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Tensor ideals for quantum groups via minimal tilting complexes

We use minimal tilting complexes to construct an explicit bijection between the set of thick tensor ideals with the two-out-of-three property in the category of finite-dimensional modules over a quantum group at a root of unity and the set of thick tensor ideals in the subcategory of tilting modules. We also explain why the analogous construction for rational representations of a reductive algebraic group over a field of positive characteristic does not give rise to a bijection.


Introduction
One way of understanding the structure of a braided monoidal category C is via its thick tensor ideals, i.e. the sets of objects of C that are closed under direct sums, retracts and tensor products with arbitrary objects of C. When C is triangulated and the tensor product bifunctor is exact in both arguments then one can further impose that tensor ideals should be well-behaved with respect to the triangulated structure (in a suitable sense), and this leads to the rich theory of tensor triangular geometry developed by P. Balmer [Bal05]. When C is abelian, and the tensor product bifunctor is exact in both arguments, we can consider the set of thick tensor ideals J with the so-called two-outof-three property, that is, with the property that for any short exact sequence 0 → A → B → C → 0 in C such that two of the three objects A, B and C belong to J , the third one also belongs to J .
In this note, we study thick tensor ideals with the two-out-of-three property in the category of modules over a quantum group at a root of unity. Let U = U ζ (g) be the quantum group at a primitive ℓ-th root of unity ζ ∈ C corresponding to a complex simple Lie algebra g, defined using divided powers as in [Jan03,Appendix H], and assume that ℓ is odd and strictly greater than the Coxeter number h of g, and not divisible by 3 if g is of type G 2 . Further, let Rep(U ) be the category of finite-dimensional U -modules of type one and let Tilt(U ) be the full subcategory of tilting U -modules, again as in [Jan03,Appendix H]. The thick tensor ideals in Tilt(U ) have been classified by V. Ostrik in [Ost97] in terms of certain Kazhdan-Lusztig cells in the affine Weyl group of U . Here, we provide an explicit bijection between the set of thick tensor ideals in Tilt(U ) and the set of thick tensor ideals with the two-outof-three property in Rep(U ), using the theory of minimal tilting complexes. As explained in [Gru22b, Subsection 2.2], we can associate to every U -module M a bounded minimal complex C min (M ) in Tilt(U ) (unique up to isomorphism), and we call C min (M ) the minimal tilting complex of M . Given a thick tensor ideal I in Tilt(U ), we define Our construction also makes perfect sense for the category Rep(G) of finite-dimensional rational representations of a simply-connected simple algebraic group G over an algebraically closed field of positive characteristic p ≥ h, but the above theorem fails in this setting. In Section 3, we show that the Frobenius twist of the Steinberg module for G generates a thick tensor ideal in Rep(G) with the two-out-of-three property which is not of the form I , for any thick tensor ideal I in the category Tilt(G) of tilting modules in Rep(G).

Minimal tilting complexes
Let us start by recalling some important facts about representations of quantum groups and minimal tilting complexes. We keep the notation from the introduction; in particular U = U ζ (g) is a quantum group at a primitive ℓ-th root of unity ζ ∈ C as in [Jan03, Appendix H], where ℓ is odd (and not divisible by 3 if g is of type G 2 ). We write Rep(U ) for the category of finite-dimensional U -modules of type one. Note that Rep(U ) is a rigid monoidal category because U is a Hopf algebra and that Rep(U ) admits a braiding by [Lus10,Chapter 32]. In the following, we simply refer to the objects of Rep(U ) as U -modules. The category Rep(U ) is also a highest weight category with weight poset (X + , ≤), where X + denotes the set of dominant weights for g (with respect to a fixed base of the root system of g) and ≤ is the usual dominance order. We write Tilt(U ) for the full subcategory of tilting modules in Rep(U ); see Section 1 and Subsection 4.3 in [Gru22b] for more details. As explained in [Gru22b, Subsection 2.2], we can assign to every U -module M a minimal tilting complex C min (M ); it is the unique minimal bounded complex in Tilt(U ) whose cohomology groups are given by Proof. See part (1) of Lemma 2.12 in [Gru22b]. Lemma 1.2. Let 0 → A → B → C → 0 be a short exact sequence of U -modules. Then Proof. This follows from Lemma 2.17 in [Gru22b].
