# Publications

Let gg be a complex simple Lie algebra. We study the family of Bethe subalgebras in the Yangian Y(g)Y(g) parameterized by the corresponding adjoint Lie group *G*. We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group G1[[t−1]]G1[[t−1]]. In particular we show that, for regular values of the parameter, these subalgebras are free polynomial algebras with the same Poincaré series as the Cartan subalgebra of the Yangian. Next, we extend the family of Bethe subalgebras to the De Concini–Procesi wonderful compactification G¯¯¯¯⊃GG¯⊃G and describe the subalgebras corresponding to generic points of any stratum in G¯¯¯¯G¯ as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in gg. In particular, we describe explicitly all Bethe subalgebras corresponding to the closure of the maximal torus in the wonderful compactification.

Yangian-like algebras associated with current R-matrices different from the Yang ones are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive or Hecke symmetries. The Yangians of RTT type are close to the corresponding RTT algebras. Some properties of these two classes of the Yangian-like algebras are studied. Thus, evaluation morphisms for them are constructed, their bi-algebra structures are described, and quantum analogs of certain symmetric polynomials, in particular, the quantum determinant, are introduced. It is proved that in any braided Yangian this determinant is always central, whereas, in general, this is not true for the Yangians of RTT type. Analogs of the Cayley-Hamilton-Newton identities in the braided Yangians are exhibited. A bosonic realization of the braided Yangians is performed.

We show that the local observables of the curved βγ system encode the sheaf of chiral differential operators using the machinery of [CG17, CG], which combine renormalization, the Batalin-Vilkovisky formalism, and factorization algebras. Our approach is in the spirit of deformation quantization via Gelfand- Kazhdan formal geometry. We begin by constructing a quantization of the βγ system with an n-dimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields Wn on the target disk, and it is naturally identified with the first Pontryagin class in Gelfand-Fuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of Wn by the closed 2-forms on the disk. By results in [CG17], we then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal βγ vertex algebra. Next, we introduce a version of Gelfand-Kazhdan formal geometry suitable for factorization algebras, and we verify that for a complex manifold X with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs of X.

We discuss relation between the cluster integrable systems and spin chains in the context of their correspondence with 5d supersymmetric gauge theories. It is shown that gl*N* XXZ-type spin chain on *M* sites is isomorphic to a cluster integrable system with *N × M* rectangular Newton polygon and *N × M* fundamental domain of a ‘fence net’ bipartite graph. The Casimir functions of the Poisson bracket, labeled by the zig-zag paths on the graph, correspond to the inhomogeneities, on-site Casimirs and twists of the chain, supplemented by total spin. The symmetricity of cluster formulation implies natural spectral duality, relating gl*N* -chain on *M* sites with the gl*M* -chain on *N* sites. For these systems we construct explicitly a subgroup of the cluster mapping class group GQ and show that it acts by permutations of zig-zags and, as a consequence, by permutations of twists and inhomogeneities. Finally, we derive Hirota bilinear equations, describing dynamics of the tau-functions or A-cluster variables under the action of some generators of GQ.

In this article we study the Gieseker–Maruyama moduli spaces B(e, n) of stable rank 2 algebraic vector bundles with Chern classes c1 = e ∈ {−1, 0} and c2 = n ≥ 1 on the projective space P3 . We construct the two new inﬁnite series Σ0 and Σ1 of irreducible components of the spaces B(e, n) for e = 0 and e = −1, respectively. General bundles of these components are obtained as cohomology sheaves of monads whose middle term is a rank 4 symplectic instanton bundle in case e = 0, respectively, twisted symplectic bundle in case e = −1. We show that the series Σ0 contains components for all big enough values of n (more precisely, at least for n ≥ 146). Σ0 yields the next example, after the series of instanton components, of an inﬁnite series of components of B(0, n) satisfying this property.

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of the topological recursion for monotone Hurwitz numbers, obtained first by Do, Dyer, and Mathews.

Let W be a complex reflection group and H_c(W) the Rational Cherednik algebra for *W* depending on a parameter c. One can consider the category O for H_c(W). We prove a conjecture of Rouquier that the categories O for H_c(W) and H_{c'}(W) are derived equivalent provided the parameters c,c' have integral difference. Two main ingredients of the proof are a connection between the Ringel duality and Harish-Chandra bimodules and an analog of a deformation technique developed by the author and Bezrukavnikov. We also show that some of the derived equivalences we construct are perverse.

