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I. Marshall

Publications and source records attributed to I. Marshall.

11 recordsLinked to original sources
arXiv2021

Trigonometric real form of the spin RS model of Krichever and Zabrodin

We investigate the trigonometric real form of the spin Ruijsenaars-Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real form emerges from Hamiltonian reduction of an obviously integrable 'free' system carried by a spin extension of the Heisenberg double of the ${\rm U}(n)$ Poisson-Lie group. The Poisson structure on the unreduced real phase space ${\rm GL}(n,\mathbb{C}) \times \mathbb{C}^{nd}$ is the direct product of that of the Heisenberg double and $d\geq 2$ copies of a ${\rm U}(n)$ covariant Poisson structure on $\mathbb{C}^n \simeq \mathbb{R}^{2n}$ found by Zakrzewski, also in 1995. We reduce by fixing a group valued moment map to a multiple of the identity, and analyze the resulting reduced system in detail. In particular, we derive on the reduced phase space the Hamiltonian structure of the trigonometric spin Ruijsenaars-Schneider system and we prove its degenerate integrability.

math-ph↗
arXiv2019

Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric $\mathrm{BC}_n$ Sutherland system

Integrable many-body systems of Ruijsenaars--Schneider--van Diejen type displaying action-angle duality are derived by Hamiltonian reduction of the Heisenberg double of the Poisson-Lie group $\mathrm{SU}(2n)$. New global models of the reduced phase space are described, revealing non-trivial features of the two systems in duality with one another. For example, after establishing that the symplectic vector space $\mathbb{C}^n\simeq\mathbb{R}^{2n}$ underlies both global models, it is seen that for both systems the action variables generate the standard torus action on $\mathbb{C}^n$, and the fixed point of this action corresponds to the unique equilibrium positions of the pertinent systems. The systems in duality are found to be non-degenerate in the sense that the functional dimension of the Poisson algebra of their conserved quantities is equal to half the dimension of the phase space. The dual of the deformed Sutherland system is shown to be a limiting case of a van Diejen system.

math-ph↗
arXiv2018

Comparison of Poisson structures and Poisson-Lie dynamical r-matrices

We construct a Poisson isomorphism between the formal Poisson manifolds g^* and G^*, where g is a finite dimensional quasitriangular Lie bialgebra. Here g^* is equipped with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G^* with its Poisson-Lie structure. We also quantize Poisson-Lie dynamical r-matrices of Balog-Feher-Palla.

math.QA↗
arXiv2017

The action-angle dual of an integrable Hamiltonian system of Ruijsenaars--Schneider--van Diejen type

Integrable deformations of the hyperbolic and trigonometric ${\mathrm{BC}}_n$ Sutherland models were recently derived via Hamiltonian reduction of certain free systems on the Heisenberg doubles of ${\mathrm{SU}}(n,n)$ and ${\mathrm{SU}}(2n)$, respectively. As a step towards constructing action-angle variables for these models, we here apply the same reduction to a different free system on the double of ${\mathrm{SU}}(2n)$ and thereby obtain a novel integrable many-body model of Ruijsenaars--Schneider--van Diejen type that is in action-angle duality with the respective deformed Sutherland model.

math-ph↗
arXiv2013

Measuring Double-Electron Capture with Liquid Xenon Experiments

We investigate the possibilities of observing the decay mode for $^{124}$Xe in which two electrons are captured, two neutrinos are emitted, and the final daughter nucleus is in its ground state, using dark matter experiments with liquid xenon. The first upper limit of the decay half-life is calculated to be 1.66$\times10^{21}$ years at a 90% confidence level (C.L.) obtained with the published background data from the XENON100 experiment. Employing a known background model from the Large Underground Xenon (LUX) experiment, we predict that the detection of double-electron capture of $^{124}$Xe to the ground state of $^{124}$Te with LUX will have approximately 115 events, assuming a half-life of 2.9 $\times$ 10$^{21}$ years. We conclude that measuring $^{124}$Xe 2$ν$ double-electron capture to the ground state of $^{124}$Te can be performed more precisely with the proposed LUX-Zeplin (LZ) experiment.

