Gluing vertex algebras

Creutzig T, Kanade S, McRae R (2022)

Publication Type: Journal article

Publication year: 2022

Journal

Advances in Mathematics Elsevier

Book Volume: 396

Article Number: 108174

DOI: 10.1016/j.aim.2021.108174

Abstract

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for C a braided tensor category, we give a detailed account of the canonical algebra construction in the Deligne product C⊠C^rev. Especially, we show that if C is semisimple but not necessarily finite or rigid, then ⨁X∈Irr(C)X^′⊠X is a commutative algebra, where X^′ is a representing object for the functor HomC(•⊗CX,1C) (assuming X^′ exists) and the sum runs over all inequivalent simple objects of U. Conversely, let A=⨁i∈IUi⊠Vi be a simple commutative algebra in a Deligne product U⊠V with U semisimple and rigid but not necessarily finite, and V rigid but not necessarily semisimple. We show that if the unit objects 1U and 1V form a commuting pair in A in a suitable sense, then there is a braid-reversed equivalence between (sub)categories of U and V that sends Ui to Vi^⁎. These results apply when U and V are braided (vertex) tensor categories of modules for simple vertex operator algebras U and V, respectively: Given τ:Irr(U)→Obj(V) such that τ(U)=V, we glue U and V along U⊠V via τ to create A=⨁X∈Irr(U)X^′⊗τ(X). Then under certain conditions, τ extends to a braid-reversed equivalence between U and V if and only if A has a simple conformal vertex algebra structure that (conformally) extends U⊗V. As examples, we glue suitable Kazhdan-Lusztig categories at generic levels to construct new vertex algebras extending the tensor product of two affine vertex subalgebras, and we prove braid-reversed equivalences between certain module subcategories for affine vertex algebras and W-algebras at admissible levels.

Involved external institutions

University of Alberta

Canada (CA) University of Denver

United States (USA) (US) Vanderbilt University

United States (USA) (US)

How to cite

APA:

Creutzig, T., Kanade, S., & McRae, R. (2022). Gluing vertex algebras. Advances in Mathematics, 396. https://doi.org/10.1016/j.aim.2021.108174

MLA:

Creutzig, Thomas, Shashank Kanade, and Robert McRae. "Gluing vertex algebras." Advances in Mathematics 396 (2022).

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