arXiv subjects

Maarten Bergvelt

Publications and source records attributed to Maarten Bergvelt.

6 recordsLinked to original sources

arXiv2019

$τ$-Functions, Birkhoff Factorizations and Difference Equations

$Q$-systems and $T$-systems are systems of integrable difference equations that have recently attracted much attention, and have wide applications in representation theory and statistical mechanics. We show that certain $τ$-functions, given as matrix elements of the action of the loop group of ${\rm GL}_{2}$ on two-component fermionic Fock space, give solutions of a $Q$-system. An obvious generalization using the loop group of ${\rm GL}_3$ acting on three-component fermionic Fock space leads to a new system of 4 difference equations.

math.RT↗

arXiv2018

Difference Hierarchies for $nT$ $τ$-Functions

We introduce hierarchies of difference equations (referred to as $nT$-systems) associated to the action of a (centrally extended, completed) infinite matrix group $GL_{\infty}^{(n)}$ on $n$-component fermionic Fock space. The solutions are given by matrix elements ($τ$-functions) for this action. We show that the $τ$-functions of type $nT$ satisfy bilinear equations of length $3,4,\dots,n+1$. The $2T$-system is, after a change of variables, the usual $3$ term $T$-system of type $A$. Restriction from $GL_{\infty}^{(n)}$ to a subgroup isomorphic to the loop group $LGL_{n}$, defines $nQ$-systems, studied earlier by the present authors for $n=2,3$.

math.RT↗

arXiv2016

Generalizations of $Q$-systems and Orthogonal Polynomials from Representation Theory

We briefly describe what tau-functions in integrable systems are. We then define a collection of tau-functions given as matrix elements for the action of $\widehat{GL_2}$ on two-component Fermionic Fock space. These tau-functions are solutions to a discrete integrable system called a $Q$-system. We can prove that our tau-functions satisfy $Q$-system relations by applying the famous "Desnanot-Jacobi identity" or by using "connection matrices", the latter of which gives rise to orthogonal polynomials. In this paper, we will provide the background information required for computing these tau-functions and obtaining the connection matrices and will then use the connection matrices to derive our difference relations and to find orthogonal polynomials. We generalize the above by considering tau-functions that are matrix elements for the action of $\widehat{GL_3}$ on three-component Fermionic Fock space, and discuss the new system of discrete equations that they satisfy. We will show how to use the connection matrices in this case to obtain "multiple orthogonal polynomials of type II".

math.RT↗

arXiv2008

$H_D$-Quantum Vertex Algebras

We discuss a class of quantum vertex algebras where not only the commutativity of the vertex algebra is broken by a braiding map $S^{(τ)}$, but also the translation covariance is broken by a translation map $S^{(γ)}$. The new class of quantum vertex operators satisfy a Braided Jacobi Identity containing both the braiding and the translation maps.

math.QA↗

arXiv2008

Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy

Pairs of $n\times n$ matrices whose commutator differ from the identity by a matrix of rank $r$ are used to construct bispectral differential operators with $r\times r$ matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case $r=1$, this reproduces well-known results of Wilson and others from the 1990's relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators. This new class of pairs $(L, Λ)$ of bispectral matrix differential operators is different than those previously studied in that $L$ acts from the left, but $Λ$ from the right on a common $r\times r$ eigenmatrix.

nlin.SI↗

arXiv2005

H_T Vertex Algebras and the Infinite Toda Lattice

Let H_T=C[T,T^{-1}] be the Hopf algebra of symmetries of a lattice of rank 1, or equivalently, H_T is the group algebra of a free Abelian group with one generator T. We construct conformal algebras, vertex Poisson algebras and vertex algebras with H_T as symmetry. For example, the Hamiltonian structure for the infinite Toda lattice gives rise to an H_T-vertex Poisson structure on a free difference algebra. Examples of H_T-vertex algebras are constructed from representations of a class of infinite dimensional Lie algebras related to H_T in the same way loop algebras are related to the Hopf algebra H_D=C[D] of infinitesimal translations used in the usual vertex algebras.

math.QA↗