David Fernández | Research

PRESENTATION

My research interests lie in a new interdisciplinary area called noncommutative algebraic geometry, which is rooted in algebraic geometry, symplectic geometry, representation theory and mathematical physics. I enjoy defining and studying new noncommutative structures, which can unveil unexpected connections in mathematics and physics.

KEYWORDS

PUBLICATIONS

Symplectic wheelgebras and noncommutative geometry

with E. Herscovich | ArXiv : 2501.17764

In this article, we explore the following statement made by V. Ginzburg and T. Schedler in [Selecta Math. (N.S.) 16 (2010), no. 4, 673-730]: "an adequate framework for doing noncommutative differential geometry is provided by the notion of wheelspace. Wheelspaces form a symmetric monoidal category". However, the category of wheelspaces turns out not to be monoidal. To address this, we introduce generalized wheelspaces, which do form a symmetric monoidal category and provide solid ground for the theory of wheelgebras. To support their first claim, Ginzburg and Schedler defined Poisson (Fock) wheelgebras in connection with Van den Bergh's double Poisson algebras via the Fock functor. We provide strong evidence to their claim by introducing symplectic wheelgebras and proving that the Fock functor sends smooth bisymplectic algebras, as defined by W. Crawley-Boevey, V. Ginzburg and P. Etingof, into our symplectic wheelgebras. In the process, we develop a Cartan calculus adapted to this wheeled context. Moreover, we present a wheeled version of the significant Van den Bergh functor, which facilitates a formalization of the Kontsevich-Rosenberg principle, bridging the noncommutative and commutative frameworks. After establishing that the classical Van den Bergh functor factors through our wheeled version, we show that symplectic Fock wheelgebras naturally induce symplectic algebras on representation schemes.

Noncommutative Poisson vertex algebras and Courant-Dorfman algebras

with L. Álvarez-Cónsul and R. Heluani | Advances in Mathematics, Volume 433 (2023) Paper No. 109269 | Journal | ArXiv : 2106.00270

We introduce the notion of double Courant-Dorfman algebra and prove that it satisfies the so-called Kontsevich-Rosenberg principle, that is, a double Courant-Dorfman algebra induces Roytenberg's Courant-Dorfman algebras on the affine schemes parametrizing finite-dimensional representations of a noncommutative algebra. The main example is given by the direct sum of double derivations and noncommutative differential 1-forms, possibly twisted by a closed Karoubi-de Rham 3-form. To show that this basic example satisfies the required axioms, we first prove a variant of the Cartan identity [L_X,L_Y] =L_{[X,Y]} for double derivations and Van den Bergh's double Schouten-Nijenhuis bracket. This new identity, together with noncommutative versions of the other Cartan identities already proved by Crawley-Boevey-Etingof-Ginzburg and Van den Bergh, establish the differential calculus on noncommutative differential forms and double derivations and should be of independent interest. Motivated by applications in the theory of noncommutative Hamiltonian PDEs, we also prove a one-to-one correspondence between double Courant-Dorfman algebras and double Poisson vertex algebras, introduced by De Sole-Kac-Valeri, that are freely generated in degrees 0 and 1.

Euler continuants in noncommutative quasi-Poisson geometry

with M. Fairon | Forum of Mathematics, Sigma, Volume 10, 2022, e88, 1-54 | Journal | ArXiv : 2105.04858

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 P1 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 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.

On the noncommutative Poisson geometry of certain wild character varieties

with M. Fairon | Submitted | ArXiv : 2103.1011

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. Moreover, he noticed that this noncommutative structure comes from a Hamiltonian double quasi-Poisson algebra constructed from the quiver; this offers a noncommutative analogue of quasi-Hamiltonian reduction. 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 monochromatic interval and the monochromatic triangle.

Double quasi-Poisson algebras are pre-Calabi-Yau

with E. Herscovich | International Mathematics Research Notices (IMRN), Vol 2022, No. 23, pp. 18291– 18345 | Journal | arXiv : 2002.10495

In this article we prove that double quasi-Poisson algebras, which are noncommutative analogues of quasi-Poisson manifolds, naturally give rise to pre-Calabi-Yau algebras. This extends one of the main results in [IK17] (see also [FH19], where a relationship between pre-Calabi-Yau algebras and double Poisson algebras was found. However, a major difference between the pre-Calabi-Yau algebra constructed in the mentioned articles and the one constructed in this work is that the higher multiplications indexed by even integers of the underlying A∞-algebra structure of the pre-Calabi-Yau algebra associated to a double quasi-Poisson algebra do not vanish, but are given by nice cyclic expressions multiplied by explicitly determined coefficients involving the Bernoulli numbers.

