Self-Inverse Yang–Baxter Operators from (Co)Algebra Structures

Nichita, Florin F.

doi:10.1006/jabr.1999.7915

Cited by 26 publications

(30 citation statements)

References 4 publications

(4 reference statements)

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“…The braided system from the third line of Table 2 yields an inclusion of the category of bialgebras in vect k into BrSyst 2 (vect k ). Nichita's work [22,2,23] can be seen in the same light. To encode associativity, he uses a generalization of the self-inverse braiding σ Ass = ν ⊗ µ + µ ⊗ ν − Id V ⊗2 , proposed by Nuss in the context of descent theory for noncommutative rings [24].…”

Section: Introductionmentioning

confidence: 78%

Braided systems: a unified treatment of algebraic structures with several operations

Lebed

2017

Homology, Homotopy and Applications

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Bialgebras and Hopf (bi)modules are typical algebraic structures with several interacting operations. Their structural and homological study is therefore quite involved. We develop the machinery of braided systems, tailored for handling such multioperation situations. Our construction covers the above examples (as well as Poisson algebras, Yetter-Drinfel ′ d modules, and several other structures, treated in separate publications). In spite of this generality, graphical tools allow an efficient study of braided systems, in particular of their representation and homology theories. These latter naturally recover, generalize, and unify standard homology theories for bialgebras and Hopf (bi)modules (due to Gerstenhaber-Schack, Panaite-Ştefan, Ospel, Taillefer); and the algebras encoding their representation theories (Heisenberg double, algebras X , Y , Z of Cibils-Rosso and Panaite). Our approach yields simplified and conceptual proofs of the properties of these objects.

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Section: Introductionmentioning

confidence: 78%

Braided systems: a unified treatment of algebraic structures with several operations

Lebed

2017

Homology, Homotopy and Applications

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“…i) The proof that ψ C is a generalised YB operator (cf. Proposition 2.3 from [12]) is left to the reader (ψ −1…”

Section: Theorem 35 (I) There Exists a Functormentioning

confidence: 99%

The Duality between Corings and Ring Extensions

Nichita

Zieliński

2012

Axioms

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We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional algebras and coalgebra. Both these duality extensions have some similarities with the Pontryagin-van Kampen duality theorem

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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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Self-Inverse Yang–Baxter Operators from (Co)Algebra Structures

Cited by 26 publications

References 4 publications

Braided systems: a unified treatment of algebraic structures with several operations

Braided systems: a unified treatment of algebraic structures with several operations

The Duality between Corings and Ring Extensions

Lifting Theorems for Tensor Functors on Module Categories

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