Tensor envelopes of regular categories

By F. Knop. 2006

64 citations recorded by the source corpus.

Abstract We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T ( A , δ ) depending on a degree function δ. Assume that all objects have only finitely many subobjects. Then our results are as follows: 1. Let N be the maximal proper tensor ideal of T ( A , δ ) . We show that T ( A , δ ) / N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories. 2. Using lattice theory, we give a simple numerical criterion for the vanishing of N . 3. We determine all degree functions for which T ( A , δ ) / N is Tannakian. As a result, we are able to interpolate the representation categories of many series of profinite groups such as the symmetric groups S n , the hyperoctahedral groups S n ⋉ Z 2 n , or the general linear groups GL ( n , F q ) over a fixed finite field. This paper generalizes work of Deligne, who first constructed the interpolating category for the symmetric groups S n .

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4ece88fa078a28a144be1c48a9171b8002191820

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Representation theory in complex rank, II (2014, unknown): “λ = Ind SQn⋉Γn V S |λV ⋉Γ n O V (πλ V ⊗ V⊗|λV |). ByanalogywithDeligne’s construction, F.Knopconstructed aKaroubian tensor category Rep(St ⋉Γt) for complex t, interpolating the categories Rep(Sn⋉Γn) ([Kn2]). More precisely, if tis a nonnegative integer, then the category Rep(St ⋉Γt) projects onto th”
Generalized trace and modified dimension functions on ribbon categories (2010, methodology): “In recent work Deligne [18] and Knop [39] used this approach to show how to construct categories which interpolate among the representation categories of the symmetric groups and GL(n,Fq), respectively.”
Gradings on walled Brauer algebras and Khovanov’s arc algebra (2011, background): “t these interesting modules, e.g. we compute their irreducible socles and heads explicitly. The work of Comes and Wilson is motivated by the study of Deligne’s tensor category Rep(GLδ); see also [D], [Kn]. For δ ∈Z, we conjecture that Deligne’s category is equivalent (on forgetting the tensor struct”