Separable extensions for crossed products over monoidal Hom-Hopf algebras

2018 ◽  
Vol 17 (09) ◽  
pp. 1850161 ◽  
Author(s):  
Zhongwei Wang ◽  
Yuanyuan Chen ◽  
Liangyun Zhang
Keyword(s):  
Hopf Algebra ◽  
Galois Extension ◽  
Hopf Algebras ◽  
Crossed Products ◽  
Separable Extensions

Let [Formula: see text] be a Frobenius monoidal Hom-Hopf algebra, and [Formula: see text] an [Formula: see text]-Hom-Hopf Galois extension of [Formula: see text]. We prove that the separability of the Hom-algebra extension [Formula: see text] is equivalent to the existence of a trace one element [Formula: see text] that centralizes [Formula: see text]. As applications, we obtain the differentiated conditions for the extension [Formula: see text] to be separable, and deduce a Doi’s result of Hom-type.

scholarly journals Comparison of two notions of weak crossed product

2020 ◽  
pp. 2150059
Author(s):  
Jorge A. Guccione ◽  
Juan J. Guccione
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Crossed Product ◽  
Weak Hopf Algebra ◽  
Crossed Products ◽  
Hopf Algebroid ◽  
Hopf Algebroids ◽  
Weak Crossed Product ◽  
Cleft Extensions ◽  
Weak Hopf Algebras

We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in [Cleft extensions of Hopf algebroids, Appl. Categor. Struct. 14(5–6) (2006) 431–469] with the notion of crossed product with a weak Hopf algebra introduced in [Crossed products for weak Hopf algebras with coalgebra splitting, J. Algebra 281(2) (2004) 731–752].

COREPRESENTATION THEORY OF MULTIPLIER HOPF ALGEBRAS II

2000 ◽  
Vol 11 (02) ◽  
pp. 233-278 ◽  
Author(s):  
HIDEKI KUROSE ◽  
ALFONS VAN DAELE ◽  
YINHUO ZHANG
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Covariant Theory ◽  
Crossed Products ◽  
Quantum Double ◽  
Multiplier Algebra ◽  
Algebraic Quantum ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Invertible Elements

We continue our development of the corepresentation theory of multiplier Hopf algebras. In this paper, we consider the corepresentations of a multiplier Hopf algebra A in a nondegenerate algebra B rather than on a vector space (cf. [25]). We concentrate ourself on those corepresentations of A in B which are invertible elements of the multiplier algebra M(B⊗A). They are called the unitary corepresentations of A. In particular, the generalized R-matrices or quasi-triangular structures of a regular multiplier Hopf algebra are unitary (bi)corepresentations. As an application the quantum double of an algebraic quantum group can be constructed by means of the universal unitary corepresentation. Moreover, a unitary corepresentation of A in B can implement an inner coaction of A on B which allows us to study the covariant theory and crossed products.

scholarly journals Crossed products of Hom-Hopf algebras

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1295-1313
Author(s):  
Daowei Lu ◽  
Yizheng Li ◽  
Shuangjian Guo
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Crossed Product ◽  
Crossed Products ◽  
Cleft Extension ◽  
Necessary And Sufficient

Let (H,?) be a Hom-Hopf algebra and (A,?) be a Hom-algebra. In this paper we will construct the Hom-crossed product (A#?H???), and prove that the extension A ? A#?H is actually a Hom-type cleft extension and vice versa. Then we will give the necessary and sufficient conditions to make (A#?H???) into a Hom-Hopf algebra. Finally we will study the lazy 2-cocycle on (H,?).

scholarly journals Finitistic dimension of weak Hopf-Galois extensions

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2825-2828
Author(s):  
Xiao-Yan Zhou ◽  
Qiang Li
Keyword(s):  
Hopf Algebra ◽  
Galois Extension ◽  
Duality Theorem ◽  
Weak Hopf Algebra ◽  
Crossed Products ◽  
Finitistic Dimension ◽  
Galois Extensions ◽  
Finite Dimensional ◽  
Dimension Conjecture

Let H be a finite dimensional weak Hopf algebra and A/B be a right faithfully flat weak H-Galois extension. Then in this note, we first show that if H is semisimple, then the finitistic dimension of A is less than or equal to that of B. Furthermore, using duality theorem, we obtain that if H and its dual H* are both semisimple, then the finitistic dimension of A is equal to that of B, which means the finitistic dimension conjecture holds for A if and only if it holds for B. Finally, as applications, we obtain these relations for the weak crossed products and weak smash products.

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.

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.

scholarly journals Crossed products for weak Hopf algebras with coalgebra splitting

2004 ◽  
Vol 281 (2) ◽  
pp. 731-752 ◽  
Author(s):  
J.N. Alonso Álvarez ◽  
R. González Rodríguez
Keyword(s):  
Hopf Algebras ◽  
Crossed Products ◽  
Weak Hopf Algebras
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