G. Cicogna, G. Gaeta
Symmetry properties are at the basis of integrability. In recent years, it appeared that so called "twisted symmetries" are as effective as standard symmetries in many respects (integrating ODEs, finding special solutions to PDEs). Here we discuss how twisted symmetries can be used to detect integrability of Lagrangian systems which are not integrable via standard symmetries.
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We review the basic ideas lying at the foundation of the recently developed theory of twisted symmetries of differential equations, and some of its developments.
The twisted Lie-algebraically deformed relativistic and nonrelativistic phase spaces are constructed with the use of Heisenberg double procedure. The corresponding Heisenberg uncertainty principles are discussed as well.
We consider the theory of \emph{twisted symmetries} of differential equations, in particular $λ$ and $μ$-symmetries, and discuss their geometrical content. We focus on their interpretation in terms of gauge transformations on the one hand, and of coverings on the other one.
The exact elementary excitations in a typical U(1) symmetry broken quantum integrable system, that is the twisted J1-J2 spin chain with nearest-neighbor, next nearest neighbor and chiral three spin interactions, are studied. The main technique is that we quantify the energy spectrum of the system by the zero roots of transfer matrix instead of the traditional Bethe roots. From the numerical calculation and singularity analysis, we obtain the patterns of zero roots. Based on them, we analytically obtain the ground state energy and the elementary excitations in the thermodynamic limit. We find t…
After the introduction of $λ$-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the familiar prolongation operation); they are as useful as standard symmetries for what concerns symmetry reduction of ODEs or determination of special (invariant) solutions for PDEs and have thus attracted attention. The geometrical relation of twisted symmetries to standard ones has already been noted: for some type of twisted symmetries (in particular, $λ$ and $μ$-…
This paper is devoted to the construction of new integrable quantum mechanical models based on certain subalgebras of the half loop algebra of gl(N). Various results about these subalgebras are proven by presenting them in the notation of the St Petersburg school. These results are then used to demonstrate the integrability, and find the symmetries, of two types of physical system: twisted Gaudin magnets, and Calogero-type models of particles on several half-lines meeting at a point.
In this paper, we discuss some specific features of symmetries of integrable systems which can be used to contruct the Fokas-Gel'fand formula for the immersion of 2D-soliton surfaces, associated with such systems, in Lie algebras. We establish the sufficient condition for the applicability of this formula. This condition requires the existence of two vector fields which generate a common symmetry of the initial system and its corresponding linear spectral problem. This means that these two fields have to be group-related and we determine an explicit form of this relation. It provides a criteri…
Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 2. Systems with dilatation and shift symmetries
研究分类了具有伸缩或平移对称性、且含二阶运动积分的三维PDM量子力学系统，给出27个系统并证明分类完备。
We prove that for any tau-symmetric bihamiltonian deformation of the tau-cover of the Principal Hierarchy associated with a semisimple Frobenius manifold, the deformed tau-cover admits an infinite set of Virasoro symmetries.
Higher Symmetries in Twisted Eleven-Dimensional Supergravity
本文研究了十一维超引力中两种可接受扭动下的更高对称性现象，揭示了$ L_\infty $代数与扭动场之间的关系。
Meromorphic differentials, twisted DR cycles and quantum integrable hierarchies
本文基于亚纯微分层交集理论定义了扭曲双重分歧层次，证明其可积性与tau对称性，并发现应用于平凡上同调场论时产生KdV层次，进而导出涉及多个特征类的交集数恒等式。
We display the construction of a twisted superalgebra for the N=1 Euclidian supergravity on 4-manifolds with an almost complex structure. It acts on a representation of twisted supersymmetry made of forms with odd and even statistics and it is covariant under a SU(2)\subset SO(4) Lorentz invariance of the manifold's tangent-space. It contains 4 twisted supersymmetry generators, one nilpotent scalar, one vector and one pseudo-scalar. The superalgebra closes on the twisted fields of supergravity in its new minimal set of auxiliary fields. Its couplings to the twisted Wess and Zumino and vector m…
We consider an unified model, called ancestor model, associated with twisted trigonometric $R$ matrix which model leads to several descendant integrable lattice models related to the U$_{q^{1/2}}(\hat{sl_2})$ symmetry. Boundary operators compatible with integrability are introduced to this model. Reflection and dual reflection equations to ensure integrability of the system are shown to be same as the untwisted case. It follows that underlying symmetry of the ancestor model with integrable boundaries is identified with the q-deformed analogue of Onsager's symmetry. The transfer matrix and its …
Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in \cite{balseiro_garcia_naranjo}, this paper investigates the global theory of integrable Hamiltonian systems on almost symplectic manifolds as an initial step to understand Hamiltonian integrability on twisted Poisson (and Dirac) manifolds. Non-commutative integrable Hamiltonian systems on almost symplectic manifolds were first defined in \cite{fasso_sansonetto}, which proved existence of local generalised action-angle coordinates in the spirit of the Liouville-Arnol'd theorem. In an…
For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the corresponding twisted partition functions
From one to infinity: symmetries of integrable systems
本文以 KdV 和 Boussinesq 方程为例，证明仅需找到一个非局部关键对称性即可保证可积性，并可由此推导出递归算子、无穷多对称性和 Lax 对。
Enhanced symmetries in minimally-twisted three-dimensional supersymmetric theories
本文证明三维N=2理论在全纯-拓扑扭变中的剩余超对称可扩展为无限维李超代数作用，并在N=4杨-米尔斯及超共形陈-西蒙斯理论中验证，描述了局部算子层面的实现。
I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.
In this work, we establish a connection between the extended Prelle-Singer procedure (Chandrasekar \textit{et al.} Proc. R. Soc. A 2005) with five other analytical methods which are widely used to identify integrable systems in the contemporary literature, especially for second order nonlinear ordinary differential equations (ODEs). By synthesizing these methods we bring out the interplay between Lie point symmetries, $λ$-symmetries, adjoint symmetries, null-forms, Darboux polynomials, integrating factors and Jacobi last multiplier in identifying the integrable systems described by second orde…
A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on $\mathbb{Z}^3$ is investigated. Our approach is a generalization of a method developed in the context of Yang-Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case-by-case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations.
Following up the work of [1] on deformed algebras, we present a class of Poincaré invariant quantum field theories with particles having deformed internal symmetries. The twisted quantum fields discussed in this work satisfy commutation relations different from the usual bosonic/fermionic commutation relations. Such twisted fields by construction are nonlocal in nature. Despite this nonlocality we show that it is possible to construct local interaction Hamiltonians which satisfy cluster decomposition principle and are Lorentz invariant. We further illustrate these ideas by considering global S…
We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein's splitting theorem for the integrable system is also studied giving some examples in which such a splitting does not exist, i.e. when the integrable system is not, locally, a product of an integrable system on the symplectic leaf and an integrable system on a transversal. The problem of splitting for integrable systems with additional symmetries is also considered.
We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted $D^{(2)}_3$ algebra (or the $D^{(2)}_3$ model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter $η$. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices …
In this paper, we study Schrödinger equations on elliptic curves called generalized Lamé equations. We suggest a method of finding integrable potentials for Schrödinger type equations. We apply this method to the Lamé equations and provide a sequence of integrable potentials for which the eigenvalue problem is solved explicitly.
In the present paper the non-Noether symmetries of the Toda model, nonlinear Schodinger equation and Korteweg-de Vries equations (KdV and mKdV) are discussed. It appears that these symmetries yield the complete sets of conservation laws in involution and lead to the bi-Hamiltonian realizations of the above mentioned models.
In this paper, based on the Fokas, Gel'fand et al approach [15,16], we provide a symmetry characterization of continuous deformations of soliton surfaces immersed in a Lie algebra using the formalism of generalized vector fields, their prolongation structure and links with the Fréchet derivatives. We express the necessary and sufficient condition for the existence of such surfaces in terms of the invariance criterion for generalized symmetries and identify additional sufficient conditions which admit an explicit integration of the immersion functions of 2D surfaces in Lie algebras. We discuss …
In this paper we construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among them a few new examples of both sine-Gordon and Liouville type equations.
Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems
研究DIM可积系统与扭曲Cherednik系统之间Hamilton算子的特征函数关系，揭示两者在对称函数上的对应关系。
We consider maps which preserve functions which are built out of the invariants of some simple vector fields. We give a reduction procedure, which can be used to derive commuting maps of the plane, which preserve the same symplectic form and first integral. We show how our method can be applied to some maps which have recently appeared in the context of Yang-Baxter maps.
