Perspectives in non-commutative algebras:
Leavitt path algebras, Hochschild (co)homology, K-theory and related topics
Anhui University
09/09/2025-20/09/2025
 























Introduction to triangulated categories and differential graded rings , Xiao-Wu CHEN
Abstract
Differential graded categories appear naturally in homological algebra, and have many applications in representation theory and noncommutative algebraic geometry. For example, following Drinfeld, noncommutative algebraic geometry might be defined to the study of differential graded categories and their homological properties. Differential graded categories provide a correct framework for algebraic triangulated categories. The aim of this series of talks is to introduce the theory of differential graded categories. The plan is as follows.
  1. The homotopy category and stable category
  2. The axioms of a triangulated category (I)
  3. The axioms of a triangulated category (II)
  4. Verdier quotients and derived categories
  5. Derived equivalences and tilting complexes
  6. Differential graded rings and their derived categories
Ample groupoid algebras , Lisa Orloff CLARK
Abstract
A groupoid is a generalisation of a group in which the operation is only partially defined. Groupoids are very general objects that appear in a variety of different mathematical settings. In this course, we will begin with an introduction to ample groupoids and their corresponding algebras. Then we will describe the groupoid model of Leavitt path algebras and demonstrate how the groupoid approach is useful.

Graphs, Monoids and Algebras, Roozbeh HAZRAT
Abstract
The notion of sandpile models or chips firing encapsulates a process by which objects may spread and evolve along a grid. The models were conceived in 1987 in the seminal paper by Bak, Tang and Wiesenfeld as an example of selforganized criticality, or the tendency of physical systems to organise themselves without any input from outside the system, toward critical but barely stable states. The models were used to describe phenomena such as forest fires, traffic jams, stock market fluctuations, etc. In subsequent major work (1990), Dhar championed the use of an abelian group naturally associated to a sandpile model as an invariant which was shown to capture many properties of the model. This abelian group is paired with a naturally-arising monoid that arises from the grid. In a different realm, the notion of Leavitt path algebras L_K(E) associated to directed graphs E, with coefficients in a field K, were introduced in 2005. These are a generalisation of algebras (denoted by L_K(1, 1+k)) introduced by William Leavitt in 1962; these "Leavitt algebras '' arise as the universal ring of type (1, 1+k) (i.e., A_1 \cong A_{1+k} as right A-modules, where k in N). In fact Leavitt established much more in the 1962 article: he showed that for any n, k in N, there is a universal ring A of type (n, n+k) (denoted L_K(n, n+k)) for which A_n \cong A_{n+k} as right A-modules. When n> 1, this universal ring is not realizable as a Leavitt path algebra. With this in mind, the notion of weighted Leavitt path algebras L_K(E, w) associated to weighted graphs (E, w) were introduced in 2011. The weighted Leavitt path algebras L_K(E, w) provide a natural (but extremely broad) context in which all of Leavitt’s algebras (corresponding to any pair n, k in N) can be realized as a specific example. The study of the commutative monoid V(B) of isomorphism classes of finitely generated projective right modules of a unital ring B (with operation \oplus) goes back to the work of Grothendieck and Serre. For a Leavitt path algebra L_K(E), the monoid V(L_K (E)) has received substantial attention since the introduction of the topic. Furthermore, the monoid V(L_K (E, w)) has been described in later works. In this course we’ll show how the notions of sandpile monoids and weighted Leavitt path algebras are quite naturally related, via the V-monoid. This relationship allows us to associate an algebra, a "sandpile algebra'', to the theory of sandpile models, thereby opening up an avenue by which to investigate sandpile models via the structure of the sandpile algebras, and vice versa. The sandpile algebras provide a natural (but significantly more focused) context in which all of Leavitt’s algebras can be realized as a specific example.
Hochschild homology and cohomology of commutative algebras , Srikanth IYENGAR
Abstract
The focus of my talk will be on the singularity category of a commutative noetherian ring. Among such rings, the simplest are the regular rings, which are abstractions of coordinate rings of smooth algebraic varieties. Among commutative noetherian rings, regular rings are characterized by the property that their singularity category is trivial. From the geometric perspective, the next level of complexity beyond smooth algebraic varieties are the subvarieties of these cut out by collection of transversal hypersurfaces. The algebraic analogues of these are the locally complete intersection rings. The singularities categories of such rings enjoy many special features. I will discuss some of these, including the theory of matrix factorizations; duality phenomenon; classification of thick subcategories; Hochschild cohomology, and connections to sheaf cohomology.
Hochschild cohomology: Methods and examples , Andrea SOLOTAR
Abstract
Given a field k, an associative k-algebra A and an A-bimodule M, the Hochschild cohomology and homology of A with coefficients in M are respectively the graded vector spaces: H^∗(A,M) = Ext^*_{ A^e}(A,M) and H_∗(A,M) =Tor^{A^e}_ ∗ (M,A). Of course any projective resolution of A as A-bimodule can be used to compute them, and the bar resolution is always available.
Unfortunately, it is not suitable for computations. Hochschild homology and cohomology– originally defined by G. Hochschild in 1945 in [H] in terms of the bar resolution– are very useful tools to study the representation theory of a given associative algebra. They are invariant under Morita equivalence and more generally under derived equivalence [R]. Just to give a flavour of what we may learn about the algebra just knowing its Hochschild cohomology, let us mention that HH^0(A) is the centre of A while HH^1(A) is isomorphic to the quotient of the k-linear derivations of A modulo the inner derivations, and HH^2(A) classifies infinitesimal deformations of A. The Hochschild cohomology is endowed with a very rich structure: • it is a graded commutative algebra via the cup product ∪, • it has a graded Lie bracket of degree -1 that endows it with a structure of graded Lie algebra, • the bracket is a graded biderivation with respect to the cup product, turning Hochschild cohomology into a Gerstenhaber algebra. • there is an action of Hochschild cohomology on Hochschild homology via the cap product. Since for any p,n ∈ N, [-,-] : HH^p(A)×HH^n(A) → HH^{p+n-1}(A), it turns out that HH^1(A) is a Lie algebra with structure given by [f,h]= f ◦h-h◦f for all f,h ∈HH^1(A), and each HH^n(A) becomes an HH^1(A)-Lie module. In other words, the Lie structure of HH^1(A) comes from looking at Der_k(A) inside the Lie algebra of k-linear endomorphisms of A, and passing to the quotient by the inner derivations. The whole structure is derived invariant [K]. In characteristic p, it is a restricted Lie algebra, and this is also derived invariant [Z]. Suppose char(k) is not 2. Given i ∈N, the squaring map sq_i : HH^{2i}→ HH^{4i-1} is defined by a → 1/ 2[a,a]. For i= 1, this map plays an important role since it describes obstructions to infinitesimal deformations. Putting all this together, we get–theoretically– a lot of information. The problem is that these invariants are not at all easy to compute, since the cup product and the Gerstenhaber bracket have been initially defined in terms of the bar resolution and the Hochschild cohomology is almost never computed using it. While the cup product is well understood, both because it coincides with the Yoneda product of classes of extensions and since it can be expressed in terms of any resolution via a diagonal map, this is not yet the case of the Gerstenhaber bracket. In spite of the interpretations given by Stasheff [St] and by Schwede [Sch], it remains mysterious. In this course I will introduce the definitions and give several examples (in the first and second lectures) and methods to compute the Hochschild calculus of a given algebra A (third and fourth lectures). Finally I will comment some applications (fifth lecture).


