Lie algebras in symmetric tensor categories

Introduction

The guiding question of this paper is how much of the architecture of classical Lie theory survives in symmetric tensor categories over fields of arbitrary characteristic. Many classical arguments depend on properties of vector spaces that need not persist in this setting. For instance, in positive characteristic, a skew-symmetric bracket satisfying the Jacobi identity need not have the PBW property [9]. Thus, extending Lie theory from vector spaces to symmetric tensor categories requires more than a formal translation.

There is a particular reason to ask this question now. Over a field of characteristic zero, Deligne proved that every pre-Tannakian category of moderate growth admits a fiber functor to supervector spaces [6], and hence can be reconstructed from ordinary or super group theory. The direct analogue fails in characteristic $p$. The fundamental counterexamples are the Verlinde categories $\mathrm{Ver}_p$, obtained by semisimplifying the category of tilting modules for $\mathrm{SL}_2$; for $p \ge5$, they admit no symmetric tensor functor to either $\mathrm{Vec}$ or $\mathrm{sVec}$. Moreover, $\mathrm{Ver}_p$ plays an important role: Ostrik proved that every symmetric fusion category in characteristic $p$ admits a symmetric tensor functor to $\mathrm{Ver}_p$ [O], and Coulembier, Etingof, and Ostrik subsequently extended this result to semisimple pre-Tannakian categories of moderate growth [4].

The study of algebraic structures in symmetric tensor categories has expanded considerably in recent years. Much of the recent work has focused on the Verlinde categories, including affine group schemes and Harish–Chandra pairs [V2, M2], representations of general linear groups [V1, K, U], categorical Chevalley–Eilenberg cohomology [11], and the construction and classification of Lie algebra objects [2, 12 N]. Contragredient Lie algebras have also been defined in arbitrary symmetric tensor categories [3].

The present paper develops some of the general tools needed for this program. Let $\mathcal{C}$ be a symmetric tensor category over an algebraically closed field $k$, and let $\mathfrak{g}$ be an operadic Lie algebra in $\mathcal{C}$. Although the natural map $\mathfrak{g} \to U(\mathfrak{g})$ need not be injective [9], its kernel is always central. If $\mathcal{C}$ is quasi-semisimple, meaning that symmetric algebras preserve monomorphisms, then the quotient by this kernel has the PBW property; see Proposition 2.3. In particular, every simple operadic Lie algebra in a quasi-semisimple symmetric tensor category has the PBW property.

Our first main result is a categorical version of Engel’s theorem: if the underlying object of a finite-dimensional Lie algebra $\mathfrak{g}$ is semisimple and every simple subobject of $\mathfrak{g}$ acts nilpotently on a finite-dimensional representation $V$, then $V$ admits a finite filtration strictly lowered by $\mathfrak{g}$; see Theorem 3.3.

We then study invariant bilinear forms. We call an operadic Lie algebra quadratic if it admits a symmetric invariant nondegenerate form, and strongly semisimple if its Killing form is nondegenerate. We prove that a finite-dimensional quadratic operadic Lie algebra with no non-zero abelian ideals is semisimple. Applied to the Killing form, this yields a categorical Cartan criterion, see Theorem 4.10. We also show that every derivation of a strongly semisimple Lie algebra is inner.

We next establish a rigidity result for Lie algebra homomorphisms. In classical theory, the vanishing of the first Chevalley–Eilenberg cohomology gives rigidity only up to conjugation. In $\mathcal{C}$, however, a non-zero object $X$ may be invariantless, meaning that $\operatorname{Hom}_{\mathcal{C}}(\mathbf{1},X) = 0$. For an invariantless $\mathfrak{a}$, we prove that the affine scheme of Lie algebra homomorphisms $\mathfrak{g} \to\mathfrak{a}$ is finite and reduced when the relevant first cohomology vanishes. In particular, this holds when $\mathfrak{g}$ is linearly reductive. This result has no nontrivial analogue in ordinary vector spaces, where the only invariantless object is zero.

We then define a graded category $\mathcal{O}$ and Verma modules $\Delta(X,d)$. As in classical highest-weight theory, every Verma module has a unique graded-simple quotient, and every graded-simple object of $\mathcal{O}$ arises in this way, see Theorems 6.3 and 6.15. In particular, for $\mathfrak{g}$ in $\mathrm{Ver}_p$, we prove $U(\mathfrak{g})$ is noetherian and $\mathcal{O}$ is abelian. The key categorical input is that the symmetric algebra of the nonclassical part of $\mathfrak{g}$ is finite-dimensional; this is Theorem 6.11.

