Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2504.06760

Applications of Poisson cohomology to the inducibility problems and study of deformation maps

Apurba Das·Ramkrishna Mandal·Anupam Sahoo
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper provides some applications of the Poisson cohomology groups introduced by Flato, Gerstenhaber and Voronov. Given an abelian extension of a Poisson algebra by a representation, we first investigate the inducibility of a pair of Poisson algebra automorphisms and show that the corresponding obstruction lies in the second Poisson cohomology group. Consequently, we obtain the Wells exact sequence connecting various automorphism groups and the second Poisson cohomology group. Subsequently, we also consider the inducibility for a pair of Poisson algebra derivations, obtain the obstruction and construct the corresponding Wells-type exact sequence. To get another application, we introduce the notion of a `deformation map' in a proto-twilled Poisson algebra. A deformation map unifies various well-known operators such as Poisson homomorphisms, Poisson derivations, crossed homomorphisms, Rota-Baxter operators of any weight, twisted Rota-Baxter operators, Reynolds operators and modified Rota-Baxter operators on Poisson algebras. We show that a deformation map $r$ induces a new Poisson algebra structure and a suitable representation of it. The corresponding Poisson cohomology is defined to be the cohomology of the deformation map $r$. Finally, we study the formal deformations of the operator $r$ in terms of the cohomology.

Explore related subjects

math.RTmath.KTmath.RA

Keep this discovery

BibTeXRIS

Apurba Das, Ramkrishna Mandal, Anupam Sahoo. 2025-04-09. Applications of Poisson cohomology to the inducibility problems and study of deformation maps. https://arxiv.org/abs/2504.06760

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Whittaker supermodules over Lie superalgebras

In this paper, we develop a general theory of quasi-Whittaker supermodules over Lie superalgebras induced from an arbitrary ideal. We determine the quasi-Whittaker vectors in universal supermodules, establish an irreducibility criterion, and classify several families of irreducible supermodules. The odd part produces a new irreducibility phenomenon absent from the Lie algebra setting. As applications, we determine all irreducible quasi-Whittaker supermodules over the $N=1$ super Schr\"odinger algebra and the $N=1$ $\frac{3}{2}$-conformal Galilei superalgebra, and over the complete spectrum-generating superalgebra in a special case.

math.RT↗
arXiv2026

Rankin--Selberg integrals of opposite conductor--one newforms

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni--Zhang's conjecture on the nonvanishing of local relative characters.

math.RT↗
arXiv2026

Obstructions to Jacobi-Finiteness of Quivers with Potentials

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod--Shafarevich--Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

math.RT↗