Extending structures II: The quantum version

Agore, A. L.; Militaru, Gigel

doi:10.1016/j.jalgebra.2011.03.033

Cited by 54 publications

(126 citation statements)

References 14 publications

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“…there exists π : E → A a normal left A-module coalgebra map such that π (a) = a, for all a ∈ A. It was proved in [2] that an extension A ⊆ E splits in the above sense if and only if E is isomorphic to a unified product between A and a certain subcoalgebra H of E. The unified product was introduced in [1] as an answer to the restricted extending structures problem for Hopf algebras. Unified products characterize Hopf algebras that factorize through a Hopf subalgebra A and a subcoalgebra H such that 1 ∈ H. As special cases of the unified product, we recover the double cross product or the crossed product of Hopf algebras (see Examples 1.1).…”

Section: Let a ⊆ E Be An Extension Of Hopf Algebras What Is The Connmentioning

confidence: 99%

“…Let (A) = (H, , , f ) be an extending datum of A. We denote by A (A) H = A H the k-module A ⊗ H together with the multiplication: (2) , g (1) ) (h (3) c (3) ) · g (2) ,…”

Section: Unified Productsmentioning

confidence: 99%

“…: (1) , h (1) )[(g (2) · h (2) ) a], (BE5) (g (1) f (h (1) , l (1) ))f (g (2) f (h (2) , l (2) ), (2) , h (2) ) = g (2) · h (2) ⊗ f (g (1) , h (1) …”

Section: Unified Productsunclassified

“…First, we introduce some new definitions as natural generalizations for the concepts of braiding and skew pairing. (1) , g (2) · t (2) )p(a (2) , f (g (1) , t (1) …”

Section: Coquasitriangular Structures On the Unified Productsmentioning

confidence: 99%

“…u(a (1) , g (3) · t (3) )p(a (2) , f (g (1) , t (1) ))τ (1, g (4) t (4) )v(1, f (g (2) , t (2) )) = u(a (1) , t)u(a (2) , g). Now using relations (15) and (17), we get (RS3).…”

Section: Theorem 36 Let a Be A Hopf Algebra And (A) = (H F ) Amentioning

confidence: 99%

See 4 more Smart Citations

Coquasitriangular Structures for Extensions of Hopf Algebras. Applications

Agore

2012

Glasgow Math. J.

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Abstract. Let A ⊆ E be an extension of Hopf algebras such that there exists a normal left A-module coalgebra map π : E → A that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra E in terms of the datum (A, E, π) as follows: first, any such extension E is isomorphic to a unified product A H, for some unitary subcoalgebra H of E (A. L. Agore and G. Militaru, Unified products and split extensions of Hopf algebras, to appear in AMS Contemp. Math.). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product A H and a certain set of datum (p, τ, u, v) related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid' Larson and Towber in [10]. These are Hopf algebras A endowed with a linear map p : A ⊗ A → k satisfying some compatibility conditions. There is, of course, a dual result concerning the quantum Yang-Baxter equation: if M is a co-representation of a coquasitriangular Hopf algebra (A, p), then the canonical mapHowever, what makes the coquasitriangular Hopf algebras so important is the fact that the converse of the above statement is also true. Namely, by the celebrated FRT theorem for any solution R of the quantum Yang-Baxter equation, there exists a quasitriangular bialgebra (A(R), p) such that R = R p ([4]).Based on this background, (co)quasitriangular Hopf algebras generated an explosion of interest and were studied for their implications in quantum groups, the construction of invariants of knots and 3-manifolds, statistical mechanics and quantum mechanics, but they also became a subject of research in its own right. Complete descriptions of the coquasitriangular structures have already been obtained for several families of Hopf algebras, see, for instance, [3,7,8] or [11].

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Section: Let a ⊆ E Be An Extension Of Hopf Algebras What Is The Connmentioning

confidence: 99%

“…Let (A) = (H, , , f ) be an extending datum of A. We denote by A (A) H = A H the k-module A ⊗ H together with the multiplication: (2) , g (1) ) (h (3) c (3) ) · g (2) ,…”

Section: Unified Productsmentioning

confidence: 99%

“…: (1) , h (1) )[(g (2) · h (2) ) a], (BE5) (g (1) f (h (1) , l (1) ))f (g (2) f (h (2) , l (2) ), (2) , h (2) ) = g (2) · h (2) ⊗ f (g (1) , h (1) …”

