In mathematics, in the theory of Hopf algebras, a Hopf algebroid is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf k-algebroids. If k is a field, a commutative k-algebroid is a cogroupoid object in the category of k-algebras; the category of such is hence dual to the category of groupoid k-schemes. This commutative version has been used in 1970-s in algebraic geometry and stable homotopy theory. The generalization of Hopf algebroids and its main part of the structure, associative bialgebroids, to the noncommutative base algebra was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry (later shown equivalent in nontrivial way to a construction of Takeuchi from the 1970s and another by Xu around the year 2000). They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf algebroid satisfying a finite projectivity condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra H is a Hopf algebroid over its separable subalgebra HL. The antipode axioms have been changed by G. Böhm and K. Szlachányi (J. Algebra) in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions. (Wikipedia).
This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/1mUo
From playlist 3D printing
In this video I shed some light on a heavily alluded to and poorly explained object, the Hopf Fibration. The Hopf Fibration commonly shows up in discussions surrounding gauge theories and fundamental physics, though its construction is not so mysterious.
From playlist Summer of Math Exposition Youtube Videos
This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/3bz5
From playlist 3D printing
Ralph Kaufmann: Graph Hopf algebras and their framework
The lecture was held within the framework of the Hausdorff Trimester Program: Periods in Number Theory, Algebraic Geometry and Physics. Abstract: I will discuss recent results linking the Hopf algebras of Goncharov for multiple zetas, the Hopf algebra of Connes and Kreimer for renormalis
From playlist Workshop: "Amplitudes and Periods"
Matt SZCZESNY - Toric Hall Algebras and infinite-dimentional Lie algebras
The process of counting extensions in categories yields an associative (and sometimes Hopf) algebra called a Hall algebra. Applied to the category of Feynman graphs, this process recovers the Connes-Kreimer Hopf algebra. Other examples abound, yielding various combinatorial Hopf algebras.
From playlist Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday
The Hofmann-Löffler-Freytag Reaction
From playlist OPEN Modules
Walter Van SUIJLEKOM - Renormalization Hopf Algebras and Gauge Theories
We give an overview of the Hopf algebraic approach to renormalization, with a focus on gauge theories. We illustrate this with Kreimer's gauge theory theorem from 2006 and sketch a proof. It relates Hopf ideals generated by Slavnov-Taylor identities to the Hochschild cocycles that are give
From playlist Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday
Jonathan Block: Singular foliations and characteristic classes
Talk by Jonathan Rosenberg in Global Noncommutative Geometry Seminar (Americas) http://www.math.wustl.edu/~xtang/NCG-Seminar.html on October 21, 2020.
From playlist Global Noncommutative Geometry Seminar (Americas)
Loïc FOISSY - Cointeracting Bialgebras
Pairs of cointeracting bialgebras recently appears in the literature of combinatorial Hopf algebras, with examples based on formal series, on trees (Calaque, Ebrahimi-Fard, Manchon), graphs (Manchon), posets... We will give several results obtained on pairs of cointeracting bialgebras: act
From playlist Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday
Markus Pflaum: The transverse index theorem for proper cocompact actions of Lie groupoids
The talk is based on joint work with H. Posthuma and X. Tang. We consider a proper cocompact action of a Lie groupoid and define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this
From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"
Stable Homotopy Seminar, 21: Computing Homotopy Groups with the Adams Spectral Sequence (Zach Himes)
Zachary Himes constructs the May spectral sequence, a tool using a filtration of of the dual Steenrod algebra that calculates the E2 page of the Adams spectral sequence. May's original insight was that the associated graded of the dual Steenrod algebra is a primitively generated Hopf algeb
From playlist Stable Homotopy Seminar
Henrique Bursztyn: Relating Morita equivalence in algebra and geometry via deformation quantization
Talk by Henrique Bursztyn in Global Noncommutative Geometry Seminar (Americas) https://globalncgseminar.org/talks/3225/ on April 2, 2021.
From playlist Global Noncommutative Geometry Seminar (Americas)
The Hopf Fibration via Higher Inductive Types - Peter Lumsdaine
Peter Lumsdaine Dalhousie University; Member, School of Mathematics February 13, 2013 For more videos, visit http://video.ias.edu
From playlist Mathematics
Ryszard Nest: Formality for algebroid stacks
The lecture was held within the framework of the Hausdorff Trimester Program Non-commutative Geometry and its Applications. (19.09.2014) This video was created and edited with kind support from eCampus Bonn and is also available at https://mediaserver.uni-bonn.de.
From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"
Cohomology of algebroids and harmonic shuffle relation - Tomohide Terasoma
Geometry and Arithmetic: 61st Birthday of Pierre Deligne Tomohide Terasoma University of Tokyo October 18, 2005 Pierre Deligne, Professor Emeritus, School of Mathematics. On the occasion of the sixty-first birthday of Pierre Deligne, the School of Mathematics will be hosting a four-day c
From playlist Pierre Deligne 61st Birthday
Sparse Identification of Nonlinear Dynamics (SINDy)
This video illustrates a new algorithm for the sparse identification of nonlinear dynamics (SINDy). In this work, we combine machine learning, sparse regression, and dynamical systems to identify nonlinear differential equations purely from measurement data. From the Paper: Discovering
From playlist Research Abstracts from Brunton Lab
Advice | Exponentializing polyseries to get triangular on-maxels, and tilde Euler polynomials
Motivated by the relation between Bernoulli numbers and Bernoulli polynomials, we introduce a very general and powerful approach to move from sequences or polyseries to families of polynomials or polynumbers. When we apply this to the Euler numbers, we obtain a variant of the usual Euler
From playlist Maxel inverses and orthogonal polynomials (non-Members)
MAE5790-13 Hopf bifurcations in aeroelastic instabilities and chemical oscillators
Supercritical vs subcritical Hopf. Airplane wing vibrations. Flutter. Chemical oscillations. Computer simulations. Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 8.2, 8.3.
From playlist Nonlinear Dynamics and Chaos - Steven Strogatz, Cornell University