We apply the spectral curve topological recursion to Dubrovin's universal Landau-Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

We present a construction of an integrable model as a projective type limit of Calogero-Sutherland models of N fermionic particles, when *N* tends to infinity. Explicit formulas for limits of Dunkl operators and of commuting Hamiltonians by means of vertex operators are given.

We present a combinatorial monomial basis (or, more precisely, a family of monomial bases) in every finite-dimensional irreducible so2n+1-module. These bases are in many ways similar to the FFLV bases for types A and C. They are also defined combinatorially via sums over Dyck paths in certain triangular grids. Our sums, however, involve weights depending on the length of the corresponding root. Accordingly, our bases also induce bases in certain degenerations of the modules but these degenerations are obtained not from the filtration by PBW degree but by a weighted version thereof.

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel--Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive, similarly to what Sottile and White conjectured in Schubert calculus. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

The universal enveloping algebra of any semisimple Lie algebra gg contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of gg. For g=glng=gln the Gelfand–Tsetlin commutative subalgebra in U(g)U(g) arises as some limit of subalgebras from this family. We study the analogous limit of shift of argument subalgebras for classical Lie algebras (g=sp2ng=sp2n or sonson). The limit subalgebra is described explicitly in terms of Bethe subalgebras in twisted Yangians Y−(2)Y−(2) and Y+(2)Y+(2), respectively. We index the eigenbasis of such limit subalgebra in any irreducible finite-dimensional representation of gg by Gelfand–Tsetlin patterns of the corresponding type, and conjecture that this indexing is, in appropriate sense, natural. According to Halacheva et al. (Crystals and monodromy of Bethe vectors. arXiv:1708.05105, 2017) such eigenbasis has a natural gg-crystal structure. We conjecture that this crystal structure coincides with that on Gelfand–Tsetlin patterns defined by Littelmann in Cones, crystals, and patterns (Transform Groups 3(2):145–179, 1998).

We introduce a family of classical integrable systems describing dynamics of M interacting gl*N* integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the GL*N**R*-matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions of the spin type Calogero-Moser models with (the classical analogues of) anisotropic spin exchange operators given in terms of the R-matrix data. In *N* = 1 case the spin Calogero-Moser model is reproduced. Explicit expressions for gl*NM* -valued Lax pair with spectral parameter and its classical dynamical *r*-matrix are obtained. Possible applications are briefly discussed.

The classical matrix-tree theorem discovered by G.Kirchhoff in 1847 expresses the principal minor of the (n x n) Laplace matrix as a sum of monomials of matrix elements indexed by directed trees with n vertices. We prove, for any k >= n, a three-parameter family of identities between degree k polynomials of matrix elements of the Laplace matrix. For k=n and special values of the parameters the identity turns to be the matrix-tree theorem.

For the same values of parameters and arbitrary k >= n the left-hand side of the identity becomes a specific polynomial of the matrix elements called higher determinant of the matrix. We study properties of the higher determinants; in particular, they have an application (due to M.Polyak) in the topology of 3-manifolds.

The notion of compatible braidings was introduced in Isaev *et al.* (1999, *J. Phys. A*, **32**, L115–L121). On the base of this notion, the authors of Isaev *et al.* (1999, *J. Phys. A*, **32**, L115–L121) defined certain quantum matrix algebras generalizing the RTT algebras and Reflection Equation ones. They also defined analogues of some symmetric polynomials in these algebras and showed that these polynomials generate commutative subalgebras, called Bethe. By using a similar approach, we introduce certain new algebras called generalized Yangians and define analogues of some symmetric polynomials in these algebras. We claim that they commute with each other and thus generate a commutative Bethe subalgebra in each generalized Yangian. Besides, we define some analogues (also arising from couples of compatible braidings) of the Knizhnik–Zamolodchikov equation—classical and quantum.

We establish the exact laws of large numbers for two time additive quantities in the raise and peel model, the number of tiles removed by avalanches and the number of global avalanches happened by given time. The validity of conjectures for the related stationary state correlation functions then follow. The proof is based on the technique of Baxter's T-Q equation applied to the associated XXZ chain and on its solution at $\Delta=-2/1$ obtained by Fridkin, Stroganov and Zagier.