Section: Unified Productsunclassified

“…First, we introduce some new definitions as natural generalizations for the concepts of braiding and skew pairing. (1) , g (2) · t (2) )p(a (2) , f (g (1) , t (1) …”

Section: Coquasitriangular Structures On the Unified Productsmentioning

confidence: 99%

“…u(a (1) , g (3) · t (3) )p(a (2) , f (g (1) , t (1) ))τ (1, g (4) t (4) )v(1, f (g (2) , t (2) )) = u(a (1) , t)u(a (2) , g). Now using relations (15) and (17), we get (RS3).…”

Section: Theorem 36 Let a Be A Hopf Algebra And (A) = (H F ) Amentioning

confidence: 99%

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Coquasitriangular Structures for Extensions of Hopf Algebras. Applications

Agore

2012

Glasgow Math. J.

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Extending Structures and Classifying Complements for Left-Symmetric Algebras

Hong

2019

Results Math

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Let A be a left-symmetric (resp. Novikov) algebra, E be a vector space containing A as a subspace and V be a complement of A in E. The extending structures problem which asks for the classification of all leftsymmetric (resp. Novikov) algebra structures on E such that A is a subalgebra of E is studied. In this paper, the definition of the unified product of left-symmetric (resp. Novikov) algebras is introduced. It is shown that there exists a left-symmetric (resp. Novikov) algebra structure on E such that A is a subalgebra of E if and only if E is isomorphic to a unified product of A and V . Two cohomological type objects H 2 A (V, A) and H 2 (V, A) are constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension A ⊂ E of left-symmetric (resp. Novikov) algebras, another cohomological type object is constructed to classify all complements of A in E. Several special examples are provided in details. 1 2 YANYONG HONGThis problem is called extending structures problem. The extending structures problems for groups, associative algebras, Hopf algebras, Lie algebras, and Leibniz algebras have been studied in [4,3,6,5,7] respectively. Since left-symmetric algebras and Novikov algebras are important algebraic structures in mathematics and mathematical physics, it is interesting and meaningful to study the extending structures problems for left-symmetric algebras and Novikov algebras. Certainly, this problem is very difficult. When A = {0}, this problem is equivalent to classify all left-symmetric algebras and Novikov algebras on an arbitrary vector space E. Of course, when the dimension of E is large, it is impossible to classify all algebraic structures. In fact, the classifications of left-symmetric algebras in dimension 2 and dimension 3 over C were obtained in [11,10], and the classifications of Novikov algebras in dimension 2 and dimension 3 over R and C were obtained in [12,16]. But, as far as we know, the complete classifications of left-symmetric algebras (resp. Novikov algebras) whose dimension are larger than 3 have not been obtained so far. There are only some papers on the classification of some special left-symmetric algebras and Novikov algebras of dimensions 4 and 5, such as [18,23,24,16]. Therefore, we will assume that A = {0} below. In fact, the extending structures problem generalizes some classic algebraic problems. For example, the extension of the left-symmetric algebra A by an abelian left-symmetric algebra C is characterized by the second cohomology group H 2 (C, A) (see [19]); the problem which asks for describing and classifying the extensions of the left-symmetric algebra A by another left-symmetric algebra B can be answered by bicrossed products of A and B (see [9]) and this problem is called the factorization problem. In this paper, our aim is to give a theoretical interpretation of the extending structures problem for left-symmetric (resp. Novikov) algebras using some cohomological type objects and provide some detailed answers to it in certain ...

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Zappa–Szép products of Garside monoids

Gebhardt

Tawn

2015

Math. Z.

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A monoid K is the internal Zappa-Szép product of two submonoids, if every element of K admits a unique factorisation as the product of one element of each of the submonoids in a given order. This definition yields actions of the submonoids on each other, which we show to be structure preserving.We prove that K is a Garside monoid if and only if both of the submonoids are Garside monoids. In this case, these factors are parabolic submonoids of K and the Garside structure of K can be described in terms of the Garside structures of the factors. We give explicit isomorphisms between the lattice structures of K and the product of the lattice structures on the factors that respect the Garside normal forms. In particular, we obtain explicit natural bijections between the normal form language of K and the product of the normal form languages of its factors.

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Extending structures II: The quantum version

Cited by 54 publications

References 14 publications

Coquasitriangular Structures for Extensions of Hopf Algebras. Applications

Coquasitriangular Structures for Extensions of Hopf Algebras. Applications

Extending Structures and Classifying Complements for Left-Symmetric Algebras

Zappa–Szép products of Garside monoids

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