The cohomology groups of X ⊗ Y can be computed via the Künneth-formula (see for instance [Big07,Theorem 4.1]): in the category of complexes in Tilt(U ).
Proof. By the Künneth formula, the tensor product complex satisfies , the bounded derived category of Rep(U ). By part (2) of Lemma 2.12 in [Gru22b], this implies that C min (M ⊗ N ) is the minimal complex of C min (M ) ⊗ C min (N ) and that there is a split monomorphism as in the statement of the lemma.

Tensor ideals for quantum groups
Recall that a thick tensor ideal in Rep(U ) is a set J of U -modules that is closed under direct sums, retracts and tensor products with arbitrary U -modules, that is, for U -modules M and N and X ∼ = M ⊕ N , we have X ∈ J if and only if M ∈ J and N ∈ J , and if M ∈ J then M ⊗ N ∈ J . Thick tensor ideals in Tilt(U ) are defined analogously. We say that a thick tensor ideal J in Rep(U ) has the 2/3-property (or two-out-of-three property) if for any short exact sequence 0 → A → B → C → 0 of U -modules such that two of the U -modules A, B and C belong to J , the third also belongs to J . Proof. First note that I is closed under direct sums and retracts because for all U -modules M 1 and M 2 , by Lemma 1.1. If M 2 ∈ I then all terms of the tensor product complex in the category of complexes in Tilt(U ) by Lemma 1.3 and as I is closed under retracts, we conclude that M 1 ⊗ M 2 ∈ I . Finally, for a short exact sequence for all i ∈ Z, by Lemma 1.2. As I is closed under retracts, we conclude that I has the 2/3property.
The following Lemma justifies the notation I .
Lemma 2.4. Let I be a thick tensor ideal in Tilt(U ). Then I is the smallest thick tensor ideal with the 2/3-property in Rep(U ) that contains I.
Proof. The inclusion I ⊆ I follows from the fact that for every tilting U -module M , we have C min (M ) = M , viewed as a complex concentrated in degree zero; see [Gru22b,Remark 2.11]. Now let J be a thick tensor ideal with the 2/3-property in Rep(U ) such that I ⊆ J , and let M be a U -module in I . We claim that M belongs to J . Writing C min (M ) as As J has the 2/3-property, it suffices to show that im(d −1 ) and ker(d 0 ) belong to J . As C min (M ) is exact in all degrees except zero, there are short exact sequences for all i ≥ 0, where T i ∈ I ⊆ J , and using the 2/3-property, we see that ker(d i ) belongs to J if and only if ker(d i+1 ) belongs to J . Now C min (M ) is bounded, so T i = 0 for some i > 0, and we conclude that ker(d 0 ) belongs to J . Analogously, we see that im(d −1 ) belongs to J , and the claim follows.  The proof in our setting is almost identical; we sketch the main ideas below and refer the reader to [Bal05] for more details. where the intersection runs over all prime thick tensor ideals P in Rep(U ) such that P has the 2/3property and J ⊆ P.
Proof. It is clear that J is contained in J ⊆P P. Now let M be a U -module that does not belong to J and consider the set S = {M ⊗n | n ∈ Z ≥0 }. Since M is a direct summand of M ⊗ M * ⊗ M (where M * denotes the dual of M ) and J is a thick tensor ideal, the set S is disjoint from J ; cf. Remark 4.3 and Proposition 4.4 in [Bal05]. As in Lemma 2.2 in [Bal05], one can use Zorn's lemma to prove that there exists a thick prime tensor ideal P in Rep(U ) such that P has the 2/3-property, J ⊆ P and S ∩ P = ∅. In particular, M does not belong to J ⊆P P and the claim follows.