We discuss quantum many-body systems with lattice translation and discrete onsite symmetries. We point out that, under a boundary condition twisted by a symmetry operation, there is an exact degeneracy of ground states if the unit cell forms a projective representation of the onsite discrete symmetry. Based on the quantum transfer matrix formalism, we show that, if the system is gapped, the ground-state degeneracy under the twisted boundary condition also implies a ground-state (quasi-)degeneracy under the periodic boundary conditions. This gives a compelling evidence for the recently proposed…
In this paper, symmetry analysis is extended to study nonlocal differential equations, in particular two integrable nonlocal equations, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg--de Vries equation. Lie point symmetries are obtained based on a general theory and used to reduce these equations to nonlocal and local ordinary differential equations separately; namely one symmetry may allow reductions to both nonlocal and local equations depending on how the invariant variables are chosen. For the nonlocal modified Korteweg--de Vries equation, analogously to the…
We investigate the relation between pluri-Lagrangian hierarchies of $2$-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in [Petrera, Suris. J. Nonlinear Math. Phys. 24:sup1, 121--145 (2017)] for ordinary differential equations. We consider hierarchies of $2$-dimensional Lagrangian PDEs (many of which have a natural $(1+1)$-dimensional space-time interpretation) and show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian …
In this paper we study Lie symmetries, Kac-Moody-Virasoro algebras, similarity reductions and particular solutions of two different recently introduced (2+1)-dimensional nonlinear evolution equations, namely (i) (2+1)-dimensional breaking soliton equation and (ii) (2+1)-dimensional nonlinear Schrödinger type equation introduced by Zakharov and studied later by Strachan. Interestingly our studies show that not all integrable higher dimensional systems admit Kac-Moody-Virasoro type sub-algebras. Particularly the two integrable systems mentioned above do not admit Virasoro type subalgebras, event…
We describe an infinite-dimensional algebra of hidden symmetries of N=4 supersymmetric Yang-Mills (SYM) theory. Our derivation is based on a generalization of the supertwistor correspondence. Using the latter, we construct an infinite sequence of flows on the solution space of the N=4 SYM equations. The dependence of the SYM fields on the parameters along the flows can be recovered by solving the equations of the hierarchy. We embed the N=4 SYM equations in the infinite system of the hierarchy equations and show that this SYM hierarchy is associated with an infinite set of graded symmetries re…
We consider the worldsheet boundary scattering and the corresponding boundary algebras for the Z=0 giant graviton and the Z=0 D7-brane in the AdS/CFT correspondence. We consider two approaches to the boundary scattering, the usual one governed by the (generalized) twisted Yangians and the q-deformed model of these boundaries governed by the quantum affine coideal subalgebras. We show that the q-deformed approach leads to boundary algebras that are of a more compact form than the corresponding twisted Yangians, and thus are favourable to use for explicit calculations. We obtain the q-deformed r…
We split the generic conformal mechanical system into a "radial" and an "angular" part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We reduce the analysis of the constants of motion of the full system to the study of certain differential equations on this orbit. For integrable mechanical systems, the conformal invariance renders them superintegrable, yielding an additional series of conserved quantities originally found by Wojciechowski in the rational Calogero model. Finally, we show that,…
We construct a five-parameter family of gauge-nonequivalent SU(2) instantons on a noncommutative four sphere $S_θ^4$ and of topological charge equal to -1. These instantons are critical points of a gauge functional and satisfy self-duality equations with respect to a Hodge star operator on forms on $S_θ^4$. They are obtained by acting with a twisted conformal symmetry on a basic instanton canonically associated with a noncommutative instanton bundle on the sphere. A completeness argument for this family is obtained by means of index theorems. The dimension of the ``tangent space'' to the modul…