We also define categorical Shapovalov maps to study the reducibility of Verma modules. In characteristic zero, a nondegenerate grading makes every Shapovalov determinant generically nonzero, which yields generic graded simplicity of Verma modules; see Theorem 6.24. In characteristic $p$, central $p$-th powers obstruct the corresponding result for ordinary Verma modules. Under a $p$-nilpotence hypothesis in $\mathrm{Ver}_p$, we prove instead that baby Verma modules are graded-simple on a nonempty Zariski open set; see Theorem 6.26. We also construct an example illustrating the category $\mathcal{O}$ and Shapovalov maps.

The final part of the paper concerns contragredient Lie algebras in quasi-semisimple symmetric tensor categories. In this setting, the Cartan matrix is replaced by representation-theoretic data [3]. We prove a simplicity criterion and show that, whenever the relevant root strings are finite, reflection of the generating data is realized by a Lie algebra isomorphism; see Theorems 7.12 and 7.17. This allows us to develop a root system, see Theorem 7.23. We end this section with the computation of an explicit example.

Lie algebras in symmetric tensor categories

A pseudo-tensor category over $\mathbf{k}$ is a $\mathbf{k}$-linear Karoubian rigid monoidal category whose tensor product is $\mathbf{k}$-bilinear, with finite-dimensional Hom spaces and $\operatorname{End}(\mathbf{1})=\mathbf{k}$. A tensor category is a pseudo-tensor category that is abelian and in which every object has finite length. When a pseudo-tensor category is endowed with a braided structure, we say the category is braided. If the braiding satisfies that the square of the braid is trivial for every pair of objects, we call the category symmetric. For the general theory of tensor categories, we follow [8].

Unless stated otherwise, $\mathcal{C}$ is a symmetric tensor category with simple unit over the algebraically closed field $\mathbf{k}$; it is not assumed semisimple. Tensor product and duality are exact. We suppress associativity constraints and symmetry morphisms, and write $\mathcal{C}^{\mathrm{ind}}$ for the ind-completion. An object of $\mathcal{C}^{\mathrm{ind}}$ is called finite-dimensional if it belongs to $\mathcal{C}$.

We call $\mathcal{C}$ quasi-semisimple if every monomorphism $X\to Y$ induces a monomorphism $S(X)\to S(Y)$ [9 Definition 7.5]. Every semisimple symmetric tensor category is quasi-semisimple. In characteristic zero, every symmetric tensor category is quasi-semisimple, since symmetric powers are the images of symmetrizer idempotents on tensor powers. In positive characteristic quasi-semisimplicity may fail, although all known counterexamples are in characteristic 2 [9 §7.3].

Schematic notation

We frequently express identities of morphisms using element notation. If $X\in\mathcal{C}$, a symbol $x\in X$ appearing in such a formula is a placeholder for a tensor factor; it is not assumed to be a classical element $\mathbf{1}\to X$. Thus this notation remains meaningful even when $\operatorname{Hom}_{\mathcal{C}}(\mathbf{1},X)=0$.

More precisely, let $A$ be a commutative algebra object of $\mathcal{C}$. The set of $A$-valued points of an object $X$ is

A morphism $f:X_1\otimes\cdots\otimes X_n\to Y$ induces a map $f_A:X_1(A)\times\cdots\times X_n(A)\to Y(A)$. It is obtained by tensoring the given $A$-valued points, using the symmetry to move the copies of $A$ next to one another, multiplying them in $A$, and then applying $f$.

At the categorical level, every such formula denotes an equality between morphisms with the same source and target, with the necessary associativity and symmetry morphisms inserted. For example, if $b:\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$ is a Lie bracket, then

denotes the morphism identity

For further examples of this notation, see Definition 4.2 and the proof of Lemma 4.3.

Basic notions

For $V\in\mathcal{C}$, write $\bigwedge^n V$ for the ordinary exterior power, i.e., the object of alternating coinvariants, and

for the divided exterior power, i.e., the object of alternating invariants. There is a natural morphism $\bigwedge^n V\to\Lambda^n V$. It is an isomorphism when $n!$ is invertible. It is also an isomorphism for every $V\in\operatorname{Ver}_p^+$ and every $n$; see [11 Lemma 2.2].

