Lie monads and dualities

In this paper, we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular, we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing\ud a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behavior of these functors in the case when the base category is braided is also considered

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“…Next result should be compared with [GV,Lemma 6.2]. Note that, in our case, the braiding of the primitive elements has not order two, in general.…”

Section: Braided Adjunctionsmentioning

confidence: 90%

Adjunctions and braided objects

Ardizzoni¹,

Menini

2014

J. Algebra Appl.

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“…Claim 2.14. Given a symmetric algebra (A, µ, u, c), one has that [−] := µ • (Id A⊗A −c) defines a braided Lie algebra structure on A (see [13,Construction 2.16]).…”

Section: The Category H(g C)mentioning

confidence: 99%

BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras

Graziani¹,

Makhlouf²,

Menini³

et al. 2015

SIGMA

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Abstract. A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α, β : A → A such that α(a)(bc) = (ab)β(c), for all a, b, c ∈ A. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).

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Milnor–Moore categories and monadic decomposition

Ardizzoni

Menini

2016

Journal of Algebra

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Abstract. In this paper Hom-Lie algebras, Lie color algebras, Lie superalgebras and other type of generalized Lie algebras are recovered by means of an iterated construction, known as monadic decomposition of functors, which is based on Eilenberg-Moore categories. To this aim we introduce the notion of Milnor-Moore category as a monoidal category for which a MilnorMoore type Theorem holds. We also show how to lift the property of being a Milnor-Moore category whenever a suitable monoidal functor is given and we apply this technique to provide examples.

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Lie monads and dualities

Cited by 3 publications

References 17 publications

Adjunctions and braided objects

Adjunctions and braided objects

BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras

Milnor–Moore categories and monadic decomposition

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