Graded cluster algebras

In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. We also obtain interesting combinatorics---namely tropical frieze patterns---on the Auslander--Reiten quivers of the categories.Comment: 23 pages, 6 figures. v2: Substantially revised with additional results. New section on graded (generalised) cluster categories. v3: added Prop. 5.5 on relationship with Grothendieck group of cluster categor

Automorphism groupoids in noncommutative projective geometry

We address a natural question in noncommutative geometry, namely the rigidity observed in many examples, whereby noncommutative spaces (or equivalently their coordinate algebras) have very few automorphisms by comparison with their commutative counterparts. In the framework of noncommutative projective geometry, we define a groupoid whose objects are noncommutative projective spaces of a given dimension and whose morphisms correspond to isomorphisms of these. This groupoid is then a natural generalization of an automorphism group. Using work of Zhang, we may translate this structure to the algebraic side, wherein we consider homogeneous coordinate algebras of noncommutative projective spaces. The morphisms in our groupoid precisely correspond to the existence of a Zhang twist relating the two coordinate algebras. We analyse this automorphism groupoid, showing that in dimension 1 it is connected, so that every noncommutative $\mathbb{P}^{1}$ is isomorphic to commutative $\mathbb{P}^{1}$. For dimension 2 and above, we use the geometry of the point scheme, as introduced by Artin-Tate-Van den Bergh, to relate morphisms in our groupoid to certain automorphisms of the point scheme. We apply our results to two important examples, quantum projective spaces and Sklyanin algebras. In both cases, we are able to use the geometry of the point schemes to fully describe the corresponding component of the automorphism groupoid. This provides a concrete description of the collection of Zhang twists of these algebras.Comment: 27 pages; v2: minor corrections and additional reference

Graded Frobenius cluster categories

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

Graded Frobenius cluster categories

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

A statistical model for isolated convective precipitation events

To study the diurnal evolution of the convective cloud field, we develop a precipitation cell tracking algorithm which records the merging and fragmentation of convective cells during their life cycles, and apply it on large eddy simulation (LES) data. Conditioning on the area covered by each cell, our algorithm is capable of analyzing an arbitrary number of auxiliary fields, such as the anomalies of temperature and moisture, convective available potential energy (CAPE) and convective inhibition (CIN). For tracks that do not merge or split (termed "solitary"), many of these quantities show generic, often nearly linear relations that hardly depend on the forcing conditions of the simulations, such as surface temperature. This finding allows us to propose a highly idealized model of rain events, where the surface precipitation area is circular and a cell's precipitation intensity falls off linearly with the distance from the respective cell center. The drop-off gradient is nearly independent of track duration and cell size, which allows for a generic description of such solitary tracks, with the only remaining parameter the peak intensity. In contrast to the simple and robust behavior of solitary tracks, tracks that result from merging of two or more cells show a much more complicated behavior. The most intense, long lasting and largest tracks indeed stem from multi-mergers - tracks involved in repeated merging. Another interesting finding is that the precipitation intensity of tracks does not strongly depend on the absolute amount of local initial CAPE, which is only partially consumed by most rain events. Rather, our results speak to boundary layer cooling, induced by rain re-evaporation, as the cause for CAPE reduction, CIN increase and shutdown of precipitation cells.Comment: Manuscript under review in Journal of Advances in Modeling Earth System

Braided Lie bialgebras associated to Kac-Moody algebras

Braided-Lie bialgebras have been introduced by Majid, as the Lie versions of Hopf algebras in braided categories. In this paper we extend previous work of Majid and of ours to show that there is a braided-Lie bialgebra associated to each inclusion of Kac-Moody bialgebras. Doing so, we obtain many new examples of infinite-dimensional braided-Lie bialgebras. We analyze further the case of untwisted affine Kac-Moody bialgebras associated to finite-dimensional simple Lie algebras. The inclusion we study is that of the finite-type algebra in the affine algebra. This braided-Lie bialgebra is isomorphic to the current algebra over the simple Lie algebra, now equipped with a braided cobracket. We give explicit expressions for this braided cobracket for the simple Lie algebra sl3