical elliptic $$R$$-matrices constitute a kind of intermediary step between vertex and dynamical $$R$$-matrices. In this paper, quantum dynamical elliptic $$R$$-matrices related to simple complex Lie groups are analyzed in detail. In order to describe the $$R$$-matrices, the characteristic classes of the underlying vector bundles are considered. The authors consider the case of the Lie group $$SL(n,\mathbb C)$$ in detail, separating the analysis in dependence of the value of $$n$$. If the latter is prime, only two types exist: the Baxter-Belavin-Drinfeld-Sklyanin vertex $$R$$-matrix and the dynamical Felder $$R$$-matrix, while for other values new types of $$R$$-matrices emerge. Their explicit construction is the main objective of this work, also verifying the validity of the QDYB equation. Interesting applications of the results to integrable systems, specifically the IRF models, are discussed.

##### MSC:
 81R12 Groups and algebras in quantum theory and relations with integrable systems 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 16T25 Yang-Baxter equations 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 55R40 Homology of classifying spaces and characteristic classes in algebraic topology 17B20 Simple, semisimple, reductive (super)algebras 22E70 Applications of Lie groups to the sciences; explicit representations
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