scholarly journals THE VAN EST SPECTRAL SEQUENCE FOR HOPF ALGEBRAS

2004 ◽  
Vol 01 (01n02) ◽  
pp. 33-48 ◽  
Author(s):  
E. J. BEGGS ◽  
TOMASZ BRZEZIŃSKI
Keyword(s):  
Hopf Algebra ◽  
Spectral Sequence ◽  
Lie Algebra ◽  
Hopf Algebras ◽  
Cohomology Group ◽  
Spectral Sequences ◽  
De Rham Cohomology ◽  
Lie Algebra Cohomology ◽  
De Rham ◽  
Definition Of

Various aspects of the de Rham cohomology of Hopf algebras are discussed. In particular, it is shown that the de Rham cohomology of an algebra with the differentiable coaction of a cosemisimple Hopf algebra with trivial 0-th cohomology group, reduces to the de Rham cohomology of (co)invariant forms. Spectral sequences are discussed and the van Est spectral sequence for Hopf algebras is introduced. A definition of Hopf–Lie algebra cohomology is also given.

Mixed Hodge Structures

2017 ◽  
Author(s):  
Fouad El Zein ◽  
Lˆe D˜ung Tr ´ang
Keyword(s):  
Spectral Sequence ◽  
Linear Algebra ◽  
Algebraic Variety ◽  
Spectral Sequences ◽  
Hodge Structures ◽  
De Rham Complex ◽  
Rham Complex ◽  
De Rham ◽  
Definition Of ◽  
Mixed Hodge Structures

This chapter discusses mixed Hodge structures (MHS). It first defines the abstract category of Hodge structures and introduces spectral sequences. The decomposition on the cohomology of Kähler manifolds is used to prove the degeneration at rank 1 of the spectral sequence defined by the filtration F on the de Rham complex in the projective nonsingular case. The chapter then introduces an abstract definition of MHS as an object of interest in linear algebra. It then attempts to develop algebraic homology techniques on filtered complexes up to filtered quasi-isomorphisms of complexes. Finally, this chapter provides the construction of the MHS on any algebraic variety.

scholarly journals Central invariants and enveloping algebras of braided Hom-Lie algebras

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 3893-3915
Author(s):  
Shengxiang Wang ◽  
Xiaohui Zhang ◽  
Shuangjian Guo
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Hopf Algebras ◽  
Enveloping Algebras ◽  
Definition Of ◽  
Generalized Lie Algebras

Let (H,?) be a monoidal Hom-Hopf algebra and HH HYD the Hom-Yetter-Drinfeld category over (H,?). Then in this paper, we first introduce the definition of braided Hom-Lie algebras and show that each monoidal Hom-algebra in HH HYD gives rise to a braided Hom-Lie algebra. Second, we prove that if (A,?) is a sum of two H-commutative monoidal Hom-subalgebras, then the commutator Hom-ideal [A,A] of A is nilpotent. Also, we study the central invariant of braided Hom-Lie algebras as a generalization of generalized Lie algebras. Finally, we obtain a construction of the enveloping algebras of braided Hom-Lie algebras and show that the enveloping algebras are H-cocommutative Hom-Hopf algebras.

scholarly journals Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p

2021 ◽  
Vol 9 ◽  
Author(s):  
Benjamin Antieau ◽  
Bhargav Bhatt ◽  
Akhil Mathew
Keyword(s):  
Spectral Sequence ◽  
Spectral Sequences ◽  
Characteristic P ◽  
De Rham

Abstract We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

On unrolled Hopf algebras

2018 ◽  
Vol 27 (10) ◽  
pp. 1850053
Author(s):  
Nicolás Andruskiewitsch ◽  
Christoph Schweigert
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Hopf Algebras ◽  
Drinfeld Module ◽  
Nichols Algebra ◽  
Small Quantum ◽  
Definition Of ◽  
Special Case

We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra [Formula: see text] of a Yetter–Drinfeld module [Formula: see text] on which a Lie algebra [Formula: see text] acts by biderivations. As a special case, we find unrolled versions of the small quantum group.

scholarly journals Notes on Absolute Hodge Classes

2017 ◽  
Author(s):  
Franc¸ois Charles ◽  
Christian Schnell
Keyword(s):  
Abelian Varieties ◽  
Hodge Conjecture ◽  
De Rham Cohomology ◽  
Galois Action ◽  
De Rham ◽  
Definition Of ◽  
Full Proof