For $V\in\mathcal{C}$, a classical element of $V$ is a morphism $x:1\to V$. We set

with its associative multiplication and commutator bracket. If $A\subseteq\operatorname{End}(V)$ is a subobject, then $A^r\subseteq\operatorname{End}(V)$ denotes the image of the $r$-fold composition map.

A representation of $\mathfrak{g}$ on $V$ is an operadic Lie algebra morphism $\mathfrak{g}\to\mathfrak{gl}(V)$. We write $V^{\mathfrak{g}}$ for the largest subobject of $V$ annihilated by $\mathfrak{g}$; equivalently,

where the arrow is adjoint to the action $\mathfrak{g}\otimes V\to V$. In particular, the adjoint representation

is a morphism of operadic Lie algebras by the Jacobi identity.

An ideal of $\mathfrak{g}$ is a subobject stable under the adjoint action.

In characteristic zero the PBW property is automatic [16, 9]. In positive characteristic it is an additional condition. For a finite-dimensional operadic Lie algebra $\mathfrak{g}\in\mathcal{C}$, let

Proof. The image of the natural map $\mathfrak{g} \to U(\mathfrak{g})$ lies in the commutator Lie algebra of $U(\mathfrak{g})$, so its kernel is an ideal. The adjoint representation (1) extends uniquely to an algebra morphism $U(\mathfrak{g}) \to\operatorname{End}(\mathfrak{g})$. It follows that $I(\mathfrak{g}) \subseteq\ker(\operatorname{ad})$.

Assume now that $C$ is quasi-semisimple. The induced map $\mathfrak{g}/I(\mathfrak{g}) \to U(\mathfrak{g})$ is a monomorphism of operadic Lie algebras. By [9 Proposition 7.7], its source has the PBW property, so it is a Lie algebra in the sense of Definition 2.1. If $\mathfrak{g}$ is simple, then the central ideal $I(\mathfrak{g})$ is zero, since $\mathfrak{g}$ is nonabelian. The final assertion follows. □

Proof. Let $\eta_L : S(L) \twoheadrightarrow\operatorname{gr} U(L)$ and $\eta_A : S(A) \twoheadrightarrow\operatorname{gr} U(A)$ be the canonical morphisms. Naturality gives

The right-hand side is a monomorphism: $\eta_A$ is an isomorphism, and $S(j)$ is split injective because the symmetric-algebra functor preserves the chosen retraction. Thus $\eta_L$ is a monomorphism, and hence an isomorphism. The last assertion uses [9 Corollary 7.3]. □

For $C = \operatorname{Ver}_p$, the PBW property is equivalent to the $p$-Jacobi condition of [9 Theorem 6.6]. The preceding proposition gives, in particular, that $I(\mathfrak{g}) \subseteq\ker(\operatorname{ad}) = Z(\mathfrak{g})$; compare [N, Proposition 2.3].

An object $X \in C$ is invariantless if $\operatorname{Hom}_C(1, X) = 0$. A finite-dimensional Lie algebra $\mathfrak{g}$ is linearly reductive if the category of its finite-dimensional modules in $C$ is semisimple.

If $C$ is quasi-semisimple, then Proposition 2.3 implies that every simple or semisimple operadic Lie algebra is a Lie algebra. A simple operadic Lie algebra is perfect and has zero center. Distinct simple ideals in a direct sum commute.

Engel's theorem

In this section $C$ is an arbitrary symmetric tensor category, and $\Bbbk$ need not be algebraically closed.

Let $V \in C$. A subobject $U \subseteq\operatorname{End}(V)$ is nilpotent if $U^n = 0$ for some $n \ge1$. We will need the following lemmas.

Proof. Let $a_i: U^{\otimes i} \otimes V \to V$ be the iterated action and let $\widetilde{a}_i: V \to(U^{\otimes i})^* \otimes V$ be its adjoint. Set $F_iV := \ker\widetilde{a}_i$. Then $F_0V = 0$, $F_NV = V$, and $U \cdot F_iV \subseteq F_{i-1}V$. Conversely, the inclusions imply that $U^N \cdot V = 0$, and hence $U^N = 0$. $\square$

Lemma 3.2. If $U \subseteq\operatorname{End}(V)$ satisfies $U^N = 0$ for some $N \geq1$, then the commutator action of $U$ on $\operatorname{End}(V)$ satisfies

Proof. Put $m = 2N - 1$. We expand