We describe a framework for encoding cluster combinatorics using categorical methods. We give a definition of an abstract cluster structure, which captures the essence of cluster mutation at a tropical level and show that cluster algebras, cluster varieties, cluster categories and surface models all have associated abstract cluster structures. For the first two classes, we also show that they can be constructed from abstract cluster structures. By defining a suitable notion of morphism of abstract cluster structures, we introduce a category of these and show that it has several desirable properties, such as initial and terminal objects and finite products and coproducts. We also prove that rooted cluster morphisms of cluster algebras give rise to morphisms of the associated abstract cluster structures, so that our framework includes a version of the extant category of cluster algebras. We can do more, however, because we can relate different types of representation of abstract cluster structures (cluster algebra, varieties, categories) directly via morphisms of their associated abstract cluster structures, even though no direct map from e.g. a cluster category to the associated cluster algebra is possible. In fact, we do much of the above in the setting of abstract quantum cluster structures, with some analysis of the difference between the category of these and that of the unquantized version. In order to show the relationship between abstract quantum cluster structures and quantum cluster algebras, we reformulate the usual construction of the latter in a way that is more amenable to our purposes and which we expect will be of independent interest and use.
In the last chapter, we first looked at representations of groups on sets, realizing that the minimal assumptions make group actions hard to study collectively. We then linearized, which opens up linear algebra for use in studying linear representations. But again, we were dissatisfied: the constructions one would like to do with an algebraic object are awkward. When we considered quiver representations, something else happened. Now we ended up with a rather different flavour of representation and it was not clear how it related to the cases of groups or algebras. That would mean having to develop two separate lots of theory. We claim that the answer to all these problems is to study modules instead. We will be able to make all the constructions we wanted, and more, and we will be able to leverage the same theory in many different settings.
This volume offers a fresh and modern introduction to one of abstract algebra’s key topics. Guiding readers through the transition between structure theory and representation theory, this textbook explores how algebraic objects like groups and rings act as symmetries of other structures. Using the accessible yet powerful language of category theory, the book reimagines standard approaches to topics such as modules and algebras in a way that unlocks modern treatments of more advanced topics such as quiver representations and even representations of Hopf algebras and categories. Aimed at undergraduate students with prior exposure to linear algebra and basic group theory, the book introduces categories early and uses them throughout, providing a cohesive framework that mirrors current mathematical research. Though technically sophisticated, it also includes examples and exercises designed to develop intuition and understanding. Grabowski’s inclusion of computational tools such as SageMath offers a valuable and traditionally underdeveloped bridge between abstract theory and hands-on exploration. This is a uniquely valuable guide for students ready to stretch their understanding of the subject’s conceptual depth and evolving frontiers.
The aim of this first chapter is to gather the definitions and some fundamental properties of the main protagonists in our story: groups, rings, fields, vector spaces, algebras and quivers. We start with those you are more likely to be familiar with and you should move through these at the pace that suits you.
The final chapter is a selection of advanced topics in representation theory, providing a brief introduction to several areas of current research.
It is very common to say that groups often arise as symmetries. The subtle but important shift from “what group encodes the symmetries of this object?” to “which objects does this group give symmetries of?” moves us from the structure theory of groups to their representation theory. We also examine representations of algebras and quivers.
In this chapter, we will introduce some new language that will help us talk about representation theory. Categories are, as I hope you will see, a very natural way to express relationships among collections of algebraic objects with a particular structure, in a way that respects the natural functions between them.
We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category 𝒞 and cluster-tilting subcategory 𝒯 satisfying only mild finiteness conditions. We show that the structure theory of 𝒞 and the representation theory of 𝒯 give rise to the rich combinatorial structures of seed data and cluster ensembles, via Grothendieck groups and homological algebra. We demonstrate that there is a natural dictionary relating cluster-tilting subcategories and their tilting theory to A-side tropical cluster combinatorics and, dually, relating modules over 𝒯 to the X-side; here 𝒯 is the image of 𝒯 in the triangulated stable category of 𝒞. Moreover, the exchange matrix associated to 𝒯 arises from a natural map p_𝒯K_0(mod𝒯)→K_0(𝒯) closely related to taking projective resolutions. Via our approach, we categorify many key identities involving mutation, g-vectors and c-vectors, including in infinite rank cases and in the presence of loops and 2-cycles. We are also able to define A- and X-cluster characters, which yield A- and X-cluster variables when there are no loops or 2-cycles, and which enable representation-theoretic proofs of cluster-theoretical statements. Continuing with the same categorical philosophy, we give a definition of a quantum cluster category, as a cluster category together with the choice of a map closely related to the adjoint of p_𝒯. Our framework enables us to show that any Hom-finite exact cluster category admits a canonical quantum structure, generalising results of Geiß–Leclerc–Schröer.
