Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.0659

Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus

Yuly Billig·Alexander Molev·Ruibin Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique.

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuly Billig, Alexander Molev, Ruibin Zhang. 2007-07-04. Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus. https://arxiv.org/abs/0707.0659

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

KEEP EXPLORING

Related papers

arXiv2026

Computing the center of a fusion category

We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a general-purpose software framework TensorCategories for tensor categories that we develop within the open-source computer algebra system OSCAR. We compute explicit models for the centers in form of the tuples $(X,γ)$ where $X$ is an object and $γ$ is a half-braiding. From these models we can compute the $F$-symbols and $R$-symbols. Using the data from the AnyonWiki, we were able to compute the center together with its $F$-symbols and $R$-symbols for all the 279 multiplicity-free fusion categories up to rank 5, and furthermore some chosen examples of rank 6, including the Haagerup subfactor (presented in a separate paper). Some of these centers yield new high-rank and not multiplicity-free examples of modular categories, e.g. rank 21 and multiplicity 2.

math.RT↗
arXiv2026

Higher Verlinde categories of reductive groups

We define tensor categories ${\sf Ver}_{p^n}(G)$ in characteristic $p$ for connected reductive groups $G$ and positive integers $n$, generalising the semisimple Verlinde categories ${\sf Ver}_p(G)$ originating from Gelfand-Kazhdan and the higher Verlinde categories ${\sf Ver}_{p^n}$ for ${\rm SL}_2$ defined by Benson-Etingof-Ostrik. The construction is based on the definition of ${\sf Ver}_{p^n}$ as an abelian envelope of a quotient of a category of tilting modules, but we also introduce an expanded construction which refines the ${\rm SL}_2$ case and gives new results. In particular, the union ${\sf Ver}_{p^\infty}(G)$ can be derived from the perfection of $G$; certain exact sequences in ${\sf Rep}G$ map to exact sequences in ${\sf Ver}_{p^n}(G)$; and the underlying abelian category of ${\sf Ver}_{p^n}$ can be expressed as a subcategory of ${\sf Rep}{\rm SL}_2$, or as a Serre quotient of a subcategory of ${\sf Rep}{\rm SL}_2$.

math.RT↗
arXiv2026

Quantized integral models of reductive monoids

We construct quantized integral models of reductive monoids and establish their finite-generation properties. We also describe the quantized coordinate rings of the basic affine spaces explicitly in terms of Lusztig's dual canonical bases. The results will be used in the companion paper \cite{CJ2} to establish a classification theorem for split reductive monoids over arbitrary connected base schemes.

math.RT↗