Yang–baxter Systems and Entwining Structures

Brzeziński, Tomasz; Nichita, Florin F.

doi:10.1081/agb-200053815

Cited by 23 publications

(43 citation statements)

References 16 publications

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“…This approach leads to similar formulas and is developed by Baez in [2]. Relations between Yang-Baxter operators and entwining structures (on vector space categories) are investigated by Brzeziński and Nichita in [15].…”

Section: Introductionmentioning

confidence: 86%

Algebras Versus Coalgebras

Wisbauer

2007

Appl Categor Struct

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Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between the base category and the category of the associated bimodules. For a set G, the endofunctor G× -on the category of sets shares these properties if and only if G admits a group structure. Finally, we report about the possibility of subsuming algebras and coalgebras in the notion of (F, G)-dimodules associated to two functors F, G : A → B between different categories. This observation, due to Tatsuya Hagino, was an outcome from the theory of categorical data types and may also be of use in classical algebra.

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Section: Introductionmentioning

confidence: 86%

Algebras Versus Coalgebras

Wisbauer

2007

Appl Categor Struct

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“…In particular it is used there to insure an algebra structure on multiple products of a monad, generalising the cases considered in Nuss [22] (see 3.8) and Menini and Stefan [19]. Yang-Baxter operators in context with algebras, coalgebras and entwinings are also studied by Brzeziński, Dǎscǎlescu and Nichita in [21], [13], [8].…”

Section: 9mentioning

confidence: 99%

Lifting Theorems for Tensor Functors on Module Categories

Wisbauer

2011

J. Algebra Appl.

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Any (co)ring R is an endofunctor with (co)multiplication on the category of abelian groups. These notions were generalised to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate's lifting theorem of functors between categories to related (co)module categories. Curiously enough some of these results were not noticed by researchers in module theory and thus notions like entwining structures and smash products between algebras and coalgebras were introduced (in the nineties) without being aware that these are special cases of the more general theory.The purpose of this survey is to explain several of these notions and recent results from general category theory in the language of elementary module theory focussing on functors between module categories given by tensoring with a bimodule. This provides a simple and systematic approach to smash products, wreath products, corings and rings over corings (C-rings). We also highlight the relevance of the Yang-Baxter equation for the structures on the threefold tensor product of algebras or coalgebras (see 3.6).

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“…This construction played an important role in the construction of Yang-Baxter systems from algebra factorizations (see [16]). …”

Section: For Any Vector Spacesmentioning

confidence: 99%

Yang-Baxter Systems, Algebra Factorizations and Braided Categories

Nichita

2013

Axioms

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The Yang-Baxter equation first appeared in a paper by the Nobel laureate, C.N. Yang, and in R.J. Baxter's work. Later, Vladimir Drinfeld, Vaughan F. R. Jones and Edward Witten were awarded Fields Medals for their work related to the Yang-Baxter equation. After a short review on this equation and the Yang-Baxter systems, we consider the problem of constructing algebra factorizations from Yang-Baxter systems. Our sketch of proof uses braided categories. Other problems are also proposed.

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Yang–baxter Systems and Entwining Structures

Cited by 23 publications

References 16 publications

Algebras Versus Coalgebras

Algebras Versus Coalgebras

Lifting Theorems for Tensor Functors on Module Categories

Yang-Baxter Systems, Algebra Factorizations and Braided Categories

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