Cleft extensions for a hopf algebra generated by a nearly primitive element

1994 ◽  
Vol 22 (11) ◽  
pp. 4537-4559 ◽  
Author(s):  
Akira Masoka
Keyword(s):  
Hopf Algebra ◽  
Primitive Element ◽  
Cleft Extensions

scholarly journals Cleft extensions for a Hopf algebrakq[X,X−1, Y]

1998 ◽  
Vol 40 (2) ◽  
pp. 147-160 ◽  
Author(s):  
Hui-Xiang Chen
Keyword(s):  
Hopf Algebra ◽  
Primitive Element ◽  
Normal Basis ◽  
Crossed Products ◽  
Galois Extensions ◽  
Infinite Dimensional ◽  
Cleft Extensions ◽  
Cocommutative Hopf Algebra

The concept of cleft extensions, or equivalently of crossed products, for a Hopf algebra is a generalization of Galois extensions with normal basis and of crossed products for a group. The study of these subjects was founded independently by Blattner-Cohen-Montgomery [1] and by Doi-Takeuchi [4]. In this paper, we determine the isomorphic classes of cleft extensions for a infinite dimensional non-commutative, non-cocommutative Hopf algebra kq[X, X–l, Y], which is generated by a group-like element X and a (1,X)-primitive element Y. We also consider the quotient algebras of the cleft extensions.

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].

scholarly journals Two-cocycles and cleft extensions in left braided categories

2020 ◽  
pp. 2140013
Author(s):  
István Heckenberger ◽  
Kevin Wolf
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Cleft Extensions ◽  
Braided Categories

We define two-cocycles and cleft extensions in categories that are not necessarily braided, but where specific objects braid from one direction, like for a Hopf algebra [Formula: see text] a Yetter–Drinfeld module braids from the left with [Formula: see text]-modules. We will generalize classical results to this context and give some application for the categories of Yetter–Drinfeld modules and [Formula: see text]-modules. In particular, we will describe liftings of coradically graded Hopf algebras in the category of Yetter–Drinfeld modules with these techniques.

Invariants of FP-Projective Dimensions Under Cleft Extensions

2013 ◽  
Vol 20 (04) ◽  
pp. 689-700
Author(s):  
Xiuli Chen ◽  
Haiyan Zhu ◽  
Fang Li
Keyword(s):  
Hopf Algebra ◽  
Projective Dimension ◽  
Fixed Field ◽  
Finite Dimensional ◽  
Cleft Extensions ◽  
Finite Dimensional Hopf Algebra

Let H be a finite dimensional Hopf algebra and A be an algebra over a fixed field k. Firstly, it is proved that the left FP-projective dimension is invariant under cleft extensions when H is semisimple and A is left coherent. Secondly, using (co)induction functors, we study the relations between FP-projective dimensions in A # H-Mod and the counterparts in AH-Mod. Finally, we characterize the FP-projective preenvelopes (resp., precovers) under H*-extensions and cleft extensions, respectively.

scholarly journals Hopf algebra methods in graph theory

1995 ◽  
Vol 101 (1) ◽  
pp. 77-90 ◽  
Author(s):  
William R. Schmitt
Keyword(s):  
Graph Theory ◽  
Hopf Algebra

scholarly journals CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

1999 ◽  
Vol 02 (01) ◽  
pp. 105-129 ◽  
Author(s):  
UWE FRANZ
Keyword(s):  
Markov Process ◽  
Hopf Algebra ◽  
Markov Processes ◽  
Lévy Processes ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Vacuum State ◽  
Levy Processes ◽  
Quantum Process ◽  
Quantum Markov Processes

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

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