Tensor ideals for quantum groups
via minimal tilting complexes

One way of understanding the structure of a braided monoidal category $\mathcal{C}$ is via its thick tensor ideals, i.e. the sets of objects of $\mathcal{C}$ that are closed under direct sums, retracts and tensor products with arbitrary objects of $\mathcal{C}$. When $\mathcal{C}$ 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 $\mathcal{C}$ is abelian, and the tensor product bifunctor is exact in both arguments, we can consider the set of thick tensor ideals $\mathcal{J}$ with the so-called two-out-of-three property, that is, with the property that for any short exact sequence $0\to A\to B\to C\to 0$ in $\mathcal{C}$ such that two of the three objects $A$, $B$ and $C$ belong to $\mathcal{J}$, the third one also belongs to $\mathcal{J}$.

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 $U=U_{\zeta}(\mathfrak{g})$ be the quantum group at a primitive $\ell$-th root of unity $\zeta\in\mathbb{C}$ corresponding to a complex simple Lie algebra $\mathfrak{g}$, defined using divided powers as in [Jan03, Appendix H], and assume that $\ell$ is odd and strictly greater than the Coxeter number $h$ of $\mathfrak{g}$, and not divisible by $3$ if $\mathfrak{g}$ is of type $\mathrm{G}_{2}$. Further, let $\mathrm{Rep}(U)$ be the category of finite-dimensional $U$-modules of type one and let $\mathrm{Tilt}(U)$ be the full subcategory of tilting $U$-modules, again as in [Jan03, Appendix H]. The thick tensor ideals in $\mathrm{Tilt}(U)$ 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 $\mathrm{Tilt}(U)$ and the set of thick tensor ideals with the two-out-of-three property in $\mathrm{Rep}(U)$, 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 $C_{\mathrm{min}}(M)$ in $\mathrm{Tilt}(U)$ (unique up to isomorphism), and we call $C_{\mathrm{min}}(M)$ the minimal tilting complex of $M$. Given a thick tensor ideal $\mathcal{I}$ in $\mathrm{Tilt}(U)$, we define

and prove that $\langle\mathcal{I}\rangle$ is a thick tensor ideal in $\mathrm{Rep}(U)$ with the two-out-of-three property. Our main result is as follows; see Theorem 2.9.

Thick tensor ideals in $\mathrm{Rep}(U)$ have also been studied by B. Boe, J. Kujawa and D. Nakano in [BKN19] via the geometry of the nilpotent cone of $\mathfrak{g}$, and we partially rely on their results.

Our construction also makes perfect sense for the category $\mathrm{Rep}(G)$ of finite-dimensional rational representations of a simply-connected simple algebraic group $G$ over an algebraically closed field of positive characteristic $p\geq h$, 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 $\mathrm{Rep}(G)$ with the two-out-of-three property which is not of the form $\langle\mathcal{I}\rangle$, for any thick tensor ideal $\mathcal{I}$ in the category $\mathrm{Tilt}(G)$ of tilting modules in $\mathrm{Rep}(G)$.

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 $U=U_{\zeta}(\mathfrak{g})$ is a quantum group at a primitive $\ell$-th root of unity $\zeta\in\mathbb{C}$ as in [Jan03, Appendix H], where $\ell$ is odd (and not divisible by $3$ if $\mathfrak{g}$ is of type $\mathrm{G}_{2}$). We write $\mathrm{Rep}(U)$ for the category of finite-dimensional $U$-modules of type one. Note that $\mathrm{Rep}(U)$ is a rigid monoidal category because $U$ is a Hopf algebra and that $\mathrm{Rep}(U)$ admits a braiding by [Lus10, Chapter 32]. In the following, we simply refer to the objects of $\mathrm{Rep}(U)$ as $U$-modules. The category $\mathrm{Rep}(U)$ is also a highest weight category with weight poset $(X^{+},\leq)$, where $X^{+}$ denotes the set of dominant weights for $\mathfrak{g}$ (with respect to a fixed base of the root system of $\mathfrak{g}$) and $\leq$ is the usual dominance order. We write $\mathrm{Tilt}(U)$ for the full subcategory of tilting modules in $\mathrm{Rep}(U)$; 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 $C_{\mathrm{min}}(M)$; it is the unique minimal bounded complex in $\mathrm{Tilt}(U)$ whose cohomology groups are given by

for $i\in\mathbb{Z}$. (See [Gru22b, Definition 2.2] for the definition of a minimal complex.) Below, we list some key properties of minimal tilting complexes.