Before we can prove our main result, we need two more preliminary lemmas.
Lemma 2.7. Let (I j ) j∈J be a collection of thick tensor ideals in Tilt(U ). Then we have Proof. For a U -module M and for i ∈ Z, the tilting module C min (M ) i belongs to j∈J I j if and only if it belongs to I j for all j ∈ J. Hence M belongs to j∈J I j if and only if M belongs to j∈J I j . All the results so far are valid without any additional assumptions on the order ℓ of ζ, apart from those that were made at the beginning of Section 1. For the rest of the section, we assume that ℓ > h, the Coxeter number of g.
Lemma 2.8. Let J be a thick tensor ideal in Rep(U ) with the 2/3-property. Then we have where the intersection runs over all prime thick tensor ideals P in Rep(U ) such that P has the 2/3property and J ⊆ P.
Proof. By Theorem 8.2.1 in [BKN19], every prime thick tensor ideal P in Rep(U ) with the 2/3property is generated by an indecomposable tilting module M . Hence if I is the thick tensor ideal in Tilt(U ) generated by M then P = I , and by Lemma 2.5, we have P ∩ Tilt(U ) = I and therefore P = P ∩ Tilt(U ) . Using Proposition 2.6 and Lemma 2.7, it follows that Remark 2.10. The existence of a bijection between the set of thick tensor ideals in Tilt(U ) and the set of thick tensor ideals in Rep(U ) with the 2/3-property can also be deduced by combining Corollary 7.7.2 and Theorem 8.1.1 in [BKN19]. However, it is not clear from the results in [BKN19] that this bijection can be described in terms of minimal tilting complexes, as we have done here.

The modular case
Let G be a simply connected simple algebraic group over an algebraically closed field k of characteristic p > 0, with the same root system as g. Then the category Rep(G) of finite-dimensional rational G-modules is a highest weight category with weight poset (X + , ≤), and we write L λ and T λ for the simple G-module and the indecomposable tilting G-module of highest weight λ ∈ X + , respectively. As before, we write Tilt(G) for the full subcategory of tilting modules in Rep(G) and we simply refer to the objects of Rep(G) as G-modules.
We can mimic the construction from Section 2 to define a map I → I from the set of thick tensor ideals in Tilt(G) to the set of thick tensor ideals in Rep(G) with the 2/3-property, and all the results from the preceding sections up to (including) Lemma 2.7 carry over to this setting verbatim. Below, we explain why Theorem 2.9 fails when we replace Rep(U ) by Rep(G). We first introduce some more notation.
Let Φ be the root system of G and fix a base Π of Φ corresponding to a positive system Φ + . Further let W be the Weyl group of G and let X and X + be the weight lattice of G and the set of dominant weights with respect to Φ + . A dominant weight is called p-restricted if it belongs to where (− , −) is a W -invariant inner product on X ⊗ Z R and α ∨ = 2α (α,α) denotes the coroot corresponding to α ∈ Φ, and we say that λ ∈ X is p-regular if (λ + ρ, β ∨ ) is not divisible by p for any β ∈ Φ, where ρ is the half-sum of all positive roots. For all λ ∈ X + , we can uniquely write λ = λ 0 + pλ 1 with λ 0 ∈ X 1 and λ 1 ∈ X + , and by Steinberg's tensor product theorem (see [Jan03,Proposition 3.16 λ 1 , where M [1] denotes the pullback of a G-module M along the Frobenius morphism Fr : G → G. For r > 0, let us further denote by G r the r-th Frobenius kernel of G (i.e. the scheme theoretic kernel of the r-th power of Fr), as in Section II.3.1 in [Jan03]. The complexity over G r of a G-module M is the dimension c Gr (M ) = dim V Gr (M ) of its support variety over G r . We do not recall any details of the definition here and instead refer the reader to Section 2 in [NPV02]. Some important properties of the complexity of G-modules are listed below. Proof. This follows from the properties of support varieties listed in (2.2.4)-(2.2.7) in [NPV02].
