Cluster variables for affine Lie-Poisson systems. (English. Russian original) Zbl 07824560
Theor. Math. Phys. 217, No. 3, 1987-2004 (2023); translation from Teor. Mat. Fiz. 217, No. 3, 672-693 (2023).
Summary: We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all \(n_1+m\) sources are separated from all \(n_2+m\) sinks, we can construct a cluster-algebra realization of elements of an affine Lie-Poisson algebra \(R(\lambda,\mu){\stackrel{1}{T}}(\lambda){\stackrel{2}{T}}(\mu) ={\stackrel{2}{T}}(\mu){\stackrel{1}{T}}(\lambda)R(\lambda,\mu)\) with \(({n_1\times n_2})\)-matrices \(T(\lambda)\). Upon satisfaction of some invertibility conditions, we can construct a realization of a quantum loop algebra. Having the quantum loop algebra, we can also construct a realization of the twisted Yangian algebra or of the quantum reflection equation. For each such a planar network, we can therefore construct a symplectic leaf of the corresponding infinite-dimensional algebra.
MSC:
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 16T25 | Yang-Baxter equations |
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