For $U$-modules $M$ and $N$, let us write $M\stackrel{{\scriptstyle\oplus}}{{\subseteq}}N$ if $M$ admits a split embedding into $N$, and denote by $C_{\mathrm{min}}(M)_{i}$ the term in homological degree $i\in\mathbb{Z}$ of $C_{\mathrm{min}}(M)$.

Given two complexes $X=(\cdots\to X_{i}\xrightarrow{d_{i}}X_{i+1}\to\cdots)$ and $Y=(\cdots\to Y_{i}\xrightarrow{d_{i}^{\prime}}Y_{i+1}\to\cdots)$ of $U$-modules, we define the tensor product complex $X\otimes Y$ with terms

for $i\in\mathbb{Z}$ and differential defined on $X_{j}\otimes Y_{k}$ via $d_{j}\otimes\mathrm{id}_{Y_{k}}+(-1)^{j}\cdot\mathrm{id}_{X_{j}}\otimes d^{\prime}_{k}$. The cohomology groups of $X\otimes Y$ can be computed via the Künneth-formula (see for instance [Big07, Theorem 4.1]):

Recall that a thick tensor ideal in $\mathrm{Rep}(U)$ is a set $\mathcal{J}$ 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 $X\cong M\oplus N$, we have $X\in\mathcal{J}$ if and only if $M\in\mathcal{J}$ and $N\in\mathcal{J}$, and if $M\in\mathcal{J}$ then $M\otimes N\in\mathcal{J}$. Thick tensor ideals in $\mathrm{Tilt}(U)$ are defined analogously. We say that a thick tensor ideal $\mathcal{J}$ in $\mathrm{Rep}(U)$ has the 2/3-property (or two-out-of-three property) if for any short exact sequence $0\to A\to B\to C\to 0$ of $U$-modules such that two of the $U$-modules $A$, $B$ and $C$ belong to $\mathcal{J}$, the third also belongs to $\mathcal{J}$.

The following Lemma justifies the notation $\langle\mathcal{I}\rangle$.

For a thick tensor ideal $\mathcal{J}$ in $\mathrm{Rep}(U)$, it is straightforward to see that $\mathcal{J}\cap\mathrm{Tilt}(U)$ (the set of tilting modules in $\mathcal{J}$) is a thick tensor ideal in $\mathrm{Tilt}(U)$. The next result shows that the map $\mathcal{J}\mapsto\mathcal{J}\cap\mathrm{Tilt}(U)$ is a section to the map $\mathcal{I}\mapsto\langle\mathcal{I}\rangle$ from the set of thick tensor ideals in $\mathrm{Tilt}(U)$ to the set of thick tensor ideals in $\mathrm{Rep}(U)$ with the 2/3-property.

Following [Bal05] and [BKN19], let us call a proper thick tensor ideal $\mathcal{P}$ in $\mathrm{Rep}(U)$ a prime ideal if $M\otimes N\in\mathcal{P}$ implies that either $M\in\mathcal{P}$ or $N\in\mathcal{P}$, for $U$-modules $M$ and $N$. The following result has been proven by P. Balmer in the framework of tensor triangulated geometry in [Bal05, Lemma 4.2]. The proof in our setting is almost identical; we sketch the main ideas below and refer the reader to [Bal05] for more details.

Before we can prove our main result, we need two more preliminary lemmas.