In this paper, to begin with, we review six different analytical methods which are widely used to derive symmetries, integrating factors, multipliers, Darboux polynomials and integrals of second order nonlinear ordinary differential equations. We illustrate the algorithm behind each method by considering a nonlinear oscillator equation as an example. In the second part of this paper, we examine the interconnections between these methods. We establish a road map between extended Prelle-Singer procedure with all other methods cited above and thereby demonstrate the interplay between Lie point sy…
Using anisotropic R-matrices associated with affine Lie algebras $\hat g$ (specifically, $A_{2n}^{(2)}, A_{2n-1}^{(2)}, B_n^{(1)}, C_n^{(1)}, D_n^{(1)}$) and suitable corresponding K-matrices, we construct families of integrable open quantum spin chains of finite length, whose transfer matrices are invariant under the quantum group corresponding to removing one node from the Dynkin diagram of $\hat g$. We show that these transfer matrices also have a duality symmetry (for the cases $C_n^{(1)}$ and $D_n^{(1)}$) and additional $Z_2$ symmetries that map complex representations to their conjugates…
In this paper, we study a generalization of twisted (groupoid) equivariant $\mathrm{K}$-theory in the sense of Freed-Moore for $\mathbb{Z}_2$-graded $\mathrm{C}^*$-algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele's $\mathrm{K}$-theory for $\mathbb{Z}_2$-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant $\mathrm{K}$-group when the $\mathrm{C}^*$-algebra is trivially graded. It is applied for the bu…
In this paper we try to find examples of integrable natural Hamiltonian systems on the sphere $S^2$ with the symmetries of each Platonic polyhedra. Although some of these systems are known, their expression is extremely complicated; we try here to find the simplest possible expressions for this kind of dynamical systems. Even in the simplest cases it is not easy to prove their integrability by direct computation of the first integrals, therefore, we make use of numerical methods to provide evidences of integrability; namely, by analyzing their Poincaré sections (surface sections). In this way …
Generalized Frobenius Manifolds with Non-flat Unity and Integrable Hierarchies
本文为具有非平坦单位元的广义Frobenius流形构造了双哈密顿可积层次（主层次），证明其具有Virasoro对称性和tau结构，推导环方程并构造拓扑形变，且给出与Volterra、q-变形KdV及Ablowitz-Ladik层次相关的实例。
We consider a certain super extension, called the super tau-cover, of a bihamiltonian integrable hierarchy which contains the Hamiltonian structures including both the local and non-local ones as odd flows. In particular, we construct the super tau-cover of the principal hierarchy associated with an arbitrary Frobenius manifold, and the super tau-cover of the Korteweg-de Vries (KdV) hierarchy. We also show that the Virasoro symmetries of these bihamiltonan integrable hierarchies can be extended to symmetries of the associated super tau-covers.
Quantum symmetries of the twisted tensor products of C*-algebras
本文研究带正交滤过的C*-代数扭张量积的量子对称群，证明满足扭形式假设时该群为原滤过量子对称群的广义Drinfeld double，给出离散群交叉积的应用及主定理不成立的反例。
Painlevé property, local and nonlocal symmetries and symmetry reductions for a (2+1)-dimensional integrable KdV equation
本文证明了(2+1)维cKP3-4方程的Painlevé性质，利用截断展开求得非局部对称性及Bäcklund变换，基于局部点对称性的Kac-Moody-Virasoro代数导出群不变约化及新Lax可积模型。
The non-Abelian symmetries of the half-infinite XXZ spin chain for all possible types of integrable boundary conditions are classified. For each type of boundary conditions, an analog of the Chevalley-type presentation is given for the corresponding symmetry algebra. In particular, two new algebras arise that are, respectively, generated by the symmetry operators of the model with triangular and special $U_q(gl_2)-$invariant integrable boundary conditions.
Automorphic symmetries and $ AdS_n $ integrable deformations
本文提出基于 $\mathcal{B}$-自同构的构造与分类方法，将二维格点可积模型推广至弦论可积模型，发现非差分形式的新 $AdS_n$ 可积形变、$R$-矩阵及其编织交叉性质。
We introduce a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs, with $N\times N$ matrices, linear in the spectral parameter. We give a classification scheme for such Lax pairs and the associated discrete integrable systems. We present two potential forms and completely classify the generic case. Many well known examples belong to our scheme for $N=2$, so many of our systems may be regarded as generalisations of these. Even at $N=3$, several new integrable systems arise. Many of our equations are mutually compatible, so can be used together to form "coloured" lattices. We also present conti…
A symmetry classification is performed for a class of differential-difference equations depending on 9 parameters. A 6-parameter subclass of these equations is an integrable discretization of the Krichever-Novikov equation. The dimension $n$ of the Lie point symmetry algebra satisfies $1 \le n \le 5$. The highest dimensions, namely $n=5$ and $n=4$ occur only in the integrable cases.
