Misha Feigin
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View article: Legendre transforms for type A<sub>n</sub> and B<sub>n</sub> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>V</mml:mtext> </mml:mrow> </mml:math> -systems
Legendre transforms for type A<sub>n</sub> and B<sub>n</sub> -systems Open
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…
View article: Free Reflection Multiarrangements and Quasi-Invariants
Free Reflection Multiarrangements and Quasi-Invariants Open
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 re…
View article: Commutativity equations and their trigonometric solutions
Commutativity equations and their trigonometric solutions Open
In the theory of Frobenius manifolds and Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations, one normally assumes that Frobenius algebras associated with a solution have an identity . Equivalently, the corresponding flat metric can be…
View article: Legendre transforms for type $A_{n}$ and $B_{n}$ $\vee$-systems
Legendre transforms for type $A_{n}$ and $B_{n}$ $\vee$-systems Open
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…
View article: Flat coordinates of algebraic Frobenius manifolds in small dimensions
Flat coordinates of algebraic Frobenius manifolds in small dimensions Open
View article: Two invariant subalgebras of rational Cherednik algebras
Two invariant subalgebras of rational Cherednik algebras Open
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 generated by all degree zero elements and the second i…
View article: $q$-Analogue of the degree zero part of a rational Cherednik algebra
$q$-Analogue of the degree zero part of a rational Cherednik algebra Open
Inside the double affine Hecke algebra of type $GL_n$, which depends on two parameters $q$ and $τ$, 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…
View article: Flat coordinates of algebraic Frobenius manifolds in small dimensions
Flat coordinates of algebraic Frobenius manifolds in small dimensions Open
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 C…
View article: Integral expressions for derivations of multiarrangements
Integral expressions for derivations of multiarrangements Open
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 ce…
View article: Bispectrality of $$AG_2$$ Calogero–Moser–Sutherland System
Bispectrality of $$AG_2$$ Calogero–Moser–Sutherland System Open
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 an…
View article: Commutativity equations and their trigonometric solutions
Commutativity equations and their trigonometric solutions Open
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…
View article: Algebra of Dunkl Laplace–Runge–Lenz vector
Algebra of Dunkl Laplace–Runge–Lenz vector Open
We introduce the Dunkl version of the Laplace–Runge–Lenz vector associated with a finite Coxeter group W acting geometrically in $$\mathbb R^N$$ and with a multiplicity function g . This vector generalizes the usual Laplace–Runge–Lenz …
View article: Free reflection multiarrangements and quasi-invariants
Free reflection multiarrangements and quasi-invariants Open
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 re…
View article: Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence
Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence Open
Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero–Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the r…
View article: The Saito determinant for Coxeter discriminant strata
The Saito determinant for Coxeter discriminant strata Open
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 t…
View article: Solutions of $BC_{n}$ Type of WDVV Equations
Solutions of $BC_{n}$ Type of WDVV Equations Open
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 s…
View article: Algebra of Dunkl Laplace-Runge-Lenz vector
Algebra of Dunkl Laplace-Runge-Lenz vector Open
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 com…
View article: $\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields
$\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields Open
It is shown that the description of certain class of representations of the holonomy Lie algebra $\mathfrak g_{\Delta}$ associated with hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems assoc…
View article: Александр Петрович Веселов (к шестидесятилетию со дня рождения)
Александр Петрович Веселов (к шестидесятилетию со дня рождения) Open
г. ноябрь -декабрь т. 71, вып.6 (432) УСПЕХИ МАТЕМАТИЧЕСКИХ НАУК Алекcандр Петрович Веселов (к шестидесятилетию со дня рождения) 4 мая 2015 г. исполнилось шестьдесят лет со дня рождения великолепного математика, яркого преподавателя, замеч…