The Arnon bases in the Steenrod algebra

2020 ◽  
Vol 27 (4) ◽  
pp. 649-654
Author(s):  
Neşet Deniz Turgay ◽  
Ismet Karaca
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Steenrod Algebra ◽  
Free Part ◽  
Mod P

AbstractLet {\mathcal{A}=\mathcal{A}_{p}} be the {\mathrm{mod}\,p} Steenrod algebra, where p is a fixed prime and let {\mathcal{A}^{\prime}} denote the Bockstein-free part of {\mathcal{A}} at odd primes. Being a connected graded Hopf algebra, {\mathcal{A}} has the canonical conjugation χ. Using this map, we introduce a relationship between the X- and Z-bases of {\mathcal{A}^{\prime}}. We show that these bases restrict to give bases to the well-known sub-Hopf algebras {\mathcal{A}(n-1)}, {n\geq 1}, of {\mathcal{A}^{\prime}}.

scholarly journals On the mod p Steenrod algebra and the Leibniz-Hopf algebra

2020 ◽  
Vol 28 (2) ◽  
pp. 951-959
Author(s):  
Neşet Deniz Turgay ◽  
Keyword(s):  
Hopf Algebra ◽  
Steenrod Algebra ◽  
Leibniz Hopf Algebra ◽  
Mod P

scholarly journals Husemoller-Witt decompositions and actions of the Steenrod algebra

1985 ◽  
Vol 28 (2) ◽  
pp. 271-288 ◽  
Author(s):  
Andrew Baker
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Steenrod Algebra ◽  
The Other ◽  
Classifying Spaces ◽  
Thom Spectra

Recently, there has been renewed interest in the homology of connective covers of the classifying spaces BU and BO, and their associated Thom spectra-see e.g. [4,6,9,10,15]. There are now numerous families of generators as well as structural results on the action of the Steenrod algebra. However, these two areas have not been well related since the methods used have tended to emphasise one goal rather than the other. In this paper we show that there are in fact canonical Hopf algebra decompositions for the sub-Hopf algebras of the homology of BU, and BO constructed by S. Kochman in [9], generalising those of [8]. Furthermore, these are clearly and consistently related to the Steenrod algebra action, and provide canonical sets of algebra generators. They should thus allow calculations of the type exemplified in [6] to be carried out in all cases, although of course the complexity of the answer increases rapidly! A by-product of our approach is that we can easily obtain results on these homologies as Hopf algebras, such as selfduality and a computation of endomorphism groups over the Steenrod algebra. We feel that the methods will also give interesting information in the case of some other familiar spaces even if their homology is not self dual (or bipolynomial); we intend to return to this in a sequel.

Stably thick subcategories of modules over Hopf algebras

2001 ◽  
Vol 130 (3) ◽  
pp. 441-474 ◽  
Author(s):  
MARK HOVEY ◽  
JOHN H. PALMIERI
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebraic Closure ◽  
Steenrod Algebra ◽  
Closed Field ◽  
Finite Dimensional ◽  
A Finite Group ◽  
Similar Classification ◽  
General Method

We discuss a general method for classifying certain subcategories of the category of finite-dimensional modules over a finite-dimensional co-commutative Hopf algebra B. Our method is based on that of Benson–Carlson–Rickard [BCR1], who classify such subcategories when B = kG, the group ring of a finite group G over an algebraically closed field k. We get a similar classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of F2. Along the way, we prove a Quillen stratification theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B.

scholarly journals Hopf algebras and skew PBW extensions

2019 ◽  
Vol 10 (2) ◽  
Author(s):  
Luis Alfonso Salcedo Plazas
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Ore Extensions ◽  
Skew Pbw Extensions

In this article we relate some Hopf algebra structures over Ore extensions and over skew PBW extensions ofa Hopf algebra. These relations are illustrated with examples. We also show that Hopf Ore extensions andgeneralized Hopf Ore extensions are Hopf skew PBW extensions.

