arXiv subjects
Publications and source records attributed to Misha Feigin.
Integrable spin Calogero-Moser type systems with non-symmetric configurations of the singularities of the potential appeared in the work of Chalykh, Goncharenko, and Veselov in 1999. We obtain various generalisations of these examples by making use of the representation theory of Cherednik algebras.
The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations have a rich structure related to the theory of Frobenius manifolds, with many known families of solutions. A Legendre transformation is a symmetry of the WDVV equations, introduced by Dubrovin. We explicitly compute the results of a Legendre transformation applied to $A_n$- and $B_n$-type multi-parameter rational solutions, relating them to known and new trigonometric solutions.
Originally motivated by connections to integrable systems, two natural subalgebras of the rational Cherednik algebra have been considered in the literature. The first is the subalgebra of all degree zero elements and the second is the Dunkl angular momentum subalgebra. In this article, we study the ring-theoretic and homological properties of these algebras. Our approach is to realise them as rings of invariants under the action of certain reductive subgroups of $\rm SL_2$. This allows us to describe their centres. Moreover, we show that they are Auslander-Gorenstein and Cohen-Macaulay and, at $t = 0$, give rise to prime PI-algebras whose PI-degree we compute. Since the degree zero subalgebra can be realized as the ring of invariants for the maximal torus $\rm T \subset SL_2$ and the action of this torus on the rational Cherednik algebra is Hamiltonian, we also consider its (quantum) Hamiltonian reduction with respect to $\rm T$. At $t = 1$, the quantum Hamiltonian reduction of the spherical subalgebra is a filtered quantization of the quotient of the minimal nilpotent orbit closure $\overline{\mathcal O}_{\min}$ in ${\mathfrak gl}(n)$ by the reflection group $W$. At $t = 0$, we get a graded Poisson deformation of the symplectic singularity $\overline{\mathcal O}_{\min}/W$.
Inside the double affine Hecke algebra of type $GL_n$, which depends on two parameters $q$ and $\tau$, we define a subalgebra $\mathbb{H}^{\mathfrak{gl}_n}$ that may be thought of as a $q$-analogue of the degree zero part of the corresponding rational Cherednik algebra. We prove that the algebra $\mathbb{H}^{\mathfrak{gl}_n}$ is a flat $\tau$-deformation of the crossed product of the group algebra of the symmetric group with the image of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{gl}_n)$ under the $q$-oscillator (Jordan-Schwinger) representation. We find all the defining relations and an explicit PBW basis for the algebra $\mathbb{H}^{\mathfrak{gl}_n}$. We describe its centre and establish a double centraliser property. As an application, we also obtain new integrable generalisations of a Macdonald-Ruijsenaars system with an external Morse (exponential) potential introduced by Van Diejen. In particular, we extend Van Diejen's Hamiltonian to the case of a system with two different types of interacting particles, and thus generalise the deformed Macdonald-Ruijsenaars operators of Chalykh-Sergeev-Veselov to the setting with an external field.
The construction of an explicit basis for a free multiarrangement is not easy in general. Inspired by the integral expressions for quasi-invariants of quantum Calogero-Moser systems, we present integral expressions for specific bases of certain multiarrangements. Our construction covers the cases of three lines in dimension $2$ (previously examined by Wakamiko) and free multiarrangements associated with complex reflection groups (Hoge, Mano, Röhrle, Stump). Furthermore, we propose a conjectural basis for the module of logarithmic vector fields of the extended Catalan arrangement of type $B_2$.
Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide polynomial Frobenius manifolds. Flat coordinates of the Frobenius metric $η$ are Saito polynomials which are distinguished basic invariants of the Coxeter group. Algebraic Frobenius manifolds are typically related to quasi-Coxeter conjugacy classes in finite Coxeter groups. We find explicit relations between flat coordinates of the Frobenius metric $η$ and flat coordinates of the intersection form $g$ for most known examples of algebraic Frobenius manifolds up to dimension 4. In all the cases, flat coordinates of the metric $η$ appear to be algebraic functions on the orbit space of the Coxeter group.
To a complex reflection arrangement with an invariant multiplicity function one can relate the space of logarithmic vector fields and the space of quasi-invariants, which are both modules over invariant polynomials. We establish a close relation between these modules. Berest-Chalykh freeness results for the module of quasi-invariants lead to new free complex reflection multiarrangements. K. Saito's primitive derivative gives a linear map between certain spaces of quasi-invariants. We also establish a close relation between non-homogeneous quasi-invariants for root systems and logarithmic vector fields for the extended Catalan arrangements. As an application, we prove the freeness of Catalan arrangements corresponding to the non-reduced root system $BC_N$.
We consider commutativity equations $F_i F_j =F_j F_i$ for a function $F(x^1, \dots, x^N),$ where $F_i$ is a matrix of the third order derivatives $F_{ikl}$. We show that under certain non-degeneracy conditions a solution $F$ satisfies the WDVV equations. Equivalently, the corresponding family of Frobenius algebras has the identity field $e$. We also study trigonometric solutions $F$ determined by a finite collection of vectors with multiplicities, and we give an explicit formula for $e$ for all the known such solutions. The corresponding collections of vectors are given by non-simply laced root systems or are related to their projections to the intersection of mirrors.