Symmetry structure of integrable hyperbolic third order equations
本文研究可积双曲三阶方程的生成对称，证明不同无限对称层级可由同一生成对称在参数不同取值处展开得到，并通过两个例子深入分析其对称结构。
Existence of Riemannian invariants for integrable systems of hydrodynamic type
本文研究了可积流体类型系统中黎曼不变量的存在性，证明在特定坐标系下系统生成元和对称性可对角化。
We construct a weight matrix for the 3D Ising model satisfying the so-called twisted tetrahedron equation. The result is based on the theory of the n-simplicial complex and the invented recursion procedure on the space of n-simplex solutions in correspondences. The weight matrix reveals some properties intrinsic for the hypercube combinatorics.
We investigate the existence of Liouville integrable cosmological models in hybrid metric-Palatini theory. Specifically we use the symmetry conditions for the existence of quadratic in the momentum conservation laws for the field equations as constraint conditions for the determination of the unknown functional form of the theory. The exact and analytic solutions of the integrable systems which found in this study are presented in terms of quadratics and Laurent expansions.
Progresses on some open problems related to infinitely many symmetries
本文探讨了可积系统中无限多对称性的开放问题，提出波参数平移对称性的线性组合可能涵盖现有对称性，并通过KdV和Burgers方程验证了这一猜想。
We construct new integrable coupled systems of N=1 supersymmetric equations and present integrable fermionic extensions of the Burgers and Boussinesq equations. Existence of infinitely many higher symmetries is demonstrated by the presence of recursion operators. Various algebraic methods are applied to the analysis of symmetries, conservation laws, recursion operators, and Hamiltonian structures. A fermionic extension of the Burgers equation is related with the Burgers flows on associative algebras. A Gardner's deformation is found for the bosonic super-field dispersionless Boussinesq equatio…
Bethe Vectors in Quantum Integrable Models with Classical Symmetries
本文为经典李代数g=gl_n, o_{2n+1}, sp_{2n}, o_{2n}的g-不变可积模型给出了离壳Bethe向量的精确简单定义，并证明了其满足Bethe方程时成为在壳向量，进而推导了作用公式、矩形递推关系和余积公式等性质。
Twisted locality-preserving automorphisms, anomaly index, and generalized Lieb-Schultz-Mattis theorems with anti-unitary symmetries
本文针对量子自旋链定义扭曲局域性保持自同构及其指标，统一描述幺正与反幺正等对称性，给出反常指标的微观定义，并证明反常对称性导致广义Lieb-Schultz-Mattis定理，揭示长程纠缠及无唯一带隙对称基态等结论。
We apply the theory of Lie point symmetries for the study of a family of partial differential equations which are integrable by the hyperbolic reductions method and are reduced to members of the Painlevé transcendents. The main results of this study is that from the application of the similarity transformations provided by the Lie point symmetries all the members of the family of the partial differential equations are reduced to second-order differential equations which are maximal symmetric and can be linearized.
We introduce a new concept of quasi-Yang-Baxter algebras. The quantum quasi-Yang-Baxter algebras being simple but non-trivial deformations of ordinary algebras of monodromy matrices realize a new type of quantum dynamical symmetries and find an unexpected and remarkable applications in quantum inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter algebras the standard procedure of QISM one obtains new wide classes of quantum models which, being integrable (i.e. having enough number of commuting integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic…
We address the novel structures arising in quantum and string integrable theories, as well as construct methods to obtain them and provide further analysis. Specifically, we implement the automorphic symmetries on periodic lattice systems to obtain integrable hierarchies, whose commutativity and integrable transformations induce a generating structure of integrable classes. This prescription is first applied to 2-dim and 4-dim setups, where we find the new $ \mathfrak{sl}_{2} $ sector, $ \mathfrak{su}(2) \oplus \mathfrak{su}(2) $ with superconductive modes, Generalised Hubbard type classes and…
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