北京雁栖湖应用数学研究院

In the 1940s Littlewood formulated three fundamental correspondences parallel to the Schur-Weyl duality. We introduce the notion of quantum immanants and establish the quantum Littlewood correspondences between quantum immanants and Schur functions for the quantum general linear group and the Hecke algebra. Via the correspondences, we have found an exact relationship between the Gelfand-Tsetlin bases of $U_{q}(gl(n))$ and Young's orthonormal bases for the Hecke algebra. This leads to a trace formula for the quantum immanants that has settled the generalization problem of $q$-analog of Kostant's formular for $\lambda $-immanants. As applications, we also derive general $q$-Littlewood-Merris-Watkins identities and $q$-Goulden-Jackson identities as special cases of the quantum Littlewood correspondence III. This is joint work with Jian Zhang.

The central extension $W_{1 + \infty}$ of the Lie algebra of differential operators on the circle appeared in mathematical physics literature in nineties, in the relation to various models of two-dimensional quantum field theory and integrable systems. A recent rise of interest in the structure and representations of this infinite-dimensional Lie algebra is due to its role in the interpretation of generating functions of certain intersection numbers on moduli spaces of stable curves as tau-functions of soliton integrable hierarchies. This in turn led to the studies of the action of $W_{1 + \infty}$ on particular families of symmetric functions. In this talk we describe the action of $W_{1 + \infty}$ operators and their B-type analogues on Schur and Schur Q-functions respectively through the techniques of formal distributions. We observe an interesting self-duality property possessed by these compact formulas.

Classification of simple smooth representations is a key problem in representation theory of vertex algebras. We will discuss their construction via twisted localization and their realizations.

We will consider central elements in the quantized enveloping algebra and quantum affine algebra associated with $gl_{n}$. A family of such elements for the quantized enveloping algebra was constructed in a landmark paper by Reshetikhin, Takhtadzhyan and Faddeev. We will calculate their eigenvalues in irreducible highest weight representations. We will also construct q-immanants which are central elements parameterized by arbitrary Young diagrams, analogous to Okounkov's quantum immanants. This construction extends to the quantum affine algebra at the critical level yielding quantum Sugawara operators which are central elements of a completed algebra. This is joint work with Naihuan Jing and Ming Liu.

Type A quantum toroidal algebras can be written as Drinfeld doubles of shuffle algebras. This can be done in two ways: using the standard shuffle algebra or the so-called “matrix” shuffle algebra, which arises from an RTT-type approach. I will discuss the matrix shuffle algebra of the quantum toroidal $\mathfrak{gl}_{n|m}$ and explain how to construct various families of commuting elements in it.

In the theory of finite-dimensional representations of affine quantum groups, the singularities of (normalized) R-matrices play an important role as they encode the non-commutativity of tensor product representations. However, computing the pole order of R-matrices is a difficult problem in general, and so far explicitly known only for fundamental and Kirillov-Reshetikhin modules. In this presentation, we restrict our attention to a certain subcategory of representations which monoidally categorifies a cluster algebra of finite type (known as Hernandez-Leclerc’s level-one subcategory), and explain that the pole order of R-matrices is computable for any irreducible representations as the dimension of E-invariants (analog of extension groups) of decorated representations of Dynkin quivers. This manifests a correspondence of numerical characteristics between monoidal and additive categorifications of cluster algebras of finite type.

In this talk, I will revisit the finite-dimensional representations of quantum affine $sl(2)$ that occur at $q$ a root of unity which are not related to the standard integer-dimension representations of $sl(2)$. These representations depend upon two points in two copies of a higher-genus algebraic curve. The R-matrix which is the intertwiner of two such representations factors into four terms - each of which encodes a Boltzmann weight of the Chiral Potts model as has long been known. In this work I will investigate this phenomenon further and show that, viewed as Borel subalgebra representations, the finite-dimensional representations themselves factorize into two very simple representations, each of which depends on a single point in the algebraic curve. This factorization will be used to both explain the R-matrix factorization and show that the transfer matrix factorizes into two operators that play the role of Q-operators in the roots of unity case. The functional relations satisfied by these Q-operators will be obtained from the study of short-exact-sequences of representations. The situation parallels the generic $q$ case, with the key difference being that only finite-dimensional representations appear in the roots-of-unity case.Based on arXiv:2412.14811.

The Onsager algebra is pivotal in the renowned solution to 2D classical Ising model by Lars Onsager in 1944. It has since played an important role in the field of exactly solvable models. As I will demonstrate, the 6-vertex model at root of unity (or equivalently the spin-1/2 XXZ model) has a hidden Onsager algebra symmetry [1], i.e. the transfer matrix of 6-vertex model commute with a specific representation of the Onsager algebra, an infinite-dimensional Lie algebra. This turns out to be important to understand the spectrum of the XXZ model at root of unity. I will explain how one can prove the existence of the Onsager algebra symmetry using Baxter's $\tau _{2}$ model [2], whose relation to the 6-vertex model dates back to the work of Bazhanov and Stroganov [3].<br>[1] Y. Miao, SciPost Phys. 11, 066 (2021).<br>[2] E. Vernier and Y. Miao, in preparation.<br>[3] V.V. Bazhanov and Yu.G. Stroganov, J. Stat. Phys. 59, 799-817 (1990).

We develop an off-diagonal approach to solve quantum integrable systems, focusing on XXX spin chains. For anti-ferromagnetic chains, we propose a new transfer matrix eigenvalue parametrization, obtaining exact solutions and ground state eigenfunctions in the thermodynamic limit. The quantum determinant's exponential decay validates inversion relations for $U(1)$-symmetric systems. For spin-1/2 chains with non-diagonal boundaries, we reveal additive boundary field contributions to surface energy and identify boundary-string excitations. Furthermore, we derive a nonlinear integral equation (NLIE) from quantum transfer matrices to compute magnetic field-dependent free energy, successfully extending the method to $SU(3)$-invariant chains with verified accuracy. This unified framework advances the understanding of quantum integrable systems' ground states, boundaries, and thermodynamics.

It is well-known that many integrable differential-difference systems possess bi-Hamiltonian structure in terms of difference operators; operators of this form for scalar system have been classified up to order $(−5,5)$ within the framework of multiplicative Poisson vertex algebras; on the other hand, many years ago Dubrovin unveiled the relation between nondegenerate operators of order $(−1,1)$ -- that can be seen as discretization of Poisson brackets of hydrodynamic type -- and the theory of Poisson-Lie groups. We extend the classification of this latter class of operators to include degenerate cases and study the Poisson cohomology of the (degenerate) Hamiltonian structure of the Toda system. This is a joint work with Daniele Valeri.

This report primarily introduces elliptic integrable systems, with a focus on the elliptic KdV and KP systems. The presentation will demonstrate how to derive these two generalized classes of elliptic integrable systems, along with their integrable features such as exact solutions and Lax pairs, by embedding elliptic curves into the well-known KdV and KP equations, respectively. Additionally, when the elliptic curves degenerate into rational curves, the elliptic KdV and KP systems reduce to the original KdV and KP equations.

In this talk, we will review our studies in different directions of partitions in representation theory related to integrable systems.

This talk provides a scheme of $r$-matrix structure for $2$ by $2$ nonlinearized spectral problem. This structure covers almost all $2$ by $2$ constrained systems. We use the r-matrix structure to prove integrability of finite-dimensional Hamiltonian system associated with the Camassa-Holm hierarchy. We will see how the CH hierarchy is related to an $r$-matrix structure and finite-dimensional integrable systems, and furthermore algebro-geometric solutions of the CH hierarchy are shown on a symplectic submanifold. Other similar peakons models, including the DP, the b-family, and cubic equations (MOCH, FORQ/MCH, Novikov etc) will be mentioned as well.

