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Keywords = cocommutator

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14 pages, 264 KB  
Article
Rota–Baxter Operators on Cocommutative Weak Hopf Algebras
by Zhongwei Wang, Zhen Guan, Yi Zhang and Liangyun Zhang
Mathematics 2022, 10(1), 95; https://doi.org/10.3390/math10010095 - 28 Dec 2021
Viewed by 1927
Abstract
In this paper, we first introduce the concept of a Rota–Baxter operator on a cocommutative weak Hopf algebra H and give some examples. We then construct Rota–Baxter operators from the normalized integral, antipode, and target map of H. Moreover, we construct a [...] Read more.
In this paper, we first introduce the concept of a Rota–Baxter operator on a cocommutative weak Hopf algebra H and give some examples. We then construct Rota–Baxter operators from the normalized integral, antipode, and target map of H. Moreover, we construct a new multiplication “∗” and an antipode SB from a Rota–Baxter operator B on H such that HB=(H,,η,Δ,ε,SB) becomes a new weak Hopf algebra. Finally, all Rota–Baxter operators on a weak Hopf algebra of a matrix algebra are given. Full article
(This article belongs to the Special Issue Rota-Baxter Algebra and Related Topics)
50 pages, 584 KB  
Article
Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras
by Javier de Lucas and Daniel Wysocki
Symmetry 2021, 13(3), 465; https://doi.org/10.3390/sym13030465 - 12 Mar 2021
Cited by 2 | Viewed by 2314
Abstract
This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. [...] Read more.
This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed. Full article
(This article belongs to the Special Issue Quantum Group Symmetry and Quantum Geometry)
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