We consider the generalised Calogero-Moser-Sutherland quantum integrable system associated to the configuration of vectors $AG_2$, which is a union of the root systems $A_2$ and $G_2$. We establish the existence of and construct a suitably defined Baker-Akhiezer function for the system, and we show that it satisfies bispectrality. We also find two corresponding dual difference operators of rational Macdonald-Ruijsenaars type in an explicit form.
We consider a class of trigonometric solutions of WDVV equations determined by collections of vectors with multiplicities. We show that such solutions can be restricted to special subspaces to produce new solutions of the same type. We find new solutions given by restrictions of root systems, as well as examples which are not of this form. Further, we consider a closely related notion of a trigonometric $\vee$-system and we show that their subsystems are also trigonometric $\vee$-systems. Finally, while reviewing the root system case we determine a version of (generalised) Coxeter number for the exterior square of the reflection representation of a Weyl group.
Let $W$ be a finite Coxeter group and $V$ its reflection representation. The orbit space $\mathcal{M}_W= V/W$ has the remarkable Saito flat metric defined as a Lie derivative of the $W$-invariant bilinear form $g$. We find determinant of the Saito metric restricted to an arbitrary Coxeter discriminant stratum in $\mathcal{M}_W$. It is shown that this determinant is proportional to a product of linear factors in the flat coordinates of the form $g$ on the stratum. We also find multiplicities of these factors in terms of Coxeter geometry of the stratum. This result may be interpreted as a generalisation to discriminant strata of the Coxeter factorisation formula for the Jacobian of the group $W$. As another interpretation, we find determinant of the operator of multiplication by the Euler vector field in the natural Frobenius structure on the strata.
We give a family of solutions of Witten-Dijkgraaf-Verlinde-Verlinde equations in $n$-dimensional space. It is defined in terms of $BC_{n}$ root system and $n+2$ independent multiplicity parameters. We also apply these solutions to define some ${\mathcal N}=4$ supersymmetric mechanical systems.
We consider Dunkl version of Laplace-Runge-Lenz vector associated with a finite Coxeter group $W$ acting geometrically in $\mathbb R^N$ with multiplicity function $g$. This vector generalizes the usual Laplace-Runge-Lenz vector and its components commute with Dunkl-Coulomb Hamiltonian given as Dunkl Laplacian with additional Coulomb potential $γ/r$. We study resulting symmetry algebra $R_{g, γ}(W)$ and show that it has Poincaré-Birkhoff-Witt property. In the absence of Coulomb potential this symmetry algebra $R_{g,0}(W)$ is a subalgebra of the rational Cherednik algebra $H_g(W)$. We show that a central quotient of the algebra $R_{g, γ}(W)$ is a quadratic algebras isomorphic to a central quotient of the corresponding Dunkl angular momenta algebra $H_g^{so(N+1)}(W)$. This gives interpretation of the algebra $H_g^{so(N+1)}(W)$ as the hidden symmetry algebra of Dunkl-Coulomb problem in $\mathbb R^N$. By specialising $R_{g, γ}(W)$ to $g=0$ we recover a quotient of the universal enveloping algebra $U(so(N+1))$ as the hidden symmetry algebra of Coulomb problem in $\mathbb R^N$. We also apply Dunkl Laplace-Runge-Lenz vector to establish maximal superintegrability of generalised Calogero-Moser systems.
We construct ${\mathcal N}=4 \,$ $\, D(2,1;α)$ superconformal quantum mechanical system for any configuration of vectors forming a V-system. In the case of a Coxeter root system the bosonic potential of the supersymmetric Hamiltonian is the corresponding generalised Calogero-Moser potential. We also construct supersymmetric generalised trigonometric Calogero-Moser-Sutherland Hamiltonians for some root systems including $BC_N$.
We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian $H$ associated with a configuration of vectors $AG_2$ on the plane which is a union of $A_2$ and $G_2$ root systems. The Hamiltonian $H$ depends on one parameter. We find an intertwining operator between $H$ and the Calogero-Moser-Sutherland Hamiltonian for the root system $G_2$. This gives a quantum integral for $H$ of order 6 in an explicit form thus establishing integrability of $H$.
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincare-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl(N) version of the subalgebra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
We consider representations of rational Cherednik algebras which are particular ideals in the ring of polynomials. We investigate convergence of the integrals which express the Gaussian inner product on these representations. We derive that the integrals converge for the minimal submodules in types B and D for the singular values suggested by Cherednik with at most one exception, hence the corresponding modules are unitary. The analogous result on unitarity of the minimal submodules in type A was obtained by Etingof and Stoica, we give a different proof of convergence of the Gaussian product in this case. We also obtain partial results on unitarity of the minimal submodule in the case of exceptional Coxeter groups and group B with unequal parameters.
We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painleve IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable.