Tensor ideals, Deligne categories and invariant theory

Abstract

We derive some tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne's universal categories RepO δ , RepGL δ and RepP . These results are then used to obtain new insight into the second fundamental theorem of invariant theory for the algebraic supergroups of types A, B, C, D, P .We also find new short proofs for the classification of tensor ideals in RepSt and in the category of tilting modules for SL2(k) with char(k) > 0 and for Uq(sl2) with q a root of unity. In general, for a simple Lie algebra g of type ADE, we show that the lattice of such tensor ideals for Uq(g) corresponds to the lattice of submodules in a parabolic Verma module for the corresponding affine Kac-Moody algebra.2010 Mathematics Subject Classification. 18D10, 17B45, 17B10, 15A72. Key words and phrases. Monoidal (super)category, tensor ideal, thick tensor ideal, Deligne category, algebraic (super)group, second fundamental theorem of invariant theory, tilting modules, quantum groups. 1 3.2. Proofs. By definition, for a submodule M of P C ½ , we have M(X) ⊂ C(½, X), for all X ∈ ObC.Proposition 3.2.1. (i) For a submodule M of P ½ , we define J M as J M (X, Y ) := ι XY M(X ∨ ⊗ Y ) , for all X, Y ∈ ObC.Then J M is a left-tensor ideal in C.

On the monoidal center of Deligne's category Re̲p(St)

“…Proof The category is a rigid braided category, whose tensor unit is given by the tensor unit in with half‐braiding corresponding to identities, so the endomorphism ring of the unit in is . Hence, by [7, Lemma 2.5.3] (see also [2, Proposition 7.1.4]), the ideal formed by all negligible morphisms is the unique maximal tensor ideal in . The kernel of the monoidal functor is a tensor ideal in giving a semisimple quotient category.…”

Symmetric tensor categories in characteristic 2

“…Then the category T is the category of modules that occur as direct summands in X ⊗i ⊗ X * ⊗j . Let t = m − n and Rep(GL t ) be the Karoubian Deligne category for parameter t. It is shown in [11,8] that the only nontrivial [8,10,20]).…”

“…The Grothendieck ring of the stable category of Ver 2 n is F p [z]/z 2 n−1 −1 (Proposition 4.67). (11) The determinant of the Cartan matrix of a block of size p r (p − 1) in Ver p n is p p r for r ≥ 0, and its entries are 0 or powers of 2 (Proposition 4.69, Corollary 4.27). (12) If p > 2 then Ver p n is a Serre subcategory in Ver p n+k (Proposition 4.58).…”

The periplectic Brauer algebra

“…If , then in is even an isomorphism by [, Theorems 4.1 and 4.5]. This has recently been extended to the case in [, Theorem 8.3.1].…”