Tag - Associative rings

Kent Vashaw: Can a Zhang twist be a cocycle twist?

26th October, 2021

The Zhang twist of a graded algebra was defined by J. Zhang in 1996, and has provend an important tool in non-commutative algebra and non-commutative algebraic geometry. On the other hand, in the world of Hopf algebras and quantum groups, the 2-cocycle twist of a Hopf algebra gives a new Hopf algebra which is Morita-Takeuchi equivalent to the original Hopf algebra. We provide sufficient conditions for a Zhang twist of a graded Hopf algebra H to be again a Hopf algebra, to be an H-cleft object, or a 2-cocycle twist of H. In particular, we introduce the notion of a twisting pair for H such that the Zhang twist of H by such a pair is a 2-cocycle twist. This new notion is investigated in the context of various examples of Hopf algebras including Manin's universal quantum groups, and the quantized coordinate rings of general linear groups.

Milen Yakimov: Non-commutative Tensor Triangular Geometry and Support Varieties for Hopf Algebras

16th October, 2021

We will describe a theory of noncommutative tensor triangular geometry for monoidal triangulated categories. It is aimed at investigating support varieties for finite dimensional Hopf algebras via non-commutative Balmer spectra. We will state effective reconstruction theorems for these spectra and an intrinsic characterization of those categories whose support variety maps satisfy the tensor product property. As an application, we obtain a treatment of the Benson-Witherspoon Hopf algebras, which previously eluded approaches of this kind, and a proof of a recent conjecture of Negron and Pevtsova that the cohomological support maps of the Borel subalgebras of all Lusztig small quantum groups possess the tensor product property. This is joint work with Daniel Nakano (University of Georgia) and Kent Vashaw (MIT).

Bethany Marsh: 𝜏-exceptional sequences

5th October, 2021

We introduce the notion of a (signed) 𝜏-exceptional sequence for a finite dimensional algebra. We show that there is a bijection between the set of complete signed exceptional sequences and ordered basic support 𝜏-tilting objects.

Benjamin Steinberg: Cartan pairs of algebras

1st October, 2021

In the seventies, Feldman and Moore studied Cartan pairs of von Neumann algebras. These pairs consist of an algebra A and a maximal commutative subalgebra B with B sitting "nicely" inside of A. They showed that all such pairs of algebras come from twisted groupoid algebras of quite special groupoids (in the measure theoretic category) and their commutative subalgebras of functions on the unit space, and that moreover the groupoid and twist were uniquely determined (up to equivalence). Kumjian and Renault developed the C*-algebra theory of Cartan pairs. Again, in this setting all Cartan pairs arise as twisted groupoid algebras, this time of effective etale groupoids, and again the groupoid and twist are unique (up to equivalence).

In recent years, Matsumoto and Matui exploited this to give C*-algebraic characterizations of continuous orbit equivalence and flow equivalence of shifts of finite type using graph C*-algebras and their commutative subalgebras of continuous functions on the shift space (which form a Cartan pair under mild assumptions on the graph). The key point was translating these dynamical conditions into groupoid language. The ring theoretic analogue of graph C*-algebras are Leavitt path algebras. Leavitt path algebras are also connected to Thompson's group V and some related simple groups considered by Matui and others. Since the Leavitt path algebra associated to a graph is the "Steinberg" algebra of the same groupoid (a ring theoretic version of groupoid C*-algebras whose study was initiated by the speaker), this led people to wonder whether these dynamical invariants can be read off the pair consisting of the Leavitt path algebra and its subalgebra of locally constant maps on the shift space. The answer is yes, and it turns out in the algebraic setting one doesn’t even need any conditions on the graph. Initially work was focused on recovering a groupoid from the pair consisting of its "Steinberg" algebra and the algebra of locally constant functions on the unit space. But no abstract theory of Cartan pairs existed and twists had not yet been considered. Our work develops the complete picture.

It turns out that a twist on a groupoid gives rise to a Cartan pair when the algebra satisfies a groupoid analogue of the Kaplansky unit conjecture. In particular, if the groupoid has a dense set of objects whose isotropy groups satisfy the Kaplansky unit conjecture (e.g., are unique product property groups or left orderable), then the groupoid gives rise to a Cartan pair. This is what happens in the case of Leavitt path algebras where the isotropy groups are either trivial or infinite cyclic and hence left orderable.

Daniel Nakano: Non-commutative Tensor Triangular Geometry and Applications

27th September, 2021

In this talk, I will show how to develop a general non-commutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (MΔC). Insights from non-commutative ring theory are used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an MΔC, K, and then to associate to K a topological space: the Balmer spectrum Spc(K). We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that Spc(K) is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an MΔC. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of K, which in turn can be applied to classify the thick two-sided ideals and Spc(K). Applications will be given for quantum groups and non-cocommutative finite-dimensional Hopf algebras studied by Benson and Witherspoon.

