arXiv subjects

Simone Castellan

Publications and source records attributed to Simone Castellan.

3 recordsLinked to original sources

arXiv2025

Non-homogeneous Koszul duality in representation theory

Motivated by the representation theory of symplectic reflection algebras, deformed preprojective algebras, and graded Hecke algebras, we consider filtered algebras $U$ whose associated graded is Koszul. The Koszul dual of $U$, as defined by Positselski, is a curved dg-algebra. We establish an exact equivalence between the unbounded derived category of $U$ and an explicit quotient of the homotopy category of injective modules over the dual curved dg-algebra. This recovers a special case of a result of Positselski. In the case where $U$ has finite global dimension, the quotient is trivial and hence the unbounded derived category of $U$ is equivalent to the homotopy category of injective modules over the dual curved dg-algebra.

math.RT↗

arXiv2023

Automorphism groups of deformations and quantizations of Kleinian singularities

It is known that, for the algebra of functions on a Kleinian singularity, the parameter space of deformations and the parameter space of quantizations coincide. We prove that, for a Kleinian singularity of type $\mathbf{A}$ or $\mathbf{D}$, isomorphisms between the quantizations are essentially the same as Poisson isomorphisms between deformations. In particular, the group of automorphisms of the deformation and the quantization corresponding to the same deformation parameter are isomorphic. We additionally describe the groups of automorphisms as abstract groups: for type $\mathbf{A}$ they have an amalgamated free product structure, for type $\mathbf{D}$ they are subgroups of the groups of Dynkin diagram automorphisms. For type $\mathbf{D}$ we also compute all the possible affine isomorphisms between deformations; this was not known before.

math.RA↗

arXiv2023

$\hbar$-Vertex algebras and chiralization of star products

We develop the theory of $\hbar$-vertex algebras, algebraic structures closely related to vertex algebras but with a deformed translation covariance axiom. We establish their structure theory, including analogues of Goddard's Uniqueness Theorem, the Reconstruction Theorem, Borcherds Identity, and the OPE Expansion Formula, and introduce the associated notions of $\hbar$-Lie conformal and $\hbar$-Poisson vertex algebras. The formalism provides a natural and simplified construction of the Zhu algebra. The main application is to the chiralization of classical star-products: we show that every star-product on the symmetric algebra of a Lie algebra (or its central extensions) admits a chiralization, and we derive explicit formulae for these chiral star-products, including the Moyal-Weyl and Gutt star-products. Setting $\hbar=0$ recovers explicit deformation quantizations of a broad class of Poisson vertex algebras, including the classical limits of free-boson, $\beta \gamma$-system, affine, and Virasoro vertex algebras.

math.QA↗