Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1209.6219

Bernstein-Gelfand-Gelfand resolutions for linear superalgebras

Kevin Coulembier
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we construct resolutions of finite dimensional irreducible gl(m|n)-modules in terms of generalized Verma modules. The resolutions are determined by the Kostant cohomology groups and extend the strong (Lepowsky-)Bernstein-Gelfand-Gelfand resolutions to the setting of Lie superalgebras. It is known that such resolutions for finite dimensional representations of Lie superalgebras do not exist in general. Thus far they have only been discovered for gl(m|n) in case the parabolic subalgebra has reductive part equal to gl(m)+gl(n) and for tensor modules. In the current paper we prove the existence of the resolutions for tensor modules of gl(m|n) or sl(m|n) and their duals for an extensive class of parabolic subalgebras including the ones already considered in the literature.

Explore related subjects

math.RTmath-phmath.MP

Keep this discovery

BibTeXRIS

Kevin Coulembier. 2012-09-27. Bernstein-Gelfand-Gelfand resolutions for linear superalgebras. https://arxiv.org/abs/1209.6219

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↗