Gluing vertex algebras

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 circle plus(x is an element of Irr(C)) X' ? X is a commutative algebra, where X' is a representing object for the functor HomC(center dot & nbsp; circle times(C) X, 1(C)) (assuming X' exists) and the sum runs over all inequivalent simple objects of U. Conversely, let A = circle plus(i is an element of I)& nbsp;U-i & nbsp;?& nbsp;V-i 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 U-i to V-i*.& nbsp;These results apply when U and V are braided (vertex) tensor categories of modules for simple vertex operator algebras U and V, respectively: Given tau : Irr(U) -> Obj(V) such that tau(U) = V, we glue U and V along U ? V via tau to create A = circle plus(x is an element of Irr(U)) X' circle times tau(X). Then under certain conditions, tau 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 circle times 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. (C)& nbsp;2021 Elsevier Inc. All rights reserved.