Cluster algebras and quantum affine algebras, after B. Leclerc Bernhard Keller

This talk, based on [14], is a report on recent work by B. Leclerc on a new type of categorification for cluster algebras.

Cluster algebras were invented by Fomin and Zelevinsky [8] at the beginning of this decade. Since then, a major effort has gone into their categorification (cf. for example [15] [1] [2] [3] [10]). Namely, in many cases, it was proved that for a given cluster algebra A, there exists a triangulated (or Frobenius) category C, such that

• the cluster variables x of A correspond to certain indecomposables T_{x} of C,

• two cluster variables x and y belong to the same cluster if and only if there are no non split extensions between the corresponding objects T_{x} and T_{y},

• the cluster monomial m = xy ⋯ z corresponds to the the object M = T_{x} ⊕ T_{y} ⊕ ⋯ T_{z} of C,

• the exchange relations xx^{∗} = m + m′ of A correspond to triangles

T_{x} → M → T_{x^{∗}} → ΣT_{x} and T_{x^{∗}} → M′ → T_{x} → ΣT_{x^{∗}}

of C.

It was shown that in certain cases, the objects T_{x} are precisely the indecomposable rigid objects of C, i.e. those without selfextensions. For example, when A has only a finite number of cluster variables, then all indecomposable objects of C are rigid and the cluster variables are in bijection with the indecomposables of C. In this case, it was also shown that the cluster algebra A can be realized as a sort of dual Hall algebra of the triangulated category C and that its commutativity reflects the fact that C is 2-Calabi-Yau, i.e. the space Ext_{C}^{1}(L, M) is in natural duality with Ext_{C}^{1}(M, L) for all objects L and M of C.

This type of categorification is very useful: it has allowed to prove properties of cluster algebras which appear to be beyond the reach of the purely combinatorial methods, cf. for example [4]. However, it is perhaps not the most natural notion of categorification which we could expect for a cluster algebra.

In order to categorify an algebra A defined over the integers and endowed with a distinguished ℤ-basis B, one would rather look for an abelian category M which is monoidal (i.e. endowed with a tensor product) and whose Grothendieck ring is isomorphic to A in such a way that the elements of B correspond to the classes of the simple objects of M, cf. for example [12]. The definition of a ‘canonical basis’ for a general cluster algebra is still an open problem (cf. for example [18]) but in many cases, this basis is known, for example when there is only a finite number of clusters or when the algebra already admits a canonical basis in the sense of Kashiwara and Lusztig. One then expects [8] that the cluster monomials, and in particular the cluster variables, belong to this canonical basis.

The natural notion of ‘tensor-indecomposability’ is primality: an object of M is prime, if it does not admit a non trivial tensor factorization. In order to categorify a cluster algebra A, one would therefore look for an abelian monoidal category M whose Grothendieck ring is A and such that

• the cluster variables x of A are the classes of certain prime simple objects S_{x} of M,

• two cluster variables x and y belong to the same cluster if and only if S_{x} ⊗ S_{y} is simple,

• the cluster monomial m = xy ⋯ z in A is the class of the simple object M = S_{x} ⊗ S_{y} ⊗ … ⊗ S_{z} of M,

• the exchange relations xx^{∗} = m + m′ come from exact sequences

0 → M → S_{x} ⊗ S_{x∗} → M′ → 0.

This last condition lacks in symmetry. But if we remember that the cluster algebra is commutative, and thus the tensor product induces a commutative multiplication in the Grothendieck group, we can save symmetry by also requiring the existence of an exact sequence

0 → M′ → S_{x^{∗}} ⊗ S_{x} → M → 0.

The natural notion which replaces rigidity in a monoidal category appears to be ‘reality’: an object of M is real if its tensor square is simple (cf. [13]). The objects S_{x} should exactly be the real prime simple objects of M. When the cluster algebra A has only finitely many cluster variables, all the prime simple objects of M should be real and the cluster variables of A should be in bijection with the prime simples.

Using classical results on representations of quantum affine algebras [5] [6] [9]

[16] [17] B. Leclerc has shown [14] that the cluster algebras of types A_{n}, n ∈ ℕ, and D_{4} (with suitable coefficients) do admit monoidal categorifications given by tensor abelian subcategories of categories of finite-dimensional representations of quantum affine algebras. He conjectures that this holds in many more cases. More precisely, the main conjecture of [14] is the following.

Conjecture (Leclerc). Let Δ be a Dynkin diagram and l ≥ 1 an integer. Let 𝔤 be the complex simple Lie algebra of type Δ, q a non zero complex number which is not a root of unity and U_{q}(𝔤̂) the corresponding quantum affine algebra. Then the category of finite-dimensional representations of U_{q}(𝔤̂) admits a monoidal abelian subcategory M_{Δ,l} which is a monoidal categorification of the cluster algebra associated with a quiver Q_{Δ,l}.