nucl-ex↗
arXiv2004

Quantization of some Poisson-Lie dynamical r-matrices and Poisson homogeneous spaces

Poisson-Lie (PL) dynamical r-matrices are generalizations of dynamical r-matrices, where the base is a Poisson-Lie group. We prove analogues of basic results for these r-matrices, namely constructions of (quasi)Poisson groupoids and of Poisson homogeneous spaces. We introduce a class of PL dynamical r-matrices, associated to nondegenerate Lie bialgebras with a splitting; this is a generalization of trigonometric r-matrices with an abelian base. We prove a composition theorem for PL dynamical r-matrices, and construct quantizations of the polarized PL dynamical r-matrices. This way, we obtain quantizations of Poisson homogeneous structures on G/L (G a semisimple Lie group, L a Levi subgroup), thereby generalizing earlier constructions.

math.QA↗
arXiv2004

The non-Abelian momentum map for Poisson-Lie symmetries on the chiral WZNW phase space

The gauge action of the Lie group $G$ on the chiral WZNW phase space ${\cal M}_{\check G}$ of quasiperiodic fields with $\check G$-valued monodromy, where $\check G\subset G$ is an open submanifold, is known to be a Poisson-Lie (PL) action with respect to any coboundary PL structure on $G$, if the Poisson bracket on ${\cal M}_{\check G}$ is defined by a suitable monodromy dependent exchange $r$-matrix. We describe the momentum map for these symmetries when $G$ is either a factorisable PL group or a compact simple Lie group with its standard PL structure. The main result is an explicit one-to-one correspondence between the monodromy variable $M \in \check G$ and a conventional variable $Ω\in G^*$. This permits us to convert the PL groupoid associated with a WZNW exchange $r$-matrix into a `canonical' PL groupoid constructed from the Heisenberg double of $G$, and consequently to obtain a natural PL generalization of the classical dynamical Yang-Baxter equation.

math.QA↗
arXiv2003

Stability analysis of some integrable Euler equations for SO(n)

A family of special cases of the integrable Euler equations on $so(n)$ introduced by Manakov in 1976 is considered. The equilibrium points are found and their stability is studied. Heteroclinic orbits are constructed that connect unstable equilibria and are given by the orbits of certain 1-parameter subgroups of SO(n). The results are complete in the case $n=4$ and incomplete for $n>4$.

math-ph↗
arXiv2002

On a Poisson-Lie analogue of the classical dynamical Yang-Baxter equation for self-dual Lie algebras

We derive a generalization of the classical dynamical Yang-Baxter equation (CDYBE) on a self-dual Lie algebra $\cal G$ by replacing the cotangent bundle T^*G in a geometric interpretation of this equation by its Poisson-Lie (PL) analogue associated with a factorizable constant r-matrix on $\cal G$. The resulting PL-CDYBE, with variables in the Lie group G equipped with the Semenov-Tian-Shansky Poisson bracket based on the constant r-matrix, coincides with an equation that appeared in an earlier study of PL symmetries in the WZNW model. In addition to its new group theoretic interpretation, we present a self-contained analysis of those solutions of the PL-CDYBE that were found in the WZNW context and characterize them by means of a uniqueness result under a certain analyticity assumption.

math.QA↗
arXiv1992

Generalized Drinfeld-Sokolov Reductions and KdV Type Hierarchies

Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al, reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra $\ell(gl_n)$, graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions of $n$ into the sum of equal numbers $n=pr$ or to equal numbers plus one $n=pr+1$. We prove that the reduction belonging to the grade $1$ regular elements in the case $n=pr$ yields the $p\times p$ matrix version of the Gelfand-Dickey $r$-KdV hierarchy, generalizing the scalar case $p=1$ considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even for $p=1$.

hep-th↗