We present a general framework for constructing polynomial integrable systems with respect to linearizations of Poisson varieties that admit log-canonical coordinate systems. Our construction is in particular applicable to Poisson varieties with compatible cluster or generalized cluster structures.We will look in more detail at explicit examples from Lie theory. This is joint work with Yanpeng Li and Yu Li.

Integrability is interesting, often associated with remarkable phenomena such as the existence of infinitely many conserved charges, factorized scattering and exact solvability. On the other hand, it is probably even more interesting to break integrability, giving rise to new phenomena such as particle production, resonance states, and notably, soliton confinements. In this talk, I will first review confinement in Ising field theory and the theoretical works to determine the associated meson spectrum. I will then present recent work in which we explore analogous phenomena in an Ising ladder system. In this setting, we observe similar confinement mechanisms but encounter a richer meson spectrum, including both interchain and intrachain mesons. I will discuss the methods used to determine these spectra and present the key findings of our study.

I will talk about (in general conjectural) construction of a $X$-cluster structure on certain Hamiltonian (coisotropic) reduction of a $X$-cluster variety. There are two main classes of examples of such constructions: moduli spaces of framed local systems with special monodromies and phase spaces of Goncharov-Kenyon integrable systems. The first class includes the phase space of open $XXZ$ chain and Ruijsenaars integrable systems. The second class includes integrable systems corresponding to the $q$-difference Painleve equations.Based on joint project with P. Gavrylenko, A. Marshakov, M. Prokushkin, M. Semenyakin, A. Shapiro, G. Schrader.

Following the development of the AGT correspondence, new relations between free field representations of quantum groups and $W$-algebras were obtained. The simplest one is the homomorphism between the level $(N,0)$ horizontal representation of the quantum toroidal $gl(1)$ algebra and (dressed) q-deformed $W_{N}$ algebras. In this talk, I will explain how to extend this type of relations to the Wakimoto representations of quantum affine $sl(N)$ algebras using the 'surface defect' deformation of the quantum toroidal $sl(N)$ algebra.

In the first part of the talk, I will describe the symplectic groupoid: a set of pairs $(B,A)$ with $A$ unipotent upper-triangular matrices and $B \in GL_{n}$ being such that the matrix $\widetilde{\mathbb{A}} = B\mathbb{A}B^T$ is itself unipotent upper triangular. Since works of J. Nelson, T. Regge, B. Dubrovin, and M. Ugaglia it was known that entries of $A$ can be identified with geodesic functions on a Riemann surface with holes; these entries then enjoy a closed Poisson algebra (reflection equation) expressible in the $r$-matrix form. In our recent work with M. Shapiro, we solved the symplectic groupoid in terms of planar networks; we used this solution to construct a complete set of geodesic functions for a closed Riemann surface. In the second part, I will present preliminary results on generalizing the above construction to the case of noncommutative symplectic groupoid subject to a Van der Bergh double Poisson bracket. Based on a forthcoming joint paper with I. Bobrova and M. Shapiro.

The talk is focused on the family of compatible quadratic Poisson brackets on $gl(n)$, generalizing the Sklyanin one. For any of the brackets in the family, the argument shift determines the compatible linear bracket. I will describe the application of the bi-Hamiltonian formalism for some pencils from this family, namely a method for constructing involutive subalgebras for a linear bracket starting by the center of the quadratic bracket. I will provide some interesting examples of families of this type.An important ingredient of the construction is the family of antidiagonal principal minors of the Lax matrix. A crucial but quite ambiguous condition of the log-canonicity of brackets of these minors with all the generators of the Poisson algebra of all the family under consideration establishes a relation of our families with cluster algebras, a similar property arises in the context of Poisson structures consistent with mutations. The talk is based on the recent joint paper with V.V. Sokolov https://arxiv.org/abs/2502.16925.

Spin Calogero-Moser systems can be formulated by the classical Hamiltonian reduction and thus the phase spaces are probably stratified spaces. In this talk, we will first give a quick review of classic Calogero-Moser systems, including the physical background, historical developments and different generalizations. We then introduce the spin Calogero-Moser systems living on quotient spaces via Hamiltonian reductions. We will then discuss their superintegrability even if the quotient spaces are stratified. A concrete example from $SU(3)$ will be provided with details as a toy model of the spin Calogero-Moser systems.

