arXiv subjects

Maxime Fairon

Publications and source records attributed to Maxime Fairon.

18 recordsLinked to original sources

arXiv2026

Double Transposed Poisson Algebras

We introduce double transposed Poisson algebras, a noncommutative analogue of the transposed Poisson algebras of Bai, Bai, Guo and Wu that is compatible with the Kontsevich--Rosenberg principle. We first consider a simplified version which we call id-adapted double transposed Poisson algebras and then explore the general definition. We prove that every such structure on a unital associative algebra $\mathbb{A}$ is governed by a single derivation $\mathbb{A}\to\mathbb{A}\otimes\operatorname{S}(\mathbb{A}/[\mathbb{A},\mathbb{A}])$. Furthermore, this induces a $\operatorname{GL}_N$-equivariant transposed Poisson structure on each representation algebra $\mathbb{A}_N=\Bbbk[\operatorname{Rep}_N(\mathbb{A})]$. We also define $H_0$-transposed Poisson structures, the transposed counterpart of Crawley-Boevey's $H_0$-Poisson structures, and use the trace map to obtain a transposed Poisson structure on the ring of $\operatorname{GL}_N$-invariants $\mathbb{A}_N^{\operatorname{GL}_N}$.

math.RT↗

arXiv2026

Quasi-Poisson varieties from double quasi-Poisson algebras in types $B,C,D$

Double (quasi-)Poisson brackets were introduced on associative algebras by Van den Bergh to induce a (quasi-)Poisson structure on their representation spaces naturally equipped with a $\mathrm{GL}$-action (type $\mathtt{A}$). If there exists a compatible involutive anti-automorphism on the underlying associative algebras, Olshanski and Safonkin proved that this construction can be upgraded to induce a Poisson structure on twisted representation spaces (types $\mathtt{B},\mathtt{C},\mathtt{D}$). We provide an analogous result for double quasi-Poisson brackets, and over an arbitrary semisimple base. We also apply our theory to quivers in order to understand the Poisson structure on twisted (localised multiplicative) quiver varieties. The formalism permits that different vertices are assigned different types. As a first application, we recover the framework of Massuyeau and Turaev for Hopf algebras with a Fox pairing, which induces in particular the Poisson structure of character varieties for the orthogonal or symplectic groups. As a second application, we introduce a modified Kontsevich system.

math.RT↗

arXiv2025

Double Poisson (vertex) algebra cohomology

A noncommutative (NC) version of Poisson geometry was initiated by Van den Bergh by introducing at the level of associative algebras the formalism of double Poisson brackets. Their key property is to induce (standard) Poisson brackets under each representation functor. Then, Pichereau and Van de Weyer developed and studied the corresponding cohomology theory under the assumption that there exists a NC bivector defining the double Poisson bracket. Our first main result is that one can remove this assumption by constructing a completed double Poisson cohomology valid in any situation, hence generalizing the approach of Pichereau-Van de Weyer. As an application, we show that the double Poisson cohomology complex associated to the path algebra of a quiver is acyclic. Furthermore, we show that this new double Poisson cohomology theory can be adapted to weaker forms of double Poisson brackets (called quasi-Poisson and gauged Poisson), and that it is compatible with representation functors. A second focus of this memoir concerns the formalism of double Poisson vertex algebras. These were introduced by De Sole, Kac and the second author, as NC versions of Poisson vertex algebras, which induce the latter structures under each representation functor. Our second main result is the development of cohomology theories for double Poisson vertex algebras. These are NC analogues of the basic, reduced and variational Poisson vertex algebra cohomologies. More importantly, we prove that under each representation functor these cohomology theories are compatible with their commutative counterparts. As an application, we compute the double Poisson vertex algebra cohomology of the generalized NC de Rham complex and of the generalized NC variational complex. Finally, we describe the relation between the double Poisson algebra and double Poisson vertex algebra cohomologies using jet and quotient functors.

math.RT↗

arXiv2024

Modified double brackets and a conjecture of S. Arthamonov

Around 20 years ago, M. Van den Bergh introduced double Poisson brackets as operations on associative algebras inducing Poisson brackets under the representation functor. Weaker versions of these operations, called modified double Poisson brackets, were later introduced by S. Arthamonov in order to induce a Poisson bracket on moduli spaces of representations of the corresponding associative algebras. Moreover, he defined two operations that he conjectured to be modified double Poisson brackets. The first case of this conjecture was recently proved by M. Goncharov and V. Gubarev motivated by the theory of Rota-Baxter operators of nonzero weight. We settle the conjecture by realising the second case as part of a new family of modified double Poisson brackets. These are obtained from mixed double Poisson algebras, a new class of algebraic structures that are introduced and studied in the present work.

