On 2nd-stage quantization of quantum cluster algebras

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

We prove the Berenstein–Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite-dimensional connected, simply connected simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [5]. We furthermore prove that the corresponding upper quantum cluster algebras coincide with the constructed quantum cluster algebras and exhibit a large number of explicit quantum seeds. Along the way a detailed study of the properties of quantum double Bruhat cells from the viewpoint of noncommutative UFDs is carried out and a quantum analog of the Fomin–Zelevinsky twist map is constructed and investigated for all double Bruhat cells. The results are valid over base fields of arbitrary characteristic and the deformation parameter is only assumed to be a non-root-of-unity.

$Q$-systems are recursion relations satisfied by the characters of the restrictions of special finite-dimensional modules of quantum affine algebras. They can also be viewed as mutations in certain cluster algebras, which have a natural quantum deformation. In this paper, we explain the relation in the simply-laced case between the resulting quantum $Q$-systems and the graded tensor product of Feigin and Loktev. We prove the graded version of the $M=N$ identities, and write expressions for these as non-commuting evaluated multi-residues of suitable products of solutions of the quantum $Q$-system. This leads to a simple reformulation of Feigin and Loktev's fusion coefficients as matrix elements in a representation of the quantum $Q$-system algebra.

In this paper we study consequences of the results of Kang et al. [ Monoidal categorification of cluster algebras , J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$ , then $[S]$ is a cluster monomial in $[\mathscr{C}]$ . If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$ , then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$ . We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [ Triangular bases in quantum cluster algebras and monoidal categorification conjectures , Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$ .

Recently it has become apparent that N = 2 supersymmetric quantum field theory has something to do with cluster algebras. I review one aspect of the connection: supersymmetric quantum field theories have associated hyperkähler moduli spaces, and these moduli spaces carry a structure that looks like an extension of the notion of cluster variety. In particular, one encounters the usual variables and mutations of the cluster story, along with more exotic extra variables and generalized mutations. I focus on a class of examples where the underlying cluster varieties are moduli spaces of flat connections on surfaces, as considered by Fock and Goncharov [Fock V, Goncharov A (2006) Publ Math Inst Hautes Études Sci 103:1-211]. The work reviewed here is largely joint with Davide Gaiotto and Greg Moore.

This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings admitting a cluster algebra structure. We then present the general definition of a cluster algebra and describe the interplay between cluster variables, coefficients, c-vectors and g-vectors. We show how c-vectors appear in the study of quantum cluster algebras and their links to the quantum dilogarithm. We then present the framework of additive categorification of cluster algebras based on the notion of quiver with potential and on the derived category of the associated Ginzburg algebra. We show how the combinatorics introduced previously lift to the categorical level and how this leads to proofs, for cluster algebras associated with quivers, of some of Fomin-Zelevinsky's fundamental conjectures.

We study deformations of cluster algebras with several quantum parameters, called toroidal cluster algebras, which naturally appear in the study of Grothendieck rings of representations of quantum affine algebras. In this context, we construct toroidal Grothendieck rings and we establish these are flat deformations of Grothendieck rings. We prove that for a family of monoidal categories \({\mathscr {C}}_1\) of simply-laced quantum affine algebras categorifying finite-type cluster algebras, the toroidal Grothendieck ring has a natural structure of a toroidal cluster algebra.

Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47). Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics). In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … [↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu. [1]: #xref-corresp-1-1

Let \(S(n)\) be the category of invariant subspaces of nilpotent operators with nilpotency index at most \(n\). Such submodule categories have been studied already in 1934 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra \(\Pi _n\) of type \(\mathbb {A}_n\); the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schroer). We are going to discuss the connection between the submodule category \(\mathcal {S}(n)\) and the module category \(\hbox {mod}\;\Pi _{n-1}\) of the preprojective algebra \(\Pi _{n-1}\). Dense functors \(\mathcal {S}(n) \rightarrow \hbox {mod}\;\Pi _{n-1}\) are known to exist: one has been constructed quite a long time ago by Auslander and Reiten, recently another one by Li and Zhang. We will show that these two functors are full, dense, objective functors with index \(2n\), thus \(\hbox {mod}\;\Pi _{n-1}\) is obtained from \(\mathcal {S}(n)\) by factoring out an ideal which is generated by \(2n\) indecomposable objects. As a byproduct we also obtain new examples of ideals in triangulated categories, namely ideals \(\mathcal {I}\) in a triangulated category \(\mathcal {T}\) which are generated by an idempotent such that the factor category \(\mathcal {T}/\mathcal {I}\) is an abelian category.

We describe a connection between the subjects of cluster algebras, polynomial identity algebras, and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac, and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we obtain an explicit formula for the discriminant of the integral form over of each quantum unipotent cell of De Concini, Kac, and Procesi for arbitrary symmetrizable Kac–Moody algebras, where is a root of unity.

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4, we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation.

Given a Poisson structure (or, equivalently, a Hamiltonian operator) $P$, we show that its Lie derivative $L_{\tau}(P)$ along a vector field $\tau$ defines another Poisson structure, which is automatically compatible with $P$, if and only if $[L_{\tau}^2(P),P]=0$, where $[\cdot,\cdot]$ is the Schouten bracket. We further prove that if $\dim\ker P\leq 1$ and $P$ is of locally constant rank, then all Poisson structures compatible with a given Poisson structure $P$ on a finite-dimensional manifold $M$ are locally of the form $L_{\tau}(P)$, where $\tau$ is a local vector field such that $L_{\tau}^2(P)=L_{\tilde\tau}(P)$ for some other local vector field $\tilde\tau$. This leads to a remarkably simple construction of bi-Hamiltonian dynamical systems. We also present a generalization of these results to the infinite-dimensional case. In particular, we provide a new description for pencils of compatible local Hamiltonian operators of Dubrovin--Novikov type and associated bi-Hamiltonian systems of hydrodynamic type. Key words: compatible Poisson structures, Hamiltonian operators, bi-Hamiltonian systems (= bihamiltonian systems), integrability, Schouten bracket, master symmetry, Lichnerowicz--Poisson cohomology, hydrodynamic type systems. MSC 2000: Primary: 37K10; Secondary: 37K05, 37J35

We discuss a discretization of the quantum Toda field theory associated with a semisimple finite-dimensional Lie algebra or a tamely-laced infinite-dimensional Kac–Moody algebra G, generalizing the previous construction of discrete quantum Liouville theory for the case G = A1. The model is defined on a discrete two-dimensional lattice, whose spatial direction is of length L. In addition we also find a ‘discretized extra dimension’ whose width is given by the rank r of G, which decompactifies in the large r limit. For the case of G = AN or , we find a symmetry exchanging L and N under appropriate spatial boundary conditions. The dynamical time evolution rule of the model is quantizations of the so-called Y-system, and the theory can be well described by the quantum cluster algebra. We discuss possible implications for recent discussions of quantum chaos, and comment on the relation with the quantum higher Teichmüller theory of type AN.

We solve the quantum version of the A 1 T-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A 1 Q-system and generalize it to the fully non-commutative case. We give the relation between the quantum T-system and the quantum lattice Liouville equation, which is the quantized Y-system.

In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category \(\mathcal {A}\) to an appropriate q-polynomial algebra. In the case that \(\mathcal {A}\) is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with \(\mathcal {A}\) is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.