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Cluster variables for affine Lie–Poisson systems

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Abstract

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.

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Acknowledgments

The author is grateful to Michael and Alexander Shapiro for the useful discussion.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    L. O. Chekhov

  2. Michigan State University, East Lansing, USA

    L. O. Chekhov

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  1. L. O. Chekhov
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Correspondence to L. O. Chekhov.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 217, pp. 672–693 https://doi.org/10.4213/tmf10632.

Appendix: Commutation relations for $$T^-_2$$

We calculate the commutation relations between two matrices

$$T^-_2:=M_{22}[M{}_{12}^{-1}]^2 M_{11}.$$
We also use \(T^-_3:=M_{22}[M{}_{12}^{-1}]^3 M_{11}\) and \(T^-_1:=M_{22}M{}_{12}^{-1} M_{11}\):
$$ \begin{aligned} \, R^{-\mathrm{T}}& \stackrel {1}{M}_{22} \stackrel {1}{M}{}_{12}^{-2} \stackrel {1}{M}_{11} \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-2} \stackrel {2}{M}_{11}={} \\ &=R^{-\mathrm{T}} \stackrel {1}{M}_{22} \stackrel {1}{M}{}_{12}^{-2} \stackrel {2}{M}_{22} \stackrel {1}{M}_{11} \stackrel {2}{M}{}_{12}^{-2} \stackrel {2}{M}_{11}+{} \\ &\hphantom{={}}+ R^{-\mathrm{T}} \stackrel {1}{M}_{22} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} (q-q^{-1}) \stackrel {2}{M}_{21} \stackrel {1}{M}_{12} P \stackrel {2}{M}{}_{12}^{-2} \stackrel {2}{M}_{11}={} \\ &= R^{-\mathrm{T}} \stackrel {1}{M}_{22} \stackrel {2}{M}_{22} [R^{-1} \stackrel {1}{M}{}_{12}^{-1}]^2 \stackrel {2}{M}{}_{12}^{-1} R^{-\mathrm{T}} \stackrel {2}{M}{}_{12}^{-1} R^{-\mathrm{T}} \stackrel {1}{M}_{11} \stackrel {2}{M}_{11}+{} \\ &\hphantom{={}}+R^{-\mathrm{T}} \stackrel {1}{M}_{22} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} (q-q^{-1}) \stackrel {2}{M}_{21} \stackrel {1}{M}_{12} P \stackrel {2}{M}{}_{12}^{-2} \stackrel {2}{M}_{11}={} \\ &= \stackrel {2}{M}_{22} \stackrel {1}{M}_{22}R^{-\mathrm{T}} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1}R^{-1} \stackrel {1}{M}{}_{12}^{-1}R^{-1} R^{-\mathrm{T}} \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}_{11} \stackrel {1}{M}_{11}R^{-\mathrm{T}} +{} \\ &\hphantom{={}}+ R^{-\mathrm{T}} \stackrel {1}{M}_{22} (q-q^{-1}) \stackrel {2}{M}_{21} P \stackrel {2}{M}{}_{12}^{-3} \stackrel {2}{M}_{11}={} \\ &= \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{22} \stackrel {1}{M}{}_{12}^{-1}R^{-1} \stackrel {1}{M}{}_{12}^{-1} [I- (q-q^{-1}) R^{-1}P] \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}_{11} \stackrel {1}{M}_{11}R^{-\mathrm{T}} +{} \\ &\hphantom{={}}+R^{-\mathrm{T}} \stackrel {1}{M}_{22} (q-q^{-1}) \stackrel {2}{M}_{21} P \stackrel {2}{M}{}_{12}^{-3} \stackrel {2}{M}_{11}={} \\ &= \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{22} \stackrel {1}{M}{}_{12}^{-1}R^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}_{11} \stackrel {1}{M}_{11}R^{-\mathrm{T}} -{} \\ &\hphantom{={}}- (q-q^{-1}) \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{22} \stackrel {1}{M}{}_{12}^{-1}R^{-1} \stackrel {1}{M}{}_{12}^{-1} R^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}_{11} \stackrel {2}{M}_{11}R^{-1}P +{} \\ &\hphantom{={}}+(q-q^{-1}) R^{-\mathrm{T}} \stackrel {1}{M}_{22} \stackrel {2}{M}_{21} P \stackrel {2}{M}{}_{12}^{-3} \stackrel {2}{M}_{11}={} \\ &= \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{22} \stackrel {1}{M}{}_{12}^{-1} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} R^{-1} \stackrel {2}{M}_{11} \stackrel {1}{M}_{11}R^{-\mathrm{T}}-{} \\ &\hphantom{={}}- (q-q^{-1}) \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{22} \stackrel {2}{M}_{11} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}_{11} P +{} \\ &\hphantom{={}}+(q-q^{-1}) R^{-\mathrm{T}} \stackrel {1}{M}_{22} \stackrel {2}{M}_{21} P \stackrel {2}{M}{}_{12}^{-3} \stackrel {2}{M}_{11}={} \\ &= \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{22} R \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} R^{-1} \stackrel {2}{M}_{11} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}_{11}R^{-\mathrm{T}} +{} \\ &\hphantom{={}}+ (q-q^{-1})^2 \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{21} \stackrel {2}{M}_{12} P [ \stackrel {1}{M}{}_{12}^{-1}]^3 \stackrel {1}{M}_{11} P+{} \\ &\hphantom{={}}+(q-q^{-1}) R^{-\mathrm{T}} \stackrel {1}{M}_{22} \stackrel {2}{M}_{21} P \stackrel {2}{M}{}_{12}^{-3} \stackrel {2}{M}_{11} - (q-q^{-1}) \stackrel {2}T{}^-_1 \stackrel {1}T{}^-_3P={} \\ &= \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{22} R \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}_{11} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}_{11}R^{-\mathrm{T}} +{} \\ &\hphantom{={}}+ (q-q^{-1})[ (q-q^{-1})P+ R^{-\mathrm{T}}] \stackrel {1}{M}_{22} \stackrel {2}{M}_{21} [ \stackrel {1}{M}{}_{12}^{-1}]^3 \stackrel {1}{M}_{11} P - (q-q^{-1}) \stackrel {2}T{}^-_1 \stackrel {1}T{}^-_3P={} \\ &= \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{22} [ (q-q^{-1})P+ R^{-\mathrm{T}}] \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}_{11} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}_{11}R^{-\mathrm{T}} +{} \\ &\hphantom{={}}+ (q-q^{-1})R \stackrel {1}{M}_{22} \stackrel {2}{M}_{21} [ \stackrel {1}{M}{}_{12}^{-1}]^3 \stackrel {1}{M}_{11} P - (q-q^{-1}) \stackrel {2}T{}^-_1 \stackrel {1}T{}^-_3P={} \end{aligned}$$
$$ \begin{aligned} &= \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{22}R^{-\mathrm{T} \stackrel {2}{M}{}_{12}^{-1} } \stackrel {2}{M}_{11} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}_{11}R^{-\mathrm{T}} +{} \\ &\hphantom{={}}+ (q-q^{-1})P \stackrel {1}{M}_{22} \stackrel {1}{M}{}_{12}^{-1}( \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}_{11}) \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}_{11}R^{-\mathrm{T}} +{} \\ &\hphantom{={}}+ (q-q^{-1})R \stackrel {1}{M}_{22} \stackrel {2}{M}_{21} [ \stackrel {1}{M}{}_{12}^{-1}]^3 \stackrel {1}{M}_{11} P - (q-q^{-1}) \stackrel {2}T{}^-_1 \stackrel {1}T{}^-_3P={} \\ &= \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{22} \stackrel {2}{M}_{11} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}_{11}R^{-\mathrm{T}} +{} \\ &\hphantom{={}}+ (q-q^{-1})P \stackrel {1}{M}_{22}[ \stackrel {1}{M}{}_{12}^{-1} ]^3 ( \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}_{11}) \stackrel {1}{M}_{11}R^{-\mathrm{T}} +{} \\ &\hphantom{={}}+ (q-q^{-1}) \stackrel {2}{M}_{21} \stackrel {1}{M}_{22}[ \stackrel {1}{M}{}_{12}^{-1}]^3 \stackrel {1}{M}_{11} P - (q-q^{-1}) \stackrel {2}T{}^-_1 \stackrel {1}T{}^-_3P={} \\ &= \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}_{11} \stackrel {1}{M}_{22} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}_{11}R^{-\mathrm{T}} -{} \\ &\hphantom{={}}-(q-q^{-1}) \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}{}_{12}^{-1} \stackrel {1}{M}_{21} \stackrel {2}{M}_{12} P \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}{}_{12}^{-1} \stackrel {1}{M}_{11}R^{-\mathrm{T}}+{} \\ &\hphantom{={}} + (q-q^{-1})P \stackrel {1}{M}_{22}[ \stackrel {1}{M}{}_{12}^{-1} ]^3 \stackrel {1}{M}_{11} \stackrel {2}{M}_{22} \stackrel {2}{M}{}_{12}^{-1} \stackrel {2}{M}_{11} -{} \\ &\hphantom{={}}- (q-q^{-1})^2P \stackrel {1}{M}_{22}[ \stackrel {1}{M}{}_{12}^{-1} ]^3 \stackrel {2}{M}_{21} \stackrel {1}{M}_{11}P- (q-q^{-1})[ \stackrel {2}T{}^-_1- \stackrel {2}{M}_{21} ] \stackrel {1}T{}^-_3P ={} \\ &={} \stackrel {2}T{}^-_{2} \stackrel {1}T{}^-_{2}R^{-\mathrm{T}} -(q-q^{-1})P \stackrel {1}{M}_{22}[ \stackrel {1}{M}{}_{12}^{-1}]^3 \stackrel {2}{M}_{21} \stackrel {1}{M}_{11} [R^{-\mathrm{T}}+{} \\ &\hphantom{={}}+(q-q^{-1})P] + (q-q^{-1})P \stackrel {1}T{}^-_{3} \stackrel {2}T{}^-_{1} - (q-q^{-1})[ \stackrel {2}T{}^-_1- \stackrel {2}{M}_{21} ] \stackrel {1}T{}^-_3P={} \\ &= {} \stackrel {2}T{}^-_{2} \stackrel {1}T{}^-_{2}R^{-\mathrm{T}} -(q-q^{-1})P \stackrel {1}{M}_{22}[ \stackrel {1}{M}{}_{12}^{-1}]^3 \stackrel {2}{M}_{21} \stackrel {1}{M}_{11} R + (q-q^{-1})P \stackrel {1}T{}^-_{3} \stackrel {2}T{}^-_{1} -{} \\ &\hphantom{={}}- (q-q^{-1})[ \stackrel {2}T{}^-_1- \stackrel {2}{M}_{21} ] \stackrel {1}T{}^-_3P={} \\ &= {}\stackrel {2}T{}^-_{2} \stackrel {1}T{}^-_{2}R^{-\mathrm{T}} -(q-q^{-1})P \stackrel {1}T{}^-_{3} \stackrel {2}{M}_{21} + (q-q^{-1})P \stackrel {1}T{}^-_{3} \stackrel {2}T{}^-_{1} -{} \\ &\hphantom{={}}- (q-q^{-1})[ \stackrel {2}T{}^-_1- \stackrel {2}{M}_{21} ] \stackrel {1}T{}^-_3P={} \\ &= {}\stackrel {2}T{}^-_{2} \stackrel {1}T{}^-_{2}R^{-\mathrm{T}} + (q-q^{-1})P \stackrel {1}T{}^-_{3}[ \stackrel {2}T{}^-_{1}- \stackrel {2}{M}_{21}] - (q-q^{-1})[ \stackrel {2}T{}^-_1- \stackrel {2}{M}_{21} ] \stackrel {1}T{}^-_3P . \end{aligned}$$
Note that if the groupoid condition
$$M_{22}M_{12}^{-1}M_{11}-M_{21}=0,$$
is satisfied, we obtain homogeneous commutation relations
$$R^{-\mathrm{T}} \stackrel{1} {T}{}^-_{2} \stackrel{2} {T}{}^-_{2} = \stackrel{2} {T}{}^-_{2} \stackrel{1} {T}{}^-_{2}R^{-\mathrm{T}}.$$

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Chekhov, L.O. Cluster variables for affine Lie–Poisson systems. Theor Math Phys 217, 1987–2004 (2023). https://doi.org/10.1134/S0040577923120140

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