Coalgebra symmetry for discrete systems

Gubbiotti, G.; Latini, D.; Tapley, B. K.

doi:10.48550/arxiv.2109.10391

Cited by 3 publications

(33 citation statements)

References 47 publications

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“…This paper is devoted to the classification and the study of a class of discretetime systems in N degrees of freedom admitting coalgebra symmetry with respect the Lie-Poisson algebra sl 2 (R). We make use of the the notion of coalgebra symmetry for discrete-time systems we recently introduced in [28]. The main outcome of this paper is that the coalgebra symmetry approach can be fruitfully used to systematically produce superintegrable discrete-time systems in an analogous way as its continuous counterpart introduced in [7,13].…”

Section: Introductionmentioning

confidence: 99%

“…When the Poisson bracket is singular, we speak of Liouville-Poisson integrability when there exist M − r functionally independent invariants in involution, where M is the number of equations and 2r is the rank of the preserved Poisson structure. For a complete overview of the subject we refer to [17,60,62], the review part of the thesis [59], and our previous paper [28].…”

Section: Introductionmentioning

confidence: 99%

“…Differently from the review [43] on continuous superintegrability we consider superintegrability to require Liouville integrability, making it a stronger property. We decide to adopt this definition because as noticed already in [28] not all discrete-time systems admitting more than N non-commuting invariants are integrable with respect to other commonly accepted notions of integrability for discrete-time systems, e.g. algebraic entropy [14].…”

Section: Introductionmentioning

confidence: 99%

“…We now briefly recall the definition of coalgebra symmetry for discrete-time systems we introduced in [28]. The concept of coalgebra was formulated in quantum group theory [18,19].…”

Section: Introductionmentioning

confidence: 99%

“…This definition allows us to provide an analogue of the construction of the invariants for Hamiltonian systems given in [7,13], a result stated in [28,Theorem 3.3].…”

Section: Introductionmentioning

confidence: 99%

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The $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems

Gubbiotti¹,

Latini²

2022

Preprint

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In this paper, we classify all the pseudo-standard variational discrete-time systems in N degrees of freedom admitting coalgebra symmetry with respect to the generic realization of the Lie-Poisson algebra sl 2 (R). This approach naturally yields several quasi-maximally and maximally superintegrable discrete-time systems, both known and new. We conjecture that this exhausts the (super)integrable cases associated with this algebraic construction. CONTENTS 1. Introduction 1 2. The sl 2 (R) Lie-Poisson coalgebra 4 3. Classification results 9 4. Polynomial invariants of the associated dynamical system 12 5. Study of the obtained systems 17 6. Conclusions 27 Acknowledgements 28 Appendix A. Explicit form of the coefficients in the case d = 3 29 Appendix B. Periodic cases of the system (3.15) for V = κξ/2 29 References 31

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We now briefly recall the definition of coalgebra symmetry for discrete-time systems we introduced in [28]. The concept of coalgebra was formulated in quantum group theory [18,19].…”

Section: Introductionmentioning

confidence: 99%

“…This definition allows us to provide an analogue of the construction of the invariants for Hamiltonian systems given in [7,13], a result stated in [28,Theorem 3.3].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems

Gubbiotti¹,

Latini²

2022

Preprint

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The sl2(R) coalgebra symmetry and the superintegrable discrete-time systems

Gubbiotti

Latini

2023

Phys. Scr.

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In this paper, we classify all the quasi-standard variational discrete-time systems in N degrees of freedom admitting coalgebra symmetry with respect to the generic realization of the Lie–Poisson algebra sl2(R). This approach naturally yields several quasi-maximally and maximally superintegrable discrete-time systems, both known and new. We conjecture that this exhausts the (super)integrable cases associated with this algebraic construction.

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Growth and integrability of some birational maps in dimension three

Graffeo¹,

Gubbiotti²

2023

Preprint

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Motivated by the study of the Kahan-Hirota-Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation c 3 ∈ Bir(P 3 ) with projectivities that permute the fixed points of c 3 and the points over which c 3 performs a divisorial contraction. More specifically, we show that three behaviour are possible: (A) integrable with quadratic degree growth and two invariants, (B) periodic with two-periodic degree sequences and more than two invariants, and (C) non-integrable with submaximal degree growth and one invariant. CONTENTS 1. Introduction 1 2. The KHK discretisation of the Euler top 7 3. The Standard Cremona Transformation in dimension three 12 4. The Cremona-cubes group C 15 5. The spaces of initial values and the algebraic entropy 19 6. Covariant linear systems 26 7. Construction of the invariants 30 8. Conclusions 37 References 42

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Coalgebra symmetry for discrete systems

Abstract: In this paper we introduce the notion of coalgebra symmetry for discrete systems. We use this concept to prove the integrability of several N -dimensional vector systems which are generalisations of well-known onedimensional discrete integrable systems.

Cited by 3 publications

References 47 publications

The $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems

The $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems

The sl2(R) coalgebra symmetry and the superintegrable discrete-time systems

Growth and integrability of some birational maps in dimension three

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