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We establish a link between the new theory of |$q$| -deformed rational numbers and the classical Burau representation of the braid group |${\mathcal {B}}_{3}$|⁠. We apply this link to the open problem of classification of faithful complex specializations of this representation. As a result we provide an answer to this problem in terms of the singular set of the |$q$| -rationals and prove the faithfulness of the Burau representation specialized at complex |$t\in {\mathbb {C}}^{*}$| outside the annulus |$3-2\sqrt 2 \leq |t| \leq 3+2\sqrt 2.$| [ABSTRACT FROM AUTHOR]

This article explores various mathematical concepts related to Markov triples, Mordell triples, and Conway's topograph. It discusses the equations and solutions associated with these concepts, as well as their connections to other mathematical ideas such as hyperbolic geometry and combinatorics of words. The article also discusses the growth patterns and shadows of these triples on the Conway topograph, providing formulas and theorems to describe their behavior. The author acknowledges the contributions of others and notes the publisher's neutral stance. [Extracted from the article]

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let |$\Gamma $| be a finite index subgroup of |$\textrm {SL}(2,\mathbb Z)$| with an action on a complex simple Lie algebra |$\mathfrak g$|⁠ , which can be extended to |$\textrm {SL}(2,{\mathbb {C}})$|⁠. We show that the Lie algebra of the corresponding |$\mathfrak {g}$| -valued modular forms is isomorphic to the extension of |$\mathfrak {g}$| over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups |$\Gamma (N), \, N\leq 6$|⁠ , is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras. [ABSTRACT FROM AUTHOR]

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 rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations A of real hyperplanes with multiplicities admitting the rational Baker–Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call A -Hermite polynomials. These polynomials form a linear basis in the space of A -quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero–Moser operator with harmonic term. In the case of the Coxeter configuration of type A N this leads to a quasi-invariant version of the Lassalle–Nekrasov correspondence and its higher order analogues. [ABSTRACT FROM AUTHOR]

It is shown that the description of certain class of representations of the holonomy Lie algebra gΔ associated with hyperplane arrangement Δ is essentially equivalent to the classification of ∨-systems associated with Δ. The flat sections of the corresponding ∨-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any ∨-system is free in Saito's sense and show this for all known ∨-systems and for a special class of ∨-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic ∨-systems. [ABSTRACT FROM AUTHOR]

We study the orbits and polynomial invariants of certain affine action of the super Weyl groupoid of Lie superalgebra gl(n,m), depending on a parameter. We show that for generic values of the parameter all the orbits are finite and separated by certain explicitly given invariants. We also describe explicitly the special set of parameters, for which the algebra of invariants is not finitely generated and does not separate the orbits, some of which are infinite. [ABSTRACT FROM AUTHOR]

We define the Dunkl and Dunkl-Heckman operators in infinite number of variables and use them to construct the quantum integrals of the Calogero-Moser-Sutherland (CMS) problems at infinity. As a corollary, we have a simple proof of integrability of the deformed quantum CMS systems related to classical Lie superalgebras. We show how this naturally leads to a quantum version of the Moser matrix, which in the deformed case was not known before. [ABSTRACT FROM AUTHOR]

Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno–Drinfeld Lie algebra $\Xmathfrak {t}_{\hspace *{.3pt}n}$. We show that Gaudin subalgebras form a variety isomorphic to the moduli space $\bar M_{0,n+1}$ of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of $\bar M_{0,n+1}$ in a Grassmannian of (n−1)-planes in an n(n−1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over $\bar M_{0,n+1}$ is isomorphic to a sheaf of twisted first-order differential operators. For each representation of the Kohno–Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of $\bar M_{0,n+1}$. [ABSTRACT FROM AUTHOR]

We give one more interpretation of the symbolic formulas U( - N) = U(N) and Sp( - 2N) = SO(2N) and show that they can be extended to the classical symmetric spaces using Macdonald duality for Jack and Jacobi symmetric functions. [ABSTRACT FROM AUTHOR]

A simple integral formula as an iterated residue is presented for the Baker–Akhiezer function related to An-type root system in both the rational and trigonometric cases. We present also a formula for the Baker–Akhiezer function as a Selberg-type integral and generalise it to the deformed An,1-case. These formulas can be interpreted as new cases of explicit evaluation of Selberg-type integrals. [ABSTRACT FROM PUBLISHER]

A simple integral formula as an iterated residue is presented for the Baker-Akhiezer function related to An-type root system in both the rational and trigonometric cases. We present also a formula for the Baker-Akhiezer function as a Selberg-type integral and generalise it to the deformed An,1-case. These formulas can be interpreted as new cases of explicit evaluation of Selberg-type integrals. [ABSTRACT FROM AUTHOR]

A connection between the Yang-Baxter relation for maps and the multidimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations. We use the classification results by Adler, Bobenko, and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multifield integrable equations. [ABSTRACT FROM AUTHOR]

An integral formula for the solutions of Knizhnik–Zamolodchikov (KZ) equation with values in an arbitrary irreducible representation of the symmetric group SN is presented for integer values of the parameter. The corresponding integrals can be computed effectively as certain iterated residues determined by a given Young diagram and give polynomials with integer coefficients. The derivation is based on Schur–Weyl duality and the results of Matsuo on the original SU(n) KZ equation. The duality between the spaces of solutions with parameters m and −m is discussed in relation with the intersection pairing in the corresponding homology groups. [ABSTRACT FROM PUBLISHER]