arXiv subjects

Alexei Latyntsev

Publications and source records attributed to Alexei Latyntsev.

6 recordsLinked to original sources

arXiv2026

Critical CoHAs, vertex coalgebras and Deformed Drinfeld coproducts

We construct a vertex coproduct on the Kontsevich--Soibelman cohomological Hall algebra (CoHA) of a quiver with potential, following Joyce (2018). We show it forms a vertex bialgebra. By applying a vertex algebraic analogue of Majid--Radford bosonisation, we form an extension of the CoHA of quivers with potential which incorporates a Cartan part. In the case of ADE quivers our vertex coproduct recovers Drinfeld's deformed coproduct on the Yangian. We compare the vertex coproduct with a localised coproduct defined by Davison and with the construction of Dotsenko--Mozgovoy when the potential is trivial. Our construction gives a new proof of the cohomological integrality theorem for symmetric quivers with trivial potential.

math.RT↗

arXiv2025

Virtual localization revisited

Let $T$ be a split torus acting on an algebraic scheme $X$ with fixed locus $Z$. Edidin and Graham showed that on localized $T$-equivariant Chow groups, (a) push-forward $i_*$ along $i : Z \to X$ is an isomorphism, and (b) when $X$ is smooth the inverse $(i_*)^{-1}$ can be described via Gysin pullback $i^!$ and cap product with $e(N)^{-1}$, the inverse of the Euler class of the normal bundle $N$. In this paper we show that (b) still holds when $X$ is a quasi-smooth derived scheme (or Deligne-Mumford stack), using virtual versions of the operations $i^!$ and $(-)\cap e(N)^{-1}$. As a corollary we prove the virtual localization formula $[X]^{vir} = i_* ([Z]^{vir} \cap e(N^{vir})^{-1})$ of Graber-Pandharipande without global resolution hypotheses and over arbitrary base fields. We include an appendix on fixed loci of group actions on (derived) stacks which should be of independent interest.

math.AG↗

arXiv2025

The stacky concentration theorem

We give a sufficient criterion for the Chow or algebraic bordism groups of an algebraic stack, localized at a set of Chern classes of line bundles, to be concentrated in some closed substack. This is a vast generalization of the torus fixed-point localization theorem in equivariant intersection theory, which is the special case of the stack quotient of a scheme $X$ by an action of a torus $T$. Taking on the one hand an algebraic stack in place of $X$, we deduce a generalization of torus localization to algebraic stacks. Taking on the other hand any algebraic group $G$ instead of $T$, we obtain a localization theorem in $G$-equivariant intersection theory.

math.AG↗

arXiv2025

Orthosymplectic modules of cohomological Hall algebras

We study modules and comodules for cohomological Hall algebras equipped with their vertex coproducts arising as objects with classical type stabilizer groups. Specifically we consider how classical type parabolic induction gives rise to actions of CoHAs of quivers with potential, of preprojective algberas, and of dimension zero sheaves on a smooth proper surface. In all cases the CoHA action is compatible with a localised (and vertex) coaction making the module a twisted Yetter-Drinfeld module over the CoHA with its localised braided bialgebra structure. In the case of dimension zero sheaves on a surface the action is related to an approach to the AGT conjecture in classical type using moduli stacks of orthosymplectic perverse coherent sheaves, a compactification of the stack of classical type bundles on a surface.

math.AG↗

arXiv2023

Factorisation quantum groups

We develop vertex and factorisation algebra analogues of the theory of quasitriangular bialgebras. Analogously to the classical theory, we prove their categories of representations are controlled by spectral R-matrices. In the vertex algebra case this generalises previous notions due to Etingof-Kazhdan and Frenkel-Reshetikhin. Finally we give examples, including Borcherds twists and homology vertex algebras.

math.AG↗

arXiv2021

Cohomological Hall algebras and vertex algebras

The moduli stack of representations of a quiver, or coherent sheaves on a proper curve, carries two structures on its cohomology: a Hall algebra and braided vertex coalgebra. We show that they are compatible, by developing a formulation of abelian localisation which works in the derived/singular setting. An application of these techniques gives an explicit formula for the cohomological Hall algebra products.

math.AG↗