The skein algebra of a surface is a noncommutative algebra that quantizes the $SL(2,C)$-character variety of the surface. It has been intensively studied in quantum topology for more than thirty years. In an influential paper from 2014, D. Thurston suggested that the skein algebra should have a natural categorification where the product in the algebra arises from a monoidal structure on a category. In this talk, I will describe such a categorification of the skein algebra of a genus zero surface with boundary. I will first review the construction of the variety of triples, a remarkable geometric object introduced by Braverman, Finkelberg, and Nakajima in their study of Coulomb branches of 3d $N = 4$ gauge theories. I will then explain how the skein algebra arises as the Grothendieck ring of the bounded derived category of equivariant coherent sheaves on the variety of triples equipped with a natural monoidal structure. This talk is based on work with Hyun Kyu Kim and Peng Shan.

In this talk, I'll describe a BV framework of effective field theories that leads to the $B$-model interpretation of dispersionless integrable hierarchy.

I will present examples of polynomial algebras that appear in context of quantum superintegrable systems. Those symmetry algebras allow to characterize spectrum and wave functions, in particular for exotic models involving Painlevé transcendents. The talk will motivate the need of other approaches to polynomial algebra beyond explicit differential operator realizations [1,2,3]. I will review recent works were the notion of commutant was applied to simple Lie algebras and their Cartan subalgebras [1,2,3]. This allowed us to present algebraic definition of superintegrability, integrals of motion and symmetry algebra. I will provide details how the Poisson-Lie bracket setting allows to construct indecomposable polynomials via systems of partial differential equations. I will present formula for the indecomposable polynomials for $sl(n)$ that lead to a polynomial algebra of degree $n−1$. I will present additional explicit formula for the quadratic and cubic algebra in the case of $sl(3)$ and $sl(4)$. I will explain how explicit realization allow recover the Racah algebra as particular case and how they can be quantized.I will point out how this method can be applied to various subalgebras chains $g \supset g^{′}$ with applications to dynamical symmetries and missing label problems. I will discuss the case of the Elliott $su(3) \supset so(3)$ and seniority $so(5) \supset su(2) \times u(1)$ chains [4] which were used in nuclear physics. I will explain how in both cases our approach gives a three generators cubic algebra. I will provide some further comments on the state of those constructions for simple Lie algebras and other subalgebras chains [5].<br>References:<br>[1] R Campoamor-Stursberg, D Latini, I Marquette, YZ Zhang, Algebraic (super-) integrability from commutants of subalgebras in universal enveloping algebras J. Phys. A: Math. and Theor. 56 (4), 045202 (2023)<br>[2] R Campoamor-Stursberg, D Latini, I Marquette, YZ Zhang, Polynomial algebras from commutants: Classical and quantum aspects of $A_3$, J. Phys. A: Conf. Series 2667 012037 (2023)<br>[3] R Campoamor-Stursberg, D Latini, I Marquette, J Zhang, YZ Zhang,, Superintegrable systems associated to commutants of Cartan subalgebras in enveloping algebras, arXiv:2406.01958<br>[4] R Campoamor-Stursberg, D Latini, I Marquette, YZ Zhang, Polynomial algebras from Lie algebras reduction chains $g \supset g’$, Ann. of Phys. 459 169496 1-19 (2023)<br>[5] R Campoamor-Stursberg, D Latini, I Marquette, J Zhang, YZ Zhang, Polynomial algebra from Lie algebra reduction chain $su(4) \supset su(2) \times su(2)$: The supermultiplet model, arXiv:2503.04108.

We classify finite-dimensional Nichols algebras of Yetter-Drinfeld modules with indecomposable support over finite solvable groups in characteristic 0, using a variety of methods including reduction to positive characteristic. As a consequence, all Nichols algebras over groups of odd order are of diagonal type, which allows us to describe all pointed Hopf algebras of odd dimension.