scholarly journals Entropic Hopf algebras

10.29007/39rd ◽  
2018 ◽  
Author(s):  
Anna Romanowska ◽  
Jonathan Smith
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Probability Distributions ◽  
Commutative Rings ◽  
Group Algebras ◽  
Tensor Algebra ◽  
Primitive Elements ◽  
Commutative Monoids ◽  
Algebraic Setting ◽  
Grouplike Element

Classically, Hopf algebras are defined on the basis of modules over commutative rings. The present study seeks to extend the Hopf algebra formalism to a more general universal-algebraic setting, entropic varieties, including (pointed) sets, barycentric algebras, semilattices, and commutative monoids. The concept of a setlike (or grouplike) element may be defined, and group algebras constructed, in any such variety. In particular, group algebras within the variety of barycentric algebras consist precisely of finitely supported probability distributions on groups. For primitive elements and group quantum doubles, the natural universal-algebraic classes are entropic Jónsson-Tarski varieties (such as semilattices or commutative monoids). There, the tensor algebra on any algebra is a bialgebra, and the set of primitive elements of a Hopf algebra forms an abelian group. Coalgebra congruences on comonoids in entropic varieties are also studied.

scholarly journals CLASSIFYING COALGEBRA SPLIT EXTENSIONS OF HOPF ALGEBRAS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250227
Author(s):  
A. L. AGORE ◽  
C. G. BONTEA ◽  
G. MILITARU
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Infinite Family ◽  
Crossed Products ◽  
Primitive Elements ◽  
Characteristic P ◽  
Generators And Relations ◽  
Split Extensions ◽  
Groups Of Automorphisms

For a given Hopf algebra A we classify all Hopf algebras E that are coalgebra split extensions of A by H4, where H4is the Sweedler's four-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras A# H4by computing explicitly two classifying objects: the cohomological "group" [Formula: see text] and CRP (H4, A) ≔ the set of types of isomorphisms of all crossed products A# H4. All crossed products A# H4are described by generators and relations and classified: they are parameterized by the set [Formula: see text] of all central primitive elements of A. Several examples are worked out in detail: in particular, over a field of characteristic p ≥ 3 an infinite family of non-isomorphic Hopf algebras of dimension 4p is constructed. The groups of automorphisms of these Hopf algebras are also described.

Primitive Generators for Algebras

1982 ◽  
Vol 34 (2) ◽  
pp. 454-465
Author(s):  
Stanley O. Kochman
Keyword(s):  
Hopf Algebra ◽  
Commutative Algebra ◽  
Hopf Algebras ◽  
Primitive Elements ◽  
Explicit Formulas ◽  
Recursive Computation

LetHbe a graded commutative algebra with a nice set of algebra generators. LetHalso be a comodule over a Hopf algebraA. In Section 2 we give conditions under which certain of these generators ofHcan be rechosen to be primitive. In addition we give explicit formulas expressing these primitive generators in terms of the original set of generators.In Section 3 we apply the theory of Section 2 to the modphomology of the Thorn spectraMO, MUandMSp.In particular we give two explicit descriptions of the image of the Hurewicz homomorphism forMO.One of these makes explicit the recursive computation of E. Brown and F. Peterson [1].In Section 4 we give a variation of the theory of Section 2 which computes primitive generators of certain Hopf algebras. This theory is applied to study the primitive elements ofH*(BU)andH*(SO;Z2).

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

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.

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.

scholarly journals Multiplier Hopf algebras: Globalization for partial actions

2019 ◽  
Vol 30 (03) ◽  
pp. 539-565
Author(s):  
Graziela Fonseca ◽  
Eneilson Fontes ◽  
Grasiela Martini
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Action Theory ◽  
Natural Extension ◽  
Partial Action ◽  
Pertinent Question ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra

In partial action theory, a pertinent question is whenever given a partial action of a Hopf algebra [Formula: see text] on an algebra [Formula: see text], it is possible to construct an enveloping action. The authors Alves and Batista, in [M. Alves and E. Batista, Globalization theorems for partial Hopf (co)actions and some of their applications, groups, algebra and applications, Contemp. Math. 537 (2011) 13–30], have shown that this is always possible if [Formula: see text] is unital. We are interested in investigating the situation, where both algebras [Formula: see text] and [Formula: see text] are not necessarily unitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [A. Van Daele, Multiplier Hopf algebras, Trans. Am. Math. Soc. 342 (1994) 917–932], which is called multiplier Hopf algebra. Therefore, we will consider partial actions of multipliers Hopf algebras on algebras with a nondegenerate product and we will present a globalization theorem for this structure. Moreover, Dockuchaev et al. in [Globalizations of partial actions on nonunital rings, Proc. Am. Math. Soc. 135 (2007) 343–352], have shown when group partial actions on nonunitary algebras are globalizable. Based on this paper, we will establish a bijection between globalizable group partial actions and partial actions of a multiplier Hopf algebra.

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