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Cluster structures on quantum coordinate rings,
 Selecta Math. (N.S.)
, 2013
"... Abstract We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric KacMoody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory C w of the module category of t ..."
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Abstract We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric KacMoody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory C w of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities which can be viewed as a qanalogue of a T system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.
The Pentagram map and YPatterns
 Adv. in Math
"... Abstract. The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its “shortest ” diagonals, and output the smaller polygon which they cut out. We employ the machinery of cluster algebras to obtain explicit formulas for the iterat ..."
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Abstract. The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its “shortest ” diagonals, and output the smaller polygon which they cut out. We employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map. Résumé. L’application pentagramme de R. Schwartz est définie par la construction suivante: on trace les diagonales “les plus courtes ” d’un polygone donné en entrée et on retourne en sortie le plus petit polygone que ces diagonales découpent. Nous employons la machinerie des algèbres “clusters ” pour obtenir des formules explicites pour les itérations de l’application pentagramme.
Thermodynamic Bethe ansatz for nonequilibrium steady states: exact energy current and fluctuations in integrable QFT
"... We evaluate the exact energy current and scaled cumulant generating function (related to the largedeviation function) in nonequilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various ..."
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We evaluate the exact energy current and scaled cumulant generating function (related to the largedeviation function) in nonequilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of D. Bernard and B. Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures Tl, Tr, and waiting for a long time. We evaluate the current J(Tl, Tr) using the exact QFT density matrix describing these nonequilibrium steady states and using Al.B. Zamolodchikov’s method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the hightemperature limit. We analyze numerically our nonequilibrium steadystate TBA equations for three models: the sinhGordon model, the roaming trajectories model, and the sineGordon model at a particular reflectionless point. Based on the numerics, we conjecture that an infinite family of nonequilibrium cfunctions, associated to the scaled cumulants, can be defined, which we interpret physically. We study the full scaled distribution function and find that it can be described by a set of independent Poisson processes. Finally, we show that the “additivity ” property of the current, which is known to hold in CFT and was proposed to hold more generally, does not hold in general IQFT, that is J(Tl, Tr) is not of the form f(Tl)−f(Tr).