Cyclic A∞-algebras and double Poisson algebras

with E. Herscovich | J. Noncommut. Geom. 15 (2021), 241-278 | Journal | arXiv : 1902.00787

In this article we prove that there exists an explicit bijection between nice d-pre-Calabi-Yau algebras and d-double Poisson differential graded algebras, where d∈ℤ, extending a result proved by N. Iyudu and M. Kontsevich. This result, showing that the existing two definitions of noncommutative Poisson structures are equivalent. We also prove that this correspondence is functorial in a quite satisfactory way, giving rise to a (partial) functor from the category of d-double Poisson dg algebras to the partial category of d-pre-Calabi-Yau algebras. Finally, we further generalize it to include double P∞-algebras, as introduced by T. Schedler.

Non-commutative Courant algebroids and quiver algebras

with L. Álvarez-Cónsul | ArXiv : 1705.04285

In this paper, we develop a differential-graded symplectic (Batalin–Vilkovisky) version of the framework of Crawley-Boevey, Etingof and Ginzburg on noncommutative differential geometry based on double derivations to construct non-commutative analogues of the Courant algebroids introduced by Liu, Weinstein and Xu. Adapting geometric constructions of Ševera and Roytenberg for (commutative) graded symplectic supermanifolds, we express the BRST charge, given in our framework by a ‘homological double derivation’, in terms of Van den Bergh’s double Poisson algebras for graded bi-symplectic non-commutative 2-forms of weight 1, and in terms of our non- commutative Courant algebroids for graded bi-symplectic non-commutative 2-forms of weight 2 (here, the grading, or ghost degree, is called weight). We then apply our formalism to obtain examples of exact non-commutative Courant algebroids, using appropriate graded quivers equipped with bi- symplectic forms of weight 2, with a possible twist by a closed Karoubi–de Rham non-commutative differential 3-form.

Noncommutative bi-symplectic NQ-algebras of weight 1

with L. Álvarez-Cónsul | Dynamical Systems, Differential Equations and Applications. AIMS Proceedings (2015) 9-28 | Journal

It is well known that symplectic NQ-manifolds of weight 1 are in 1-1 correspondence with Poisson manifolds. In this article, we prove a version of this correspondence in the framework of noncommutative algebraic geometry based on double derivations, as introduced by Crawley-Boevey, Etingof and Ginzburg. More precisely, we define noncommutative bi-symplectic NQ-algebras and prove that bi-symplectic NQ-algebras of weight 1 are in 1-1 correspondence with double Poisson algebras, as previously defined by Van den Bergh. In this way, we have the following diagram, where the horizontal arrows are representation functors, realizing the Kontsevich--Rosenberg principle:

PhD. THESIS

Non-commutative symplectic NQ-geometry and Courant algebroids

PhD. Thesis | Universidad Autónoma de Madrid | 2016.

Courant algebroids, introduced in differential geometry by Liu, Weinstein and Xu, generalize the notion of the Drinfeld double to Lie bialgebroids. The aim of this thesis is to define a notion of non-commutative Courant algebroid on the path algebra of a doubled graded quiver, satisfying the Kontsevich-Rosenberg principle. A direct approach by axiomatizing its properties fails because the Cartan identities are unknown in this setting. nevertheless, following ideas of Ševera, Roytenberg proved that symplectic NQ-manifolds (that encode higher lie algebroid structures in the Batalin-Vilkovisky formalism) of weight 2 are in 1-1 correspondence with Courant algebroids. Our method to construct non-commutative Courant algebroids is to adapt this result to a graded version of the formalism of Crawley-Boevey, Etingof and Ginzburg, displaying the following diagram:

where the horizontal arrows are representation functors, realizing the Kontsevich--Rosenberg principle. This thesis required the introduction of new concepts, interesting by their own right, which (by the Kontsevich–Rosenberg principle) are non-commutative versions of Lie algebroids, Atiyah algebroids, symplectic homological fields, derived brackets and the Batalin–Vilkovisky (or Maurer–Cartan) master equation, among others. Also, it is proved a non-commutative version of the Darboux theorem for graded symplectic forms.