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Jonathan Nilsson

Publications and source records attributed to Jonathan Nilsson.

10 recordsLinked to original sources
arXiv2025

Lifting free modules to generalized Weyl algebras

We study modules over a generalized Weyl algebra $R(\sigma,a)$ which are free when restricted to the base ring $R$. When $R$ is an integral domain, we construct all such finite-rank modules up to isomorphism, leading to new simple modules over a variety of algebras. In particular, we show that free modules that have rank $1$ over $R$ can be parametrized as $V_{\mathsf{p}}$ where $\mathsf{p}$ is a divisor of $a$. We give simplicity criteria for $V_{\mathsf{p}}$ and, additionally, when $R$ is a PID, provide a complete combinatorial description of the submodule structure of $V_{\mathsf{p}}$ and of the weight modules occurring as subquotients. We also show that, under some mild conditions on $R(\sigma,a)$, there exist simple $R$-free modules of arbitrary finite rank. We apply our results to $\mathfrak{sl}_2$ in order to construct new families of simple Cartan-free modules of all finite ranks.

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arXiv2019

Simple $\mathfrak{sl}(V)$-modules which are free over an abelian subalgebra

Let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{sl}(V)$ of maximal dimension and let $\mathfrak{n} \subset \mathfrak{p}$ be the corresponding nilradical. In this paper we classify the set of $\mathfrak{sl}(V)$-modules whose restriction to $U(\mathfrak{n})$ is free of rank $1$. It turns out that isomorphism classes of such modules are parametrized by polynomials in $\dim V-1$ variables. We determine the submodule structure for these modules and we show that they generically are simple.

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arXiv2019

Gauge modules for the Lie algebras of vector fields on affine varieties

For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra $A$ of functions and the Lie algebra $\mathcal{V}$ of vector fields on the variety. We prove that a gauge module corresponding to a simple $\mathfrak{gl}_N$-module is irreducible as a module over the Lie algebra of vector fields unless it appears in the de Rham complex.

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arXiv2017

Representations of Lie algebras of vector fields on affine varieties

For an irreducible affine variety $X$ over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on $X$ - gauge modules and Rudakov modules, which admit a compatible action of the algebra of functions. Gauge modules are generalizations of modules of tensor densities whose construction was inspired by non-abelian gauge theory. Rudakov modules are generalizations of a family of induced modules over the Lie algebra of derivations of a polynomial ring studied by Rudakov. We prove general simplicity theorems for these two types of modules and establish a pairing between them.

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arXiv2017

Representations of the Lie algebra of vector fields on a sphere

For an affine algebraic variety $X$ we study a category of modules that admit compatible actions of both the algebra of functions on $X$ and the Lie algebra of vector fields on $X$. In particular, for the case when $X$ is the sphere $\mathbb{S}^2$, we construct a set of simple modules that are finitely generated over $A$. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational $\mathrm{GL}_2$-modules.

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arXiv2016

$\mathcal{U}(\mathfrak{h})$-free modules and coherent families

We investigate the category of U(h)-free g-modules. Using a functor from this category to the category of coherent families, we show that U(h)-free modules only can exist when g is of type A or C. We then proceed to classify isomorphism classes of U(h)-free modules of rank 1 in type C, which includes an explicit construction of new simple sp(2n)-modules. Finally, we show how translation functors can be used to obtain simple U(h)-free modules of higher rank.

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arXiv2015

New family of simple $\mathfrak{gl}_{2n}(\mathbb{C})$-modules

We construct a new family of simple $\mathfrak{gl}_{2n}$-modules which depends on $n^2$ generic parameters. Each such module is isomorphic to the regular $U(\mathfrak{gl}_{n})$-module when restricted the $\mathfrak{gl}_{n}$-subalgebra naturally embedded into the top-left corner.

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arXiv2013

Simple sl_{n+1}-module structures on U(h)

We study the category M consisting of U(sl_{n+1})-modules whose restriction to U(h) is free of rank 1, in particular we classify isomorphism classes of objects in M and determine their submodule structure. This leads to new sl_{n+1}-modules. For n=1 we also find the central characters and derive an explicit formula for taking tensor product with a simple finite dimensional module.

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arXiv2012

Enumeration of basic ideals in type B

The number of ad-nilpotent ideals of the Borel subalgebra of the classical Lie algebra of type B_n is determined using combinatorial arguments involving a generalization of Dyck-paths. We also solve a similar problem for the untwisted affine Lie algebra of type ~B_n, where we instead enumerate a certain class of ideals called basic ideals. This leads to an explicit formula for the number of basic ideals in ~B_n, which gives rise to a new integer sequence.

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