References:
[H] G. Hochschild, On the cohomology groups of an associative algebra, Ann. Math. (2), 46, 1945, pp. 58–67.
[K] B. Keller, Hochschild cohomology and derived Picard groups, J. Pure Appl. Algebra 190, (1–3), 2004, pp. 177–196.
[R] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2), 43 (1), 1991, pp. 37–48.
[Sch] S. Schwede, An exact sequence interpretation of the Lie bracket in Hochschild cohomology, J. Reine Angew. Math. 498, 1998, pp. 153–172.
[St] J. Stasheff, The intrinsic bracket on the deformation complex of an associative algebra, J. Pure Appl. Algebra 89 (1–2), 1993, pp. 231–235.
[Z] A. Zimmermann, Fine Hochschild invariants of derived categories for symmetric algebras, J. Algebra 308 (1), 2007, pp. 350–367.
Gorenstein-projective modules, monomorphism categories, and model categories , Pu ZHANG
Abstract
In these 6 hours lectures, I will talk about Gorenstein-projective modules, monomorphism categories, and model categories, and we will focus on their relations.
Lecture 1: Motivation, definition, and basic Properties of Gorenstein-projective modules; independence of the axioms of Gorenstein-projective modules.
Lecture 2: Gorenstein-projective modules and singularity categories.
Lecture 3: Monomorphism categories; constructions of Gorenstein-projective modules via monomorphism categories.
Lecture 4: Model categories; Quillen's homotopy categories; the model structure induced by Gorenstein-projective modules.
Lecture 5: The one-one correspondence between abelian model structures and Hovey triples; Gillespie-Hovey triples: Gillespie's construction of hereditary Hovey triples. Lecture 6: The model structure induced by one hereditary complete cotrosion pair, based on the work of A. Beligiannis and I. Reiten.
Groebner-Shirshov bases for algebras and operads and their applications , Guodong ZHOU
Abstract
Groebner-Shirshov bases were invented in the sixties of last century. Since then, this theory becomes an important tool in algebra and computer sciences. This course is an introduction to the theory of GroebnerShirshov bases for noncommutative algebras as well as for operads. Some applications to construct resolutions and to prove Koszulness property for algebras and operads will be presented.
Kuelshammer structures on Hochschild homology and cohomology , Alexander ZIMMERMANN
Abstract
For a symmetric algebra A over a perfect field of finite characteristic p the p-power map on the centre induces a well defined map on the cocentre A/[A,A]. It can be shown that it is a derived invariant. This p-power map proved to be quite subtle and allowed to distinguish derived equivalence classes of algebras and settle quite subtle questions on parameters of classification problems for certain classes of algebras. Since the centre and the cocentre are just degree 0 Hochschild cohomology and homology, it can be asked if this map can be extended to the Hochschild cohomology and homology in all degrees. This is the case, as it was proved by Volkov and Zvonareva in particular. The Gerstenhaber Lie structure becomes a restricted Lie algebra, and this structure is actually a derived invariant as well, as it was shown by Briggs and Rubio y Degrassi.