math.RA↗

arXiv2023

Compatible Poisson structures on multiplicative quiver varieties

Any multiplicative quiver variety is endowed with a Poisson structure constructed by Van den Bergh through reduction from a Hamiltonian quasi-Poisson structure. The smooth locus carries a corresponding symplectic form defined by Yamakawa through quasi-Hamiltonian reduction. In this note, we include the Poisson structure as part of a pencil of compatible Poisson structures on the multiplicative quiver variety. The pencil is defined by reduction from a pencil of Hamiltonian quasi-Poisson structures which has dimension $\ell(\ell-1)/2$, where $\ell$ is the number of arrows in the underlying quiver. For each element of the pencil, we exhibit the corresponding compatible symplectic or quasi-Hamiltonian structure. We comment on analogous constructions for character varieties and quiver varieties. This formalism is applied to the spin Ruijsenaars-Schneider phase space in order to explain the compatibility of two Poisson structures that have recently appeared in the literature.

math.SG↗

arXiv2023

Functorial constructions related to double Poisson vertex algebras

For any double Poisson algebra, we produce a double Poisson vertex algebra using the jet algebra construction. We show that this construction is compatible with the representation functor which associates to any double Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra. We also consider related constructions, such as Poisson reductions and Hamiltonian reductions, with the aim of comparing the different corresponding categories. This allows us to provide various interesting examples of double Poisson vertex algebras, in particular from double quivers.

math.RT↗

arXiv2022

Around Van den Bergh's double brackets for different bimodule structures

A double Poisson bracket, in the sense of M. Van den Bergh, is an operation on an associative algebra $A$ which induces a Poisson bracket on each representation space $\operatorname{Rep}(A,n)$ in an explicit way. In this note, we study the impact of changing the Leibniz rules underlying a double bracket. This change amounts to make a suitable choice of $A$-bimodule structure on $A\otimes A$. In the most important cases, we describe how the choice of $A$-bimodule structure fixes an analogue to Jacobi identity, and we obtain induced Poisson brackets on representation spaces. The present theory also encodes a formalisation of the widespread tensor notation used to write Poisson brackets of matrices in mathematical physics.

math.RT↗

arXiv2021

Double Multiplicative Poisson Vertex Algebras

We develop the theory of double multiplicative Poisson vertex algebras. These structures, defined at the level of associative algebras, are shown to be such that they induce a classical structure of multiplicative Poisson vertex algebra on the corresponding representation spaces. Moreover, we prove that they are in one-to-one correspondence with local lattice double Poisson algebras, a new important class among Van den Bergh's double Poisson algebras. We derive several classification results, and we exhibit their relation to non-abelian integrable differential-difference equations. A rigorous definition of double multiplicative Poisson vertex algebras in the non-local and rational cases is also provided.

math.RT↗

arXiv2021

Integrable systems on multiplicative quiver varieties from cyclic quivers

We consider a class of complex manifolds constructed as multiplicative quiver varieties associated with a cyclic quiver extended by an arbitrary number of arrows starting at a new vertex. Such varieties admit a Poisson structure, which is obtained by quasi-Hamiltonian reduction. We construct several families of Poisson subalgebras inside the coordinate ring of these spaces, which we use to obtain degenerately integrable systems. We also extend the Poisson centre of these algebras to maximal abelian Poisson algebras, hence defining Liouville integrable systems. By considering a suitable set of local coordinates on the multiplicative quiver varieties, we can derive the local Poisson structure explicitly. This allows us to interpret the integrable systems that we have constructed as new generalisations of the spin Ruijsenaars-Schneider system with several types of spin variables.

nlin.SI↗

arXiv2021

Euler continuants in noncommutative quasi-Poisson geometry

It was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild character varieties described as moduli spaces of points on $\mathbb{P}^1$ by Sibuya. Furthermore, Boalch noticed that these varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which are attached to the quiver $\Gamma_n$ on two vertices and $n$ equioriented arrows. In this article, we go a step further by unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modeled on the quiver $\Gamma_n$. We prove that the Poisson structure carried by these varieties is induced, via the Kontsevich-Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver $\Gamma_n$ such that its noncommutative multiplicative moment map is given in terms of Euler continuants. This result generalises the Hamiltonian double quasi-Poisson algebra associated with the quiver $\Gamma_1$ by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to $\Gamma_n$ admits a factorisation in terms of $n$ copies of the algebra attached to $\Gamma_1$.

