The Looijenga–Lunts–Verbitsky Algebra and Verbitsky’s Theorem

In these notes we review some basic facts about the LLV Lie algebra. It is a rational Lie algebra, introduced by Looijenga–Lunts and Verbitsky, acting on the rational cohomology of a compact Kähler manifold. We study its structure and describe one irreducible component of the rational cohomology in the case of a compact hyperkähler manifold.

We say that the triple (e, h, f ) is an sl 2 -triple. The reason is that we can define a representation sl 2 (F) / / End(V ) of the Lie algebra sl 2 (F) on the vector space V as follows In the rest of these notes, we will mostly be interested in the graded rational vector space V = H * (X, Q) [N ], where X is a compact Kähler manifold of dimension N . Here [m] indicates the shift by m, so that V 0 = H N (X, Q). To any class a ∈ H 2 (X, Q) we can associate the operator in cohomology obtained by taking cup product e a : H * (X, Q) / / H * (X, Q), ω / / a.ω.
The operator h becomes h| H k (X,Q) := (k − N )id.
From Theorem 1.3 we see that if e a has the Lefschetz property (for example if a is a Kähler class), there is an operator f a of degree −2 that makes (e a , h, f a ) an sl 2 -triple. Moreover, the map Remark 1.4. If a ∈ H 1,1 (X, Q) is a Kähler class, it follows from standard Hodge theory that everything can be defined at the level of forms. The dual operator is f a = * −1 e a * , where * is the Hodge star operator. The sl 2 -action preserves the harmonic forms, so it induces an action on cohomology.
Definition 1.5. ( [7,12]) Let X be a compact Kähler manifold. The total Lie algebra g tot (X) of X is the Lie algebra generated by the sl 2 -triples where a ∈ H 2 (X, Q) is a class with the Lefschetz property.
The following is a general result about this Lie algebra for compact Kähler manifolds. Denote by φ the pairing on H * (X, C) given by if α has degree N + 2q or N + 2q + 1. Proposition 1.6. ([7, Proposition 1.6]) The Lie algebra g tot (X) is semisimple and preserves φ infinitesimally. Moreover, the degree-0 part g tot (X) 0 is reductive.

