Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1212.0531

The Hall module of an exact category with duality

Matthew B. Young
View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct from a finitary exact category with duality a module over its Hall algebra, called the Hall module, encoding the first order self-dual extension structure of the category. We study in detail Hall modules arising from the representation theory of a quiver with involution. In this case we show that the Hall module is naturally a module over the specialized reduced sigma-analogue of the quantum Kac-Moody algebra attached to the quiver. For finite type quivers, we explicitly determine the decomposition of the Hall module into irreducible highest weight modules.

Explore related subjects

math.RTmath.QA

Keep this discovery

BibTeXRIS

Matthew B. Young. 2014-07-11. The Hall module of an exact category with duality. https://arxiv.org/abs/1212.0531

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↗