scholarly journals FULLY BOUNDED NOETHERIAN RINGS AND FROBENIUS EXTENSIONS

2007 ◽  
Vol 06 (02) ◽  
pp. 189-206 ◽  
Author(s):  
S. CAENEPEEL ◽  
T. GUÉDÉNON
Keyword(s):  
Hopf Algebras ◽  
Noetherian Rings ◽  
Linear Map ◽  
Trace Map ◽  
Frobenius Algebras ◽  
Grouplike Element ◽  
Special Cases ◽  
Del Rio

Let i: A → R be a ring morphism, and χ: R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1R) = 1A. If R is a Frobenius A-ring, then we can define a trace map tr: A → AR. If there exists an element of trace 1 in A, then A is right FBN if and only if AR is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring. We recover results of García and del Río, and Dǎscǎlescu, Kelarev and Torrecillas on actions of group and Hopf algebras on FBN rings as special cases. We also obtain applications to extensions of Frobenius algebras, and to Frobenius corings with a grouplike element.

Generalized Frobenius Algebras and Hopf Algebras

2014 ◽  
Vol 66 (1) ◽  
pp. 205-240 ◽  
Author(s):  
Miodrag Cristian Iovanov
Keyword(s):  
Hopf Algebras ◽  
Topological Algebra ◽  
Homological Algebra ◽  
General Concept ◽  
Frobenius Algebra ◽  
Theoretic Approach ◽  
Topological Algebras ◽  
Infinite Dimensional ◽  
Finite Case ◽  
Frobenius Algebras

Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

scholarly journals A CLASS OF QUASITRIANGULAR GROUP-COGRADED MULTIPLIER HOPF ALGEBRAS

2018 ◽  
Vol 62 (1) ◽  
pp. 43-57
Author(s):  
TAO YANG ◽  
XUAN ZHOU ◽  
HAIXING ZHU
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Finite Dimensional ◽  
Special Cases ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Classical Construction

AbstractFor a multiplier Hopf algebra pairing 〈A,B〉, we construct a class of group-cograded multiplier Hopf algebras D(A,B), generalizing the classical construction of finite dimensional Hopf algebras introduced by Panaite and Staic Mihai [Isr. J. Math. 158 (2007), 349–365]. Furthermore, if the multiplier Hopf algebra pairing admits a canonical multiplier in M(B⊗A) we show the existence of quasitriangular structure on D(A,B). As an application, some special cases and examples are provided.

scholarly journals Constructions of L∞ Algebras and Their Field Theory Realizations

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Olaf Hohm ◽  
Vladislav Kupriyanov ◽  
Dieter Lüst ◽  
Matthias Traube
Keyword(s):  
Field Theory ◽  
Initial Data ◽  
Vector Space ◽  
Jacobi Identity ◽  
General Theorem ◽  
Commutator Algebra ◽  
Courant Algebroid ◽  
Linear Map ◽  
General Initial Data ◽  
Special Cases

We construct L∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these L∞ algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define L∞ algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term L∞ algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid.

scholarly journals Braided Frobenius algebras from certain Hopf algebras

2021 ◽  
pp. 2350012
Author(s):  
Masahico Saito ◽  
Emanuele Zappala
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Frobenius Algebra ◽  
Compatibility Conditions ◽  
Ribbon Graphs ◽  
Frobenius Algebras ◽  
Ternary Operation ◽  
Compact Surfaces ◽  
Baxter Operator

A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation [Formula: see text], that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations.

Weak multiplier Hopf algebras I. The main theory

2015 ◽  
Vol 2015 (705) ◽  
Author(s):  
Alfons Van Daele ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebras ◽  
Special Cases ◽  
Multiplier Hopf Algebras

AbstractAFrom these conditions we develop the theory. In particular,are special cases of such

THE QUASITRIANGULAR STRUCTURES FOR ω-SMASH COPRODUCT HOPF ALGEBRAS

2009 ◽  
Vol 08 (05) ◽  
pp. 673-687 ◽  
Author(s):  
ZHENGMING JIAO
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
Necessary And Sufficient

In this paper, the quasitriangular structures of ω-smash coproduct Hopf algebras Bω ⋈ H as constructed by Caenepeel, Ion, Militaru and Zhu were studied. Necessary and sufficient conditions for ω-smash coproduct Hopf algebras to be quasitriangular Hopf algebras are given in terms of properties of their components. As applications of our results, some special cases are discussed. Especially, The quasitriangular structures for D(H)* and H4ω ⋈ kℤ2 are constructed.

scholarly journals When Hopf algebras are Frobenius algebras

1971 ◽  
Vol 18 (4) ◽  
pp. 588-596 ◽  
Author(s):  
Bodo Pareigis
Keyword(s):  
Hopf Algebras ◽  
Frobenius Algebras

Decomposition of totally transcendental modules

1980 ◽  
Vol 45 (1) ◽  
pp. 155-164 ◽  
Author(s):  
Steven Garavaglia
Keyword(s):  
Quantifier Elimination ◽  
Decomposition Theorem ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Artinian Modules ◽  
Injective Modules ◽  
Unique Decomposition ◽  
Special Cases ◽  
Decomposition Theorems ◽  
Coherent Rings

The main theorem of this paper states that if R is a ring and is a totally transcendental R-module, then has a unique decomposition as a direct sum of indecomposable R-modules. Natural examples of totally transcendental modules are injective modules over noetherian rings, artinian modules over commutative rings, projective modules over left-perfect, right-coherent rings, and arbitrary modules over Σ – α-gens rings. Therefore, our decomposition theorem yields as special cases the purely algebraic unique decomposition theorems for these four classes of modules due to Matlis; Warfield; Mueller, Eklof, and Sabbagh; and Shelah and Fisher. These results and a number of other corollaries about totally transcendental modules are covered in §1. In §2, I show how the results of § 1 can be used to give an improvement of Baur's classification of ω-categorical modules over countable rings. In §3, the decomposition theorem is used to study modules with quantifier elimination over noetherian rings.The goals of this section are to prove the decomposition theorem and to derive some of its immediate corollaries. I will begin with some notational conventions. R will denote a ring with an identity element. LR is the language of left R-modules described in [4, p. 251] and TR is the theory of left R-modules. “R-module” will mean “unital left R-module”. A formula will mean an LR-formula.

Sign in / Sign up

Export Citation Format

Share Document