Pointed Hopf algebras are a wide class of Hopf algebras, including group algebras and enveloping algebras of Lie algebras. In this talk, based on a recent work with Susan Montgomery, we study actions of pointed Hopf algebras on simple algebras. These actions are known to be inner, as in the case of Skolem-Noether theorem. We try to give explicit descriptions, whenever possible, and consider Taft algebras, their Drinfeld doubles and some quantum groups.
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.
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.
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).
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.
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.
In this talk, we will consider the support variety theory for Hopf algebras and finite tensor categories. We will start by presenting the basic definitions and properties of these algebraic structures (Hopf algebras and tensor categories) and then we will introduce the theory of support varieties for them. We will analyze questions about projectivity and tensor products by using support theory and we will illustrate this via some examples. If time allows, we will discuss some deeper results involving complexity, realization, and connectedness of the varieties.
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