For r, c > 0, the preceding lemma implies that the set is a thick tensor ideal in Rep(G) with the 2/3-property. In the remainder of this section, we prove that the tensor ideal J 2,≤|Φ| is not of the form I for any thick tensor ideal I in Tilt(G) when p ≥ h.
Suppose from now on that p ≥ h, and note that J 2,≤|Φ| is a proper tensor ideal in Rep(G) because the complexity of the trivial G-module k over G 2 is the dimension c G 2 (k) = dim G of the variety of pairs of commuting nilpotent elements in the Lie algebra g k of G; see Lemma 1.7 in [SFB97a], Theorem 5.2 in [SFB97b] and the introduction to [Pre03]. (p−1)·ρ belongs to J 2,≤|Φ| . Proof. By Theorem 2.4 in [Nak95], we have where c G 1 (k) = |Φ| is the dimension of the nilpotent cone of G and c G 1 (L (p−1)·ρ ) = 0 because the Steinberg module L (p−1)·ρ is projective as a G 1 -module. See Sections II.12.14 and II.10.2 in [Jan03].
Lemma 3.3. Let λ ∈ X + be a p-regular weight. Then there is no proper thick tensor ideal I in Tilt(G) such that L λ belongs to I .
Proof. By Proposition 12 and the remarks after Proposition 9 in [And04], the category Tilt(G) has a unique maximal thick tensor ideal, whose objects are the direct sums of tilting modules T λ with (λ + ρ, α ∨ 0 ) ≥ p, where α ∨ 0 is the highest coroot. (It is called the ideal of negligible tilting modules, cf. [EO22].) Therefore, it suffices to show that there exist i ∈ Z and µ ∈ X + with (µ + ρ, α ∨ 0 ) < p such that T µ is a direct summand of C min (L λ ) i .
Let us write d(λ) for the number of hyperplanes of the form H r,β = {x ∈ X ⊗ Z R | (x + ρ, β ∨ ) = rp} that separate λ and 0, for β ∈ Φ + and r > 0. Then d(λ) equals the good filtration dimension of L λ by Corollary 4.5 in [Par03], whence C min (L λ ) d(λ) = 0 by Lemma 2.15 in [Gru22b]. Using Lemma 1.12 and Proposition 3.3 in [Gru22b] and the linkage principle from Section II.7 in [Jan03], it is straightforward to see that for all i ∈ Z and ν ∈ X + such that T ν is a direct summand of C min (L λ ) i , we have |i| ≤ d(λ) − d(ν). In particular, any weight µ ∈ X + such that T µ is a direct summand of C min (L λ ) d(λ) satisfies d(µ) = 0 and therefore (µ + ρ, α ∨ 0 ) < p (again using the linkage principle), as required. Remark 3.4. More detailed results about the minimal tilting complexes of simple G-modules are proven in Proposition II.2.6 in [Gru22a]; they will be published in a forthcoming article.
Corollary 3.5. The thick tensor ideal J 2,≤|Φ| in Rep(G) is not of the form I for any thick tensor ideal I in Tilt(G).
Proof. By Lemma 3.2, the simple G-module L (p 2 −p)·ρ belongs to J 2,≤|Φ| , but it does not belong to I for any proper thick tensor ideal I in Tilt(G) by Lemma 3.3 since the weight (p 2 − p) · ρ is p-regular.
(Recall that we assume that p ≥ h = (ρ, α ∨ 0 ) + 1.) As J 2,≤|Φ| is a proper thick tensor ideal in Rep(G), it follows that J 2,≤|Φ| is not of the form I for any thick tensor ideal I in Tilt(G).