In [14], Leclerc explicitly describes the subcategory M_{Δ,l} and the quiver Q_{Δ,l}. For example, if Δ = D_{5} and l = 3, then the quiver Q_{Δ,l} is as follows

[figure: ∘; ∘; ∘; •; ∘; ∘; ∘; •; ∘; ∘; ∘; •; ∘; ∘; ∘; •; ∘; ∘; ∘; •]

The vertices marked by • correspond to ‘frozen variables’ of the initial cluster. For Δ = A_{1} and l = 3, the quiver Q_{Δ,l} is

\xymatrix{∘ & ∘ \ar[l] \ar[r] & ∘ & • \ar[l]}

In this last case, the subcategory M_{Δ,l} is the full subcategory on the finitedimensional U_{q}(ŝl_{2})-modules all of whose simple subfactors have Drinfeld polynomials with roots in q^{4}, q^{2}, q^{0}, q^{−2}. The isomorphism between the cluster algebra A(Q_{A_{1},3}) and the Grothendieck group K_{0}(M_{A_{1},3}) ⊗_{ℤ} ℚ sends the variables x_{1}, x_{2}, x_{3}, x_{4} of the initial cluster to the classes of the Kirillov-Reshetikhin modules W_{1,q^{0}}, W_{2,q^{−2}}, W_{3,q^{−2}} and W_{4,q^{−4}}. The complete list of the prime simples (up to isomorphism) is

[figure: W_{4,q^{−4}}; α; W_{3,q^{−4}}; W_{3,q^{−2}}; β; W_{2,q^{−4}}; W_{2,q^{−2}}; W_{2,q^{0}}; β; W_{1,q^{−4}}; W_{1,q^{−2}}; W_{1,q^{0}}; W_{1,q^{2}}; α]

The arrows do not indicate morphisms but serve to identify the vertices other than W_{4,q^{−4}} with those of the Auslander-Reiten quiver of the cluster category of type A_{3} (the arrows on the left and on the right of the diagram are identified as indicated by their labels). Every simple module in M_{Δ,l} is a tensor product of modules in this list. A given tensor product of modules in the list other than W_{4,q^{−4}} is simple iff the corresponding direct sum of indecomposables of the cluster category is rigid.

Thus, at least in certain examples, one obtains two rather different categorifications of a given cluster algebra. Table 1 sums up the correspondences. The category C is much ‘smaller’ than M and M is much less well understood than C. It does not seem to be known whether it has enough projectives, for example. The table suggests that M should be an ‘exponential’ of C or C a ‘linearisation’ of M …

cluster algebra A additive categorification C monoidal categorification M
+  ?  ⊕
×  ⊕  ⊗
cluster monomial  rigid object  real simple object
cluster variable  rigid indecomposable  real prime simple

Table 1. Correspondences between categorifications

Finally, let us point out [11] [7] for a very different link between cluster algebras and quantum affine algebras, which does not seem to be related to categorification.

References

[1] Aslak Bakke Buan, Robert J. Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov, Tilting theory and cluster combinatorics, Advances in Mathematics 204 (2) (2006), 572– 618.

[2] Aslak Bakke Buan, Robert J. Marsh, Idun Reiten, and Gordana Todorov, Clusters and seeds in acyclic cluster algebras, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3049–3060 (electronic), With an appendix coauthored in addition by P. Caldero and B. Keller.

[3] Philippe Caldero and Bernhard Keller, From triangulated categories to cluster algebras, arXiv:math.RT/0506018, to appear in Inv. Math.

[4] Philippe Caldero and Markus Reineke, On the quiver Grassmannian in the acyclic case, arXiv:math/0611074.

[5] Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras, Comm. Math. Phys. 142 (1991), no. 2, 261–283.

[6] , A guide to quantum groups, Cambridge University Press, Cambridge, 1994.

[7] Philippe Di Francesco and Rinat Kedem, Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property, arXiv:0803.0362.

[8] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529 (electronic).

[9] Edward Frenkel and Evgeny Mukhin, Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), no. 1, 23–57.

[10] Christof Geiß, Bernard Leclerc, and Jan Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589–632.

[11] Rinat Kedem, Q-systems as cluster algebras, arXiv:0712.2695.

[12] Mikhail Khovanov, Volodymyr Mazorchuk, and Catharina Stroppel, A brief review of abelian categorification, arXiv:math.RT/0702746.

[13] B. Leclerc, Imaginary vectors in the dual canonical basis of U_{q}(𝔫), Transform. Groups 8 (2003), no. 1, 95–104.

[14] Bernard Leclerc, Algèbres affines quantiques et algèbres amassées, in French, Notes from a talk at the Algebra Seminar at the Institut Henri Poincaré, January 14, 2008, available at

www.institut.math.jussieu.fr/projets/tg/Seminaires/2007-08/LeclercExpoIHP08.pdf.

[15] Robert Marsh, Markus Reineke, and Andrei Zelevinsky, Generalized associahedra via quiver representations, arXiv:math.RT/0205152.

[16] Hiraku Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238 (electronic).

[17] , Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math. (2) 160 (2004), no. 3, 1057–1097.

[18] Paul Sherman and Andrei Zelevinsky, Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Mosc. Math. J. 4 (2004), no. 4, 947–974, 982.