scholarly journals Antipode Formulas for some Combinatorial Hopf Algebras

10.37236/5949 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Rebecca Patrias
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Explicit Formulas ◽  
Quasisymmetric Functions ◽  
Noncommutative Symmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Combinatorial Formulas ◽  
K Theory

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.

scholarly journals QUANTUM RANDOM WALKS AND TIME REVERSAL

1993 ◽  
Vol 08 (25) ◽  
pp. 4521-4545 ◽  
Author(s):  
S. MAJID
Keyword(s):  
Hopf Algebra ◽  
Random Walks ◽  
Hopf Algebras ◽  
Evolution Operator ◽  
Original System ◽  
Quantum Random Walks ◽  
Quantum Random Walk ◽  
Time Evolution Operator ◽  
Simple Physical Interpretation ◽  
Primitive Notion

Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated with a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If H is any Hopf algebra, it may be realized in Lin(H) in such a way that Δh=W(h⊗1)W−1 for an operator W. This W is interpreted as the time evolution operator for the system at time t coupled quantum-mechanically to the system at time t+δ. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a novel kind of CTP theorem.

Gocommutative Hopf Algebras

1967 ◽  
Vol 19 ◽  
pp. 350-360 ◽  
Author(s):  
Richard G. Larson
Keyword(s):  
Hopf Algebra ◽  
Vector Space ◽  
Hopf Algebras ◽  
Image Position ◽  
Algebra Homomorphisms

A coalgebra over the field F is a vector space A over F, with maps δ: A → A ⊗ A and ∊: A → F such that1and2The notion of coalgebra is dual to the notion of algebra with unit, with δ as coproduct (equation (1) says that δ is associative) and ∊ as the unit map (equation (2) is just the statement that ∊ is a unit for the coproduct δ). If A is also an algebra with unit and δ and ∊ are algebra homomorphisms, A is a Hopf algebra.

scholarly journals The cohomology of the Steenrod algebra and the mod p Lannes-Zarati homomorphism

2020 ◽  
Vol 556 ◽  
pp. 656-695
Author(s):  
Phan Hoàng Chơn ◽  
Phạm Bích Như
Keyword(s):  
Steenrod Algebra ◽  
Mod P

A braided T-category over weak monoidal Hom-Hopf algebras

2019 ◽  
Vol 19 (08) ◽  
pp. 2050159
Author(s):  
Guohua Liu ◽  
Wei Wang ◽  
Shuanhong Wang ◽  
Xiaohui Zhang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Weak Hopf Algebras ◽  
T Category

In this paper, we define and study weak monoidal Hom-Hopf algebras, which generalize both weak Hopf algebras and monoidal Hom-Hopf algebras. Let [Formula: see text] be a weak monoidal Hom-Hopf algebra with bijective antipode and let [Formula: see text] be the set of all automorphisms of [Formula: see text], we introduce a category [Formula: see text] with [Formula: see text] and construct a braided [Formula: see text]-category [Formula: see text] having all the categories [Formula: see text] as components.

The shuffle Hopf algebra and noncommutative full completeness

1998 ◽  
Vol 63 (4) ◽  
pp. 1413-1436 ◽  
Author(s):  
R. F. Blute ◽  
P. J. Scott
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Natural Extension ◽  
Additive Group ◽  
Completeness Theorem ◽  
Vector Spaces ◽  
Invariance Condition ◽  
Shuffle Algebra ◽  
Topological Vector Spaces

AbstractWe present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL + MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces.This paper is a natural extension of the authors' previous work, “Linear Läuchli Semantics”, where a similar theorem is obtained for the commutative logic MLL + MIX. In that paper, we interpret proofs as dinaturals which are invariant under certain actions of the additive group of integers. Here we also present a simplification of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras in this paper corresponds to the passage from commutative to noncommutative logic. However, in our noncommutative setting, one must still keep the invariance condition on dinaturals.

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