This chapter surveys the theory of absolute Hodge classes. First, the chapter recalls the construction of cycle maps in de Rham cohomology, which is then used in the definition of absolute Hodge classes. The chapter then deals with variational properties of absolute Hodge classes. After stating the variational Hodge conjecture, the chapter proves Deligne's principle B and discusses consequences of the algebraicity of Hodge bundles and of the Galois action on relative de Rham cohomology. Finally, the chapter provides some important examples of absolute Hodge classes: a discussion of the Kuga–Satake correspondence as well as a full proof of Deligne's theorem which states that Hodge classes on abelian varieties are absolute.

scholarly journals COMPILATION OF RELATIONS FOR THE ANTISYMMETRIC TENSORS DEFINED BY THE LIE ALGEBRA COCYCLES OF su(n)

2001 ◽  
Vol 16 (08) ◽  
pp. 1377-1405 ◽  
Author(s):  
J. A. DE AZCÁRRAGA ◽  
A. J. MACFARLANE
Keyword(s):  
Lie Algebra ◽  
Essential Role ◽  
Lie Algebra Cohomology ◽  
Powerful Means ◽  
Casimir Operators ◽  
Definition Of ◽  
Comprehensive Compilation

This paper attempts to provide a comprehensive compilation of results, many new here, involving the invariant totally antisymmetric tensors (Omega tensors) which define the Lie algebra cohomology cocycles of su (n), and that play an essential role in the optimal definition of Racah–Casimir operators of su (n). Since the Omega tensors occur naturally within the algebra of totally antisymmetrized products of λ-matrices of su (n), relations within this algebra are studied in detail, and then employed to provide a powerful means of deriving important Omega tensor/cocycle identities. The results include formulas for the squares of all the Omega tensors of su (n). Various key derivations are given to illustrate the methods employed.

scholarly journals On a spectral sequence for the cohomology of a nilpotent Lie algebra

2014 ◽  
Vol 14 (01) ◽  
pp. 1450078
Author(s):  
Viviana del Barco
Keyword(s):  
Spectral Sequence ◽  
Lie Algebra ◽  
Cohomology Groups ◽  
Nilpotent Lie Algebra ◽  
Lie Algebra Cohomology ◽  
Direct Sum ◽  
Differential Complex ◽  
Dimension One

Given a nilpotent Lie algebra 𝔫 we construct a spectral sequence which is derived from a filtration of its Chevalley–Eilenberg differential complex and converges to the Lie algebra cohomology of 𝔫. The limit of this spectral sequence gives a grading for the Lie algebra cohomology, except for the cohomology groups of degree 0, 1, dim 𝔫 - 1 and dim 𝔫 as we shall prove. We describe the spectral sequence associated to a nilpotent Lie algebra which is a direct sum of two ideals, one of them of dimension one, in terms of the spectral sequence of the co-dimension one ideal. Also, we compute the spectral sequence corresponding to each real nilpotent Lie algebra of dimension less than or equal to six.

scholarly journals T-Ω Formulation with Higher-Order Hierarchical Basis Functions for Nonsimply Connected Conductors

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Ahmed Khebir ◽  
Paweł Dłotko ◽  
Bernard Kapidani ◽  
Ammar Kouki ◽  
Ruben Specogna
Keyword(s):  
Cohomology Group ◽  
Single Unit ◽  
Eddy Currents ◽  
Higher Order ◽  
Basis Functions ◽  
De Rham Cohomology ◽  
Arbitrary Topology ◽  
Hierarchical Basis ◽  
De Rham

This paper presents in detail the extension of the T-Ω formulation for eddy currents based on higher-order hierarchical basis functions so that it can automatically deal with conductors of arbitrary topology. To this aim, we supplement the classical hierarchical basis functions with nonlocal basis functions spanning the first de Rham cohomology group of the insulating region. Such nonlocal basis functions may be efficiently and automatically found in negligible time with the recently introduced Dłotko–Specogna (DS) algorithm. The approach presented in this paper merges techniques together which to some extent already existed in literature but they were never grouped together and tested as a single unit.

Sign in / Sign up

Export Citation Format

Share Document