Richard Borcherds: Rings and Modules

27th September, 2021

This is a 22-lecture course, with each lecture being about 30 minutes or so, given online by Richard Borcherds. It gives an introduction to rings and modules.

Tamanna Chatterjee: Parity Sheaves Arising from Graded Lie Algebras II

13th September, 2021

Let G be a complex, connected, reductive, algebraic group, and χ : ℂ× → G be a fixed cocharacter that defines a grading on 𝔤, the Lie algebra of G. In my first talk I have talked about the grading, derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. In this talk I will define parabolic induction and restriction both on nilpotent cone and graded setting. We will dive into the results of Lusztig in characteristic 0 in the graded setting. Under some assumptions on the field k and the group G we will recover some results of Lusztig in characteristic 0. These assumptions together with Mautner's cleanness conjecture will play a vital role. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. Lusztig's work on ℤ-graded Lie algebras is related to representations of affine Hecke algebras, so a long term goal of this work will be to interpret parity sheaves in the context of affine Hecke algebras.

Maria Ofelia Ronco: Generalization of dendriform algebras

9th September, 2021

In a joint work with D. López N. and L.-F. Préville-Ratelle in 2015 we introduce a family of non-symmetric operads Dyckm, which satisfies that:

1. Dyck0 is the operad of associative algebras,

2. Dyck1 is the operad Dend of dendriform algebras, introduced by J.-L. Loday,

3. the vector space spanned by the set of m-Dyck paths has a natural structure of free Dyckm algebra over one element,

4. for any k ≥ 1, there exist degeneracy operators si : Dyckm → Dyckm-1 and face operators dj: Dyckm → Dyckm+1, which defines a simplicial complex in the category of non-symmetric operads.

The main examples of Dyckm algebra are the vector spaces spanned by the m-simplices of certain combinatorial Hopf algebras, like the Malvenuto-Reutenauer algebras and the algebra of packed words.

A well-known result on associative algebras states that, as an 𝒮-module, the operad of Ass of associative algebras is the composition Ass = Com ∘ Lie, where Com is the operad of commutative algebras and Lie is the operad of Lie algebras. The version of this result for dendriform algebras is that Dend = Ass ∘ Brace, where Brace is the operad of brace algebras.

Our goal is to introduce the notion of m-brace algebra, for m ≥ 2, and prove that there exists a Poincaré-Birkoff-Witt Theorem in this context, stating that Dyckm = Ass ∘ m-Brace.

Onofrio Di Vincenzo: Algebras and superalgebras with (super-)involutions and their polynomial identities

29th July, 2021

In this talk we consider the *-polynomial identities of algebras with involutions. The positive solution of Specht's problem, given by Aljadeff, Giambruno and Karasik, for the T*-ideals of the free algebra with involution, show the decisive role of the identities of finite-dimensional superalgebras with superinvolution. In this talk we consider block-triangular matrix algebras related to any sequence of such *-simple superalgebras. These *-simple superalgebras are also involved in determining the exact value of the correponding exponent. We review the results in this area and we show that that every minimal affine variety of superalgebras with superinvolution is generated by one of the block-triangular matrix algebras we introduced.

Mikhail Kotchetov: Fine gradings on classical simple Lie algebras

8th July, 2021

Gradings by abelian groups have played an important role in the theory of Lie algebras since its beginning: the best known example is the root space decomposition of a semisimple complex Lie algebra, which is a grading by a free abelian group (the root lattice). Involutive automorphisms or, equivalently, gradings by the cyclic group of order 2, appear in the classification of real forms of these Lie algebras. Gradings by all cyclic groups were classified by V. Kac in the late 1960s and applied to the study of symmetric spaces and affine Kac-Moody Lie algebras.

In the past two decades there has been considerable interest in classifying gradings by arbitrary groups on algebras of different varieties including associative, Lie and Jordan. Of particular importance are the so-called fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism, although not in a unique way. If the ground field is algebraically closed and of characteristic 0, then the classification of fine abelian group gradings on an algebra (up to equivalence) is the same as the classification of maximal quasitori in the algebraic group of automorphisms (up to conjugation). Such a classification is now known for all finite-dimensional simple complex Lie algebras.

In this talk I will review the above mentioned classification and present a recent joint work with A. Elduque and A. Rodrigo-Escudero in which we classify fine gradings on classical simple real Lie algebras.