1.3
Now let X be a compact hyperkähler manifold of complex dimension 2n. In this case, the Lie algebra g tot (X) is also called the Looijenga-Lunts-Verbitsky Lie algebra. It is well known that for each hyperkähler metric g on X we get an action of the quaternion algebra H on the real tangent bundle T X. This means that we have three complex structures I, J, K such that where h is the shifted degree operator.
In particular, this definition makes sense for the rank one module H equipped with the standard inner product. This gives a Lie algebra g(H) ⊂ End( • H * ). We denote by g(H) 0 the degree-0 component of g(H) (here the degree is viewed as an endomorphism of the graded vector space). It is a Lie subalgebra, and we denote it by g(H) 0 := [g(H) 0 , g(H) 0 ] its derived Lie algebra. Theorem 2.3. With the above notation we have the following.
(3) The algebra decomposes with respect to the degree as Furthermore, g(H) ±2 H 0 as Lie algebras, and g(H) 0 = g(H) 0 ⊕ Rh with g(H) 0 H 0 ; this last isomorphism is compatible with the actions on Taking exterior powers we get This gives an injective map g(H) / / End( • V * ), given by the natural tensor product representation. It is a direct check that the image of this morphism is exactly the algebra g(V ).
(2) Consider the subrepresentation W ⊂ • H * given by Vol. 90 (2022) The Looijenga-Lunts-Verbitsky Algebra 407 We equip it with the quadratic form given by setting 0 H * ⊕ 4 H * to be a hyperbolic plane, orthogonal to the 3-plane, and {ω I , ω J , ω K } to be an orthonormal basis of the 3-plane. By a direct computation we can see that the action of g(H) respects infinitesimally this quadratic form. This gives a map g(H) / / so(W ) so(4, 1), (2.1) that we next show to be an isomorphism. Since W has dimension 5, the Lie algebra so(W ) has dimension 10. Now consider the following 10 elements of g(H): [11] showed that K IJ acts like the Weil operator associated with the Hodge structure on • H * given by K, and similarly K JK and K IK . This means that it acts on a (p, q) form with respect to K as multiplication by i(p − q). It follows that the ten operators above are linearly independent over W , hence the map is surjective. Moreover they generate g(H) as a vector space. In fact, they generate g(H) as a Lie algebra, and one has the following relations (see [11]): where λ, μ, ν ∈ {I, J, K} and ν = λ, ν = μ. This implies that they are a basis of g(H), hence the map (2.1) is an isomorphism. Point (3) follows using this explicit basis. Indeed we have In particular, we have: Since I, J, K ∈ H 0 act on • H * as Weil operators for the corresponding complex structures on H, the isomorphism is compatible with the actions. Now we can compute the Lie algebra g g . As above we denote by (g g ) 0 the degree-0 part, and by (g g ) 0 := [(g g ) 0 , (g g ) 0 ] its derived Lie algebra.
Proposition 2.4. Let (X, g) be a hyperkähler manifold with a fixed hyperkähler metric.
(1) There is a natural isomorphism of graded Lie algebras g g g(H). In particular g g so(4, 1). (2) The semisimple part (g g ) 0 acts on H * (X, R) via derivations.
Proof. (1). Consider the Lie subalgebraĝ g ⊂ End(Ω • X ), generated by the sl 2 -triples (e a , h, f a ) with a ∈ F (g), at the level of forms (in particular f a = * −1 e a * ). From Theorem 2.3, we see that for every point x ∈ X there is an inclusion g(H) / / End(Ω • X,x ). This gives an inclusion g(H) / / x∈X End(Ω • X,x ). It follows from the definitions that the two algebras g(H) andĝ g are equal as subalgebras of x∈X End(Ω • X,x ). Since the metric g is fixed, the sl 2 -triples (e a , h, f a ) preserve the harmonic forms H * (X), and so doesĝ g . Since H * (X) H * (X, R) we get a morphism This map is surjective, because the image contains the sl 2 -triples that generate g g . Moreover, by explicit computations similar to the proof of Theorem 2.3, we can see that dim g g ≥ 10. Hence the map is an isomorphism.
(2). From the previous proposition we have an isomorphism compatible with the actions on cohomology Hence, it suffices to prove the statement for the action of I, J, K. Each of them gives a complex structure, and acts as the Weil operator on the associated Hodge decomposition. So, the action on (p, q) forms is given by multiplication by i(p − q), which is a derivation.

The Total Lie Algebra
The goal of this section is to prove the following result due to Looijenga and Lunts [7, Proposition 4.5] and Verbitsky [12, Theorem 1.6].
Theorem 3.1. Let X be a hyperkähler manifold. With the above notation, we have the following.
The main geometric input in the proof is the following lemma.
The proof relies on the following fact. In turn, this follows from a celebrated theorem by Yau. While the statement of Theorem 3.1 is over Q, we will give the proof over R following [7].