All the results so far are valid without any additional assumptions on the order $\ell$ of $\zeta$, apart from those that were made at the beginning of Section 1. For the rest of the section, we assume that $\ell>h$, the Coxeter number of $\mathfrak{g}$.

Now we are ready to prove our main result:

Let $G$ be a simply connected simple algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p>0$, with the same root system as $\mathfrak{g}$. Then the category $\mathrm{Rep}(G)$ of finite-dimensional rational $G$-modules is a highest weight category with weight poset $(X^{+},\leq)$, and we write $L_{\lambda}$ and $T_{\lambda}$ for the simple $G$-module and the indecomposable tilting $G$-module of highest weight $\lambda\in X^{+}$, respectively. As before, we write $\mathrm{Tilt}(G)$ for the full subcategory of tilting modules in $\mathrm{Rep}(G)$ and we simply refer to the objects of $\mathrm{Rep}(G)$ as $G$-modules.

We can mimic the construction from Section 2 to define a map $\mathcal{I}\mapsto\langle\mathcal{I}\rangle$ from the set of thick tensor ideals in $\mathrm{Tilt}(G)$ to the set of thick tensor ideals in $\mathrm{Rep}(G)$ 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 $\mathrm{Rep}(U)$ by $\mathrm{Rep}(G)$. We first introduce some more notation.

Let $\Phi$ be the root system of $G$ and fix a base $\Pi$ of $\Phi$ corresponding to a positive system $\Phi^{+}$. 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 $\Phi^{+}$. A dominant weight is called $p$-restricted if it belongs to

where $(-\,,-)$ is a $W$-invariant inner product on $X\otimes_{\mathbb{Z}}\mathbb{R}$ and $\alpha^{\vee}=\frac{2\alpha}{(\alpha,\alpha)}$ denotes the coroot corresponding to $\alpha\in\Phi$, and we say that $\lambda\in X$ is $p$-regular if $(\lambda+\rho,\beta^{\vee})$ is not divisible by $p$ for any $\beta\in\Phi$, where $\rho$ is the half-sum of all positive roots. For all $\lambda\in X^{+}$, we can uniquely write $\lambda=\lambda_{0}+p\lambda_{1}$ with $\lambda_{0}\in X_{1}$ and $\lambda_{1}\in X^{+}$, and by Steinberg’s tensor product theorem (see [Jan03, Proposition 3.16]), we have $L_{\lambda}\cong L_{\lambda_{0}}\otimes L_{\lambda_{1}}^{[1]}$, where $M^{[1]}$ denotes the pullback of a $G$-module $M$ along the Frobenius morphism $\mathrm{Fr}\colon G\to G$. 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 $\mathrm{Fr}$), as in Section II.3.1 in [Jan03]. The complexity over $G_{r}$ of a $G$-module $M$ is the dimension $c_{G_{r}}(M)=\dim V_{G_{r}}(M)$ 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.

For $r,c>0$, the preceding lemma implies that the set

is a thick tensor ideal in $\mathrm{Rep}(G)$ with the 2/3-property. In the remainder of this section, we prove that the tensor ideal $\mathcal{J}_{2,\leq\lvert\Phi\rvert}$ is not of the form $\langle\mathcal{I}\rangle$ for any thick tensor ideal $\mathcal{I}$ in $\mathrm{Tilt}(G)$ when $p\geq h$.

Suppose from now on that $p\geq h$, and note that $\mathcal{J}_{2,\leq\lvert\Phi\rvert}$ is a proper tensor ideal in $\mathrm{Rep}(G)$ because the complexity of the trivial $G$-module $\Bbbk$ over $G_{2}$ is the dimension $c_{G_{2}}(\Bbbk)=\dim G$ of the variety of pairs of commuting nilpotent elements in the Lie algebra $\mathfrak{g}_{\Bbbk}$ of $G$; see Lemma 1.7 in [SFB97a], Theorem 5.2 in [SFB97b] and the introduction to [Pre03].