We extend recent work of the third author and Kouloukas by constructing deformations of integrable cluster maps corresponding to the Dynkin types $A_{2N}$, lifting these to higher-dimensional maps possessing the Laurent property and demonstrating integrality of the deformations for $N\leq 3$. This provides the first infinite class of examples (in arbitrarily high rank) of such maps and gives information on the associated discrete integrable systems. Key to our approach is a ``local expansion'' operation on quivers which allows us to construct and study mutations in type $A_{2N}$ from those in type $A_{2(N-1)}$.
We show that under mild assumptions the Segre product of two graded cluster algebras has a natural cluster algebra structure.
We show that under mild assumptions the Segre product of two graded cluster algebras has a natural cluster algebra structure.
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, using 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, the twodimensional quantum projective space 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.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
W e study the triple ofa quasitriangular Lie bialgebra as a natural extension oftheDrinfeld double.Thetripleisitselfa quasitriangular Liebialgebra.W eproveseveralresultsaboutthealgebraicstructureof thetriple,analogoustoknown resultsforthedouble.Am ongthem ,we provethatin thefactorisablecasethetripleisisom orphicto a twisting ofg g g by a certain cocycle. W e also considerrealform softhe tripleand the triangularcase.
Given a category, one may construct slices of it. That is, one builds a new category whose objects are the morphisms from the category with a fixed codomain and morphisms certain commutative triangles. If the category is a groupoid, so that every morphism is invertible, then its slices are (connected) groupoids. We give a number of constructions that show how slices of groupoids have properties even closer to those of groups than the groupoids they come from. These include natural notions of kernels and coset spaces. MSC (2020): 18B40 (Primary), 20N02 (Secondary)
Given a category, one may construct slices of it. That is, one builds a new category whose objects are the morphisms from the category with a fixed codomain and morphisms certain commutative triangles. If the category is a groupoid, so that every morphism is invertible, then its slices are (connected) groupoids. We give a number of constructions that show how slices of groupoids have properties even closer to those of groups than the groupoids they come from. These include natural notions of kernels and coset spaces.
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.
We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum cluster algebras of finite rank. As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, extending an earlier construction of the authors for k=2.
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 P is isomorphic to commutative P. 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. MSC (2010): 16S38 (Primary), 16D90, 16W50, 20G42 (Secondary) Email: nicholas.cooney@uca.fr . Email: j.grabowski@lancaster.ac.uk. Website: http://www.maths.lancs.ac.uk/~grabowsj/
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.
Cluster algebras are currently playing a prominent role in several areas of mathematics, including Lie theory, representation theory and geometry. We will introduce cluster algebras and explain some basic algebraic constructions for them, including morphisms, subalgebras and quotients. We will also study gradings, show how they can be classified and use them to produce twisted cluster algebra structures. We also explain some of the major theorems relating to cluster algebras, regarding their combinatorial and algebraic structure and in particular the main classification theorem of Fomin–Zelevinsky. Quantum cluster algebras are a natural noncommutative generalisation of cluster algebras. This noncommutativity is relatively mild: the main feature is that elements in the same cluster must commute up to a power of q, although elements from different clusters may have more complicated relationships. Many results for cluster algebras have direct parallels for the quantum case so we will treat these alongside the classical case. We will conclude with looking at two important classes of quantum cluster algebras in detail: quantum matrices, following the representation-theoretic approach of Geiß–Leclerc– Schröer, and quantum Grassmannians. †Email: j.grabowski@lancaster.ac.uk. Website: http://www.maths.lancs.ac.uk/~grabowsj/