math.RT↗

arXiv2021

On the noncommutative Poisson geometry of certain wild character varieties

To show that certain wild character varieties are multiplicative analogues of quiver varieties, Boalch introduced colored multiplicative quiver varieties. They form a class of (nondegenerate) Poisson varieties attached to colored quivers whose representation theory is controlled by fission algebras: noncommutative algebras generalizing the multiplicative preprojective algebras of Crawley-Boevey and Shaw. Previously, Van den Bergh exploited the Kontsevich-Rosenberg principle to prove that the natural Poisson structure of any (non-colored) multiplicative quiver variety is induced by an $H_0$-Poisson structure on the underlying multiplicative preprojective algebra; indeed, it turns out that this noncommutative structure comes from a Hamiltonian double quasi-Poisson algebra constructed from the quiver itself. In this article we conjecture that, via the Kontsevich-Rosenberg principle, the natural Poisson structure on each colored multiplicative quiver variety is induced by an $H_0$-Poisson structure on the underlying fission algebra which, in turn, is obtained from a Hamiltonian double quasi-Poisson algebra attached to the colored quiver. We study some consequences of this conjecture and we prove it in two significant cases: the interval and the triangle.

math.RT↗

arXiv2021

Superintegrability of Calogero-Moser systems associated with the cyclic quiver

We study complex integrable systems on quiver varieties associated with the cyclic quiver, and prove their superintegrability by explicitly constructing first integrals. We interpret them as rational Calogero-Moser systems endowed with internal degrees of freedom called spins. They encompass the usual systems in type $A_{n-1}$ and $B_n$, as well as generalisations introduced by Chalykh and Silantyev in connection with the multicomponent KP hierarchy. We also prove that superintegrability is preserved when a harmonic oscillator potential is added.

nlin.SI↗

arXiv2020

Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems

Double (quasi-)Poisson algebras were introduced by Van den Bergh as non-commutative analogues of algebras endowed with a (quasi-)Poisson bracket. In this work, we provide a study of morphisms of double (quasi-)Poisson algebras, which we relate to the $H_0$-Poisson structures of Crawley-Boevey. We prove in particular that the double (quasi-)Poisson algebra structure defined by Van den Bergh for an arbitrary quiver only depends upon the quiver seen as an undirected graph, up to isomorphism. We derive from our results a representation theoretic description of action-angle duality for several classical integrable systems.

math.QA↗

arXiv2019

Double quasi-Poisson brackets : fusion and new examples

We exhibit new examples of double quasi-Poisson brackets, based on some classification results and the method of fusion. This method was introduced by Van den Bergh for a large class of double quasi-Poisson brackets which are said differential, and our main result is that it can be extended to arbitrary double quasi-Poisson brackets. We also provide an alternative construction for the double quasi-Poisson brackets of Van den Bergh associated to quivers, and of Massuyeau-Turaev associated to the fundamental groups of surfaces.

math.QA↗

arXiv2019

Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers

We study multiplicative quiver varieties associated to specific extensions of cyclic quivers with $m\geq 2$ vertices. Their global Poisson structure is characterised by quasi-Hamiltonian algebras related to these quivers, which were studied by Van den Bergh for an arbitrary quiver. We show that the spaces are generically isomorphic to the case $m=1$ corresponding to an extended Jordan quiver. This provides a set of local coordinates, which we use to interpret integrable systems as spin variants of the trigonometric Ruijsenaars-Schneider system. This generalises to new spin cases recent works on classical integrable systems in the Ruijsenaars-Schneider family.

math-ph↗

arXiv2018

On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system

We suggest a Hamiltonian formulation for the spin Ruijsenaars-Schneider system in the trigonometric case. Within this interpretation, the phase space is obtained by a quasi-Hamiltonian reduction performed on (the cotangent bundle to) a representation space of a framed Jordan quiver. For arbitrary quivers, analogous varieties were introduced by Crawley-Boevey and Shaw, and their interpretation as quasi-Hamiltonian quotients was given by Van den Bergh. Using Van den Bergh's formalism, we construct commuting Hamiltonian functions on the phase space and identify one of the flows with the spin Ruijsenaars-Schneider system. We then calculate all the Poisson brackets between local coordinates, thus answering an old question of Arutyunov and Frolov. We also construct a complete set of commuting Hamiltonians and integrate all the flows explicitly.

math-ph↗

arXiv2017

Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models

We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with $m$ vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case $m=1$ corresponds to the tadpole quiver and the Ruijsenaars-Schneider system and its variants, while for $m>1$ we obtain new integrable systems that generalise the Ruijsenaars-Schneider system. These systems and their quantum versions also appeared recently in the context of supersymmetric gauge theory and cyclotomic DAHAs, as well as in the context of the Macdonald theory.

math.QA↗

arXiv2017

Introduction to graded geometry

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential Z-graded manifolds and algebraic structures.

math.DG↗