Atoms in the monoid of monomial ideals
Nikola Bogdanovic
Factorization theory is a subfield of algebra concerned with how the elements of a given ring or multiplicative monoid factor into irreducible elements (or "atoms"), especially in settings where such factorizations are not unique. Historically, the subject has its roots in the study of rings of integers in number fields. In this talk, after introducing the essential notions of the area, I will focus on a seemingly elementary yet still not fully understood object, namely the monoid of nonzero monomial ideals of a polynomial ring in several indeterminates over a field. In particular, I will characterize the atoms among monomial ideals with a small number of minimal generators and show how this characterization can be used to compute their density in an appropriate sense.
Explicit Factorization Algorithms for Banff Cluster Algebras
Mara Pompili
Rings arising as intersections of Laurent polynomial rings need not have unique factorization: in A = ℤ[x, a/x], the element 6 factors both as 2·3 and as x·(6/x). We call such rings FLIRs, and show that although element factorization can fail, ideal factorization always behaves, these are Krull domains, with the class group Cl(A) measuring the gap between the two. We give an explicit presentation of Cl(A) directly from the data defining a FLIR. Cluster algebras provide a rich source of examples: we introduce Banff cluster algebras, for which explicit Laurent charts can be computed algorithmically, making class groups, the UFD property, and complete factorizations effectively computable. This is a joint work with D. Smertnig.
Associative algebras: on polynomial identities and beyond
Ginevra Giordani
In this talk we will introduce the theory of polynomial identities (PI-theory) for associative algebras. After stating the main classical results of PI-theory, we will talk about the main algebraic structures studied from the PI point of view. Finally, we will introduce their generalizations and we will close the talk by presenting research advances.
Faithful matrix representations of diagram monoids
Marianne Johnson
Briefly, I will present some of my recent joint work with James East and Mark Kambites -- no prior knowledge is assumed, all words in the title will be explained, and there will be pictures.
In more detail: Diagram algebras arise naturally across mathematics and science. These algebras have bases consisting of various kinds of set partitions, represented and multiplied diagrammatically. Key examples include partition algebras, Brauer algebras and Temperley–Lieb algebras. Underlying the structure of these algebras are the partition, Brauer and Temperley–Lieb (a.k.a. Jones) monoids, which in turn occur as endomorphism monoids of associated tensor categories. The first main result I will present is that the partition (Brauer, Temperley–Lieb) category admits a faithful involutive tensor category representation by zero-one matrices over an arbitrary additively idempotent semiring. The dimensions of the matrices involved are powers of two, which correspond to the minimal possible size of a faithful involutive tensor representation of these categories by matrices over any semiring. Intriguingly, the matrices involved in our representation encode the number of 'floating components' formed when composing partitions and the second main result is that this idea can then be adapted to construct faithful representations of twisted partition monoids (by using matrices over a ring of appropriately chosen characteristic). We also exhibit lower dimensional faithful involutive representations of the Brauer and Temperley–Lieb categories (and monoids); in the case of Temperley–Lieb, the dimensions are Fibonacci numbers.
Noncommutative Stone dualities
The talk will present an introduction to the noncommutative duality theory that extends the classical Stone duality between generalized Boolean algebras and locally compact Stone spaces. In these dualities, generalized Boolean algebras are replaced by Boolean inverse, restriction or range semigroups, while locally compact Stone spaces are replaced by ample groupoids or categories. The latter are \'etale groupoids or categories whose unit space is a locally compact Stone space. I will discuss the dualities developed over the last 20 years by Lawson, Resende, Cockett and Garner, and the presenter.
Irreducible representation of free algebras
Francesca Mantese
We present a complete classification of finite dimensional irreducible representations of the free algebra K⟨x₁, ..., xₙ⟩ in terms of irreducible polynomials in the non-commutative variables x₁, ..., xₙ. The main tool is the study of simple modules over the Leavitt algebra Lₖ(1, n), which arises as universal localization of K⟨x₁, ..., xₙ⟩. This approach leads to new insights in the study of the factorization of polynomials in non-commutative variables and other related topics, such as an algorithm to divide polynomials or to compute the greatest common divisor between them. This is part of a joint project with P. N. Anh.