An alternative notion of Hopf Algebroid

Using the language of𝔥-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group,ℱell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra𝔰𝔩n. We apply the generalized FRST construction and obtain an𝔥-bialgebroidℱell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the𝔥-Hopf algebroidℱell(GL(n)).

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SMASH PRODUCTS, SEPARABLE EXTENSIONS AND A MORITA CONTEXT OVER HOPF ALGEBROIDS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813501247 ◽

2014 ◽

Vol 13 (04) ◽

pp. 1350124

Author(s):

YONG WANG ◽

GUANGQUAN GUO

Keyword(s):

Type Theorem ◽

Smash Product ◽

Module Algebra ◽

Morita Context ◽

Hopf Algebroid ◽

Hopf Algebroids ◽

Maschke Type Theorem ◽

Module Algebras ◽

Product Algebra ◽

Separable Extensions

Let [Formula: see text] be a Hopf algebroid, and A a left [Formula: see text]-module algebra. This paper is concerned with the smash product algebra A#H over Hopf algebroids. In this paper, we investigate separable extensions for module algebras over Hopf algebroids. As an application, we obtain a Maschke-type theorem for A#H-modules over Hopf algebroids, which generalizes the corresponding result given by Cohen and Fischman in [Hopf algebra actions, J. Algebra100 (1986) 363–379]. Furthermore, based on the work of Kadison and Szlachányi in [Bialgebroid actions on depth two extensions and duality, Adv. Math.179 (2003) 75–121], we construct a Morita context connecting A#H and [Formula: see text] the invariant subalgebra of [Formula: see text] on A.

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Batalin–Vilkovisky structures on Ext and Tor

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2012-0086 ◽

2014 ◽

Vol 2014 (697) ◽

Cited By ~ 2

Author(s):

Niels Kowalzig ◽

Ulrich Krähmer

Keyword(s):

Algebraic Structure ◽

Hopf Algebroid

AbstractThis article studies the algebraic structure of homology theories defined by a left Hopf algebroid

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On the finite dual of a cocommutative Hopf algebroid. Application to linear differential matrix equations and Picard-Vessiot theory.

Bulletin of the Belgian Mathematical Society - Simon Stevin ◽

10.36045/j.bbms.200218 ◽

2021 ◽

Vol 28 (1) ◽

Author(s):

Laiachi El Kaoutit ◽

José Gómez-Torrecillas

Keyword(s):

Matrix Equations ◽

Hopf Algebroid ◽

Linear Differential ◽

Differential Matrix

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Gauge groups and bialgebroids

Letters in Mathematical Physics ◽

10.1007/s11005-021-01482-2 ◽

2021 ◽

Vol 111 (6) ◽

Author(s):

Xiao Han ◽

Giovanni Landi

Keyword(s):

Gauge Group ◽

Galois Extension ◽

Principal Bundle ◽

Module Structure ◽

Crossed Module ◽

Gauge Groups ◽

Quantum Sphere ◽

Hopf Algebroid ◽

Commutative Algebras

AbstractWe study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include: Galois objects of Taft algebras, a monopole bundle over a quantum sphere and a not faithfully flat Hopf–Galois extension of commutative algebras. For each of the latter two examples, there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.

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Comparison of two notions of weak crossed product

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500596 ◽

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].

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