Sandpile models are about how things spread along a grid (think of Covid!) and Leavitt algebras are algebras associated to graphs. We relate these two subjects!
Tag - Associative rings
Motivated by work on the Steenrod algebra, Moore and Peterson introduced the notion of (graded) nearly Frobenius algebras; these were later renamed P-algebras by Margolis. This is a preliminary report on the development of an analogous theory for non-graded Hopf algebras which as far as I know is not in the algebra literature.
I will give a very brief overview of the graded theory, then explain one approach to emulating it based on filtered colimits of finite-dimensional Hopf algebras which are Frobenius extensions of each other.
Balmer initiated the study of separable commutative algebras (tt-rings in short) in tt-geometry: these are commutative algebras for which the multiplication map admits a bimodule section. Their importance has grown in recent years due to the fact that the category of modules over a tt-ring is again a tt-category, and that tt-rings allow to prove strong descent results. However, the classification of all tt-rings in a tt-category is an open problem in many cases of interest. In this talk, I will relate the notion of tt-ring to the notion of finite cover due to Mathew, and use this connection to provide classification results for tt-rings in some special cases of interest.
In this talk, we will report on ongoing joint work with Gustavo Jasso. The goal is to show that algebraic triangulated categories satisfying the assumptions in the title have a unique DG-enhancement over a ground perfect field, up to Morita equivalence. This extends previous work on finite triangulated categories. The key step is the connection with Geiss-Keller-Oppermann's notion of n-angulated categories, which are like triangulated categories but with longer ‘triangles'.
The Hecke algebra is in general not quasi-hereditary, meaning that its module category is not a highest weight category; while it admits a quasi-hereditary cover by the category 𝒪 of a certain rational Cherednik algebra due to Ginzburg-Guay-Opdam-Rouquier. It was later shown in type A that this category 𝒪 can be realized concretely as the module category of Dipper-James's q-Schur algebra, but this realization problem remains open beyond types A and B. An essential step for type D, i.e., the complex reflection group G(2,2,n), is to study Hu's Hecke subalgebra, which deforms a wreath product that is not a Coxeter group. In this talk, I'll introduce a new theory allowing us to take the wreath product of an algebra by a Hecke algebra. Before our work, wreath products related to Hecke algebras were worked out at the degenerate level by Wan-Wang. Our wreath product produces the Ariki-Koike algebras as special cases as well as new 'Hecke algebras' of wreath products between symmetric groups. These are the first steps towards answering the realization problem for complex reflection groups.
The upper triangular matrix algebras are important in Linear Algebra, and represent a powerful tool in Ring Theory. They also appear in the theory of PI algebras.
In addition to the usual associative product, one can consider the Lie bracket and also the symmetric (Jordan) product on the upper triangular matrices.
We discuss the group gradings on the upper triangular matrices viewed as an associative, Lie and Jordan algebra, respectively. Valenti and Zaicev proved that the associative gradings are, in a sense, given by gradings on the matrix units. Di Vincenzo, Valenti and Koshlukov classified such gradings. Later on, Yukihide and Koshlukov, described the Lie and the Jordan gradings. In this talk we recall some of these results as well as a new development in a rather general setting, obtained by Yukihide and Koshlukov.
The natural permutation representation of the symmetric group admits a q-analogue known as the Burau representation. The symmetric group admits two natural covering groups: the braid group of Artin and the twin group of Khovanov, obtained respectively by forgetting the cubic and quadratic relations in the Coxeter presentation of the symmetric group. By computing centralizers of tensor powers of the Burau representation, we obtain new instances of Schur-Weyl duality for braid groups and twin groups, in terms of the partial permutation and partial Brauer algebras. The method produces many representations of each group that can be understood combinatorially.
Weighted KLRW algebras are diagram algebras that depend on continuous parameters. Varying these parameters gives a way to interpolate between various algebras that appear in (categorical) representation theory such as semisimple algebras, KLR algebras, quiver Schur algebras and diagrammatic Cherednik algebras. This talk is a friendly (and diagrammatic!) introduction explaining these algebras, with no prior knowledge about any of these assumed.
The Kazhdan–Lusztig (KL) cells of a Coxeter group are subsets of the group defined using the KL basis of the associated Iwahori–Hecke algebra. The cells of symmetric groups can be computed via the Robinson–Schensted correspondence, but for general Coxeter groups combinatorial descriptions of KL cells are largely unknown except for cells of a-value 0 or 1, where a refers to an ℕ-valued function defined by Lusztig that is constant on each cell. In this talk, we will report some recent progress on KL cells of a-value 2. In particular, we classify Coxeter groups with finitely many elements of a-value 2, and for such groups we characterize and count all cells of a-value 2 via certain posets called heaps. We will also mention some applications of these results for cell modules.
For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. We then explain several important roles of complexes of flat cotorsion modules and give some applications.
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