Proof of Theorem 3.1. Consider the subspace
where V 2 is the abelian Lie subalgebra generated by e a with a ∈ H 2 (X, R), V −2 is the abelian Lie subalgebra generated by the f a with a ∈ H 2 (X, R) where f a is defined, and V 0 is the Lie subalgebra generated by [e a , f b ]. To prove (1) and (2), it is enough to show that V is a Lie subalgebra of g tot (X). Indeed, since g tot (X) is generated by elements contained in V this would imply V = g tot (X). Since V 2 and V −2 are abelian, it suffices to show that Proof of the claim. Proposition 3.3 implies that the set {(a, b) ∈ H 2 (X, R) × H 2 (X, R) | a, b ∈ F (g) for some metric g} is open. Arguing as in the proof of Lemma 3.2 we see that V 0 is generated by the elements [e a , f b ] with a, b ∈ F (g) for some metric g. If we fix a hyperkähler metric g, the elements [e a , f b ] with a, b ∈ F (g) generate the Lie algebra (g g ) 0 and their brackets the Lie subalgebra (g g ) 0 . Thus, V 0 is generated by the Lie algebras (g g ) 0 and their brackets. Since the Lie algebras (g g ) 0 act on cohomology via derivations, the same is true for their brackets, hence V 0 acts via derivations. Moreover, from point (3) of Theorem 2.3 we get the decomposition V 0 = V 0 + Rh. Since g tot (X) 0 is reductive (Proposition 1.6) and h is in the center, we get h ∈ V 0 ⊂ g tot (X) 0 , so the sum is direct. The inclusion [V 0 , V −2 ] ⊂ V −2 is more difficult. Let G 0 ⊂ GL(H * (X, R)) be the closed Lie subgroup with Lie algebra V 0 . For every t ∈ G 0 we have te a t −1 = e t(a) and tht −1 = h, by integrating the analogous relations at the level of Lie algebras. Since the third element of a sl 2 -triple is unique, we get that tf a t −1 = f t(a) . This implies that the adjoint action of G 0 leaves V −2 invariant, hence so does the Lie algebra V 0 .
To summarize, at this point we showed (1) and (2), and also that g tot (X) 0 acts via derivations. It remains to show that g tot (X) 0 so(H 2 (X, R), q).
We begin by defining the map g tot (X) 0 / / so(H 2 (X, R), q). For this, we consider the restriction of the action of g tot (X) 0 to H 2 (X, R), and show that it preserves infinitesimally the Beauville-Bogomolov-Fujiki form q. We can fix a hyperkähler metric g and check this for (g g ) 0 , because these Lie subalgebras generate g tot (X) 0 . From Theorem 2.3 it is enough to check it for the Weil operators associated to the three complex structures I, J, K induced from g. Fix one of them, say I; we have to verify that q(Iα, β) + q(α, Iβ) = 0, for every α, β ∈ H 2 (X, R). This follows from a direct verification using the qorthogonal decomposition induced by the Hodge decomposition with respect to the complex structure I.
To conclude the proof it remains to show that this map is bijective; we begin with the surjectivity. Fix a hyperkähler metric g, the image of the Lie algebra (g g ) 0 in so(H 2 (X, R), q) is generated (as a vector space) by the Weil operators associated to I, J, K. Using this, it is easy to see that (g g ) 0 kills the q-orthogonal complement to the characteristic 3-plane F (g), and it maps onto so(F (g), q| F (g) ).
One can check that varying the metric g the Lie subalgebras so(F (g), q| F (g) ) generate so(H 2 (X, R)), hence the surjectivity.
For the injectivity we proceed as follows. Let SH 2 (X, R) ⊂ H * (X, R) be the graded subalgebra generated by H 2 (X, R); it is a g tot (X) representation for Corollary 4.6. By Lemma 4.7, the map g tot (X) / / gl(SH 2 (X, R)) is injective. Since g tot (X) 0 acts via derivations, the map must be injective already at the level of H 2 (X, R). Proof. Recall that for a rational quadratic space (V, q) there is an isomorphism The desired isomorphism follows from this, at least at the level of vector spaces. The computations to show that it is in fact an isomorphism of Lie algebras are carried out in [4,Proposition 2.7].
Example 3.7. If X is a K3 surface, then the Mukai completionH(X, Q) is the rational cohomology H * (X, Q) with the usual Mukai pairing. This identification is compatible with the action of g tot (X). Proof. Let I, J, K be the three natural complex structures associated to a hyperkähler metric g, and assume I is the given one. As recalled before, the com- Recall that if g is a Lie algebra, the universal enveloping algebra U g of g is the smallest associative algebra extending the bracket on g. It is defined as the quotient of the tensor algebra by the elements of the form: In particular, if g is abelian, then U g = Sym * g.

Corollary 3.9. There is a natural decomposition
where · denotes the multiplication in U g tot (X).
Proof. We have to show that every element in x ∈ U g tot (X) can be written as a sum of elements of the form x 2 · x 0 · x −2 with x i ∈ U g tot (X) i . It is enough to check this on the images of pure tensors. On these it follows from the fact that the bracket is graded and the decomposition in Theorem 3.1.

Primitive Decomposition
In this section, we study the relationship between the actions of g tot (X) and g tot (X) 0 on H * (X, Q), where X is a compact hyperkähler manifold of dimension dim(X) = 2n. The main reference is [7], see also [8,Theorem 4.4]. Funding Open access funding provided by Universitá degli Studi di Roma Tor Vergata within the CRUI-CARE Agreement.
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