Jan E. Grabowski

Quantum Cluster Algebra Structures on Quantum Grassmannians and their Quantum Schubert Cells: The Finite-type Cases

We exhibit quantum cluster algebra structures on quantum GrassmanniansKq[Gr(2;n)] and their quantum Schubert cells, as well as onKq[Gr(3; 6)],Kq[Gr(3; 7)] andKq[Gr(3; 8)]. These cases are precisely those where the quantum cluster algebra is of nite type and the structures we describe quantize those found by Scott for the classical situation.

Graded quantum cluster algebras and an application to quantum Grassmannians

We introduce a framework for Z-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation. The resulting grading has the property that every (quantum) cluster variable is homogeneous. In the quantum setting, we use this grading framework to give a construction that behaves somewhat like twisting, in that it produces a new quantum cluster algebra with the same cluster combinatorics but with different quasi-commutation relations between the cluster variables. We apply these results to show that the quantum Grassmannians

Cluster algebras of infinite rank

K_q[Gr(k, n)]

Braided Enveloping Algebras Associated to Quantum Parabolic Subalgebras

admit quantum cluster algebra structures, as quantizations of the cluster algebra structures on the classical Grassmannian coordinate ring found by Scott. This is done by lifting the quantum cluster algebra structure on quantum matrices due to Geis–Leclerc–Schroer and completes earlier work of the authors on the finite-type cases.

On Lie induction and the exceptional series

Holm and Jorgensen have shown the existence of a cluster structure on a certain category D that shares many properties with finite type A cluster categories and that can be fruitfully considered as an infinite analogue of these. In this work we determine fully the combinatorics of this cluster structure and show that these are the cluster combinatorics of cluster algebras of infinite rank. That is, the clusters of these algebras contain infinitely many variables, although one is only permitted to make finite sequences of mutations. The cluster combinatorics of the category D are described by triangulations of an ∞-gon and we see that these have a natural correspondence with the behaviour of Plucker coordinates in the coordinate ring of a doubly-infinite Grassmannian, and hence the latter is where a concrete realization of these cluster algebra structures may be found. We also give the quantum analogue of these results, generalising work of the first author and Launois. An appendix by Michael Groechenig provides a construction of the coordinate ring of interest here, generalizing the well-known scheme-theoretic constructions for Grassmannians of finite-dimensional vector spaces.

Examples of quantum cluster algebras associated to partial flag varieties

Associated to each subset J of the nodes I of a Dynkin diagram is a triangular decomposition of the corresponding Lie algebra 𝔤 into three subalgebras (generated by e j , f j for j ∈ J and h i for i ∈ I), (generated by f d , d ∈ D = I∖J) and its dual . We demonstrate a quantum counterpart, generalising work of Majid and Rosso, by exhibiting analogous triangular decompositions of U q (𝔤) and identifying a graded braided Hopf algebra that quantizes . This algebra has many similar properties to , in many cases being a Nichols algebra and therefore completely determined by its associated braiding.

Graded Frobenius cluster categories

Lie bialgebras occur as the principal objects in the infinitesimalization of the theory of quantum groups — the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonization construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyze the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra. We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E9, F5 and G3. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E9.

Graded cluster algebras

We give several explicit examples of quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky, on quantized coordinate rings of partial flag varieties and their associated unipotent radicals. These structures are shown to be quantizations of the cluster algebra structures found on the corresponding classical objects by Geis, Leclerc and Schroer, whose work generalizes that of several other authors. We also exhibit quantum cluster algebra structures on the quantized enveloping algebras of the Lie algebras of the unipotent radicals.

A triple construction for Lie bialgebras

Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other. In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients. We interpret the grading K-theoretically and prove similar results to the triangulated case, in particular obtaining that degrees are additive on exact sequences. We show that the categories of Buan, Iyama, Reiten and Scott, some of which were used by Geiss, Leclerc and Schroer to categorify cells in partial flag varieties, and those of Jensen, King and Su, categorifying Grassmannians, are examples of graded Frobenius cluster categories.