Berceanu, Barbu R.; Nichita, Florin F.; Popescu, CalinAlgebra structures arising from Yang-Baxter systems.Commun. Algebra 41, No. 12, 4442–4452 (2013).

Mathematics Subject Classification 2010: *16T20, 16T25, 17B37, 81R50

Keywords: Yang-Baxter operator; braid relation; quantum Yang-Baxter equation;algebra; coalgebra; Yang-Baxter system; Yang-Baxter commutator; entwining struc-ture; braiding; braid system

Reviewer: Jorg Feldvoss (8086)

A Yang-Baxter operator is an invertible linear endomorphism of the second ten-sor power of a vector space that satisfies the braid relation. It is well known thatYang-Baxter operators are closely related to solutions of the quantum Yang-Baxterequation, and vice versa. Dascalescu and the second author of the paper under reviewshowed that any algebra or coalgebra structure on a vector space gives rise to a two-parameter family of Yang-Baxter operators [Commun. Algebra 27, No. 12, 5833-5845(1999; Zbl. 0952.16034)].

A triple of linear endomorphisms on tensor products of two vector spaces withcertain compatibility conditions expressed in terms of certain Yang-Baxter commu-tators is called a Yang-Baxter system. An algebra A is said to be entwined with acoalgebra C if there exists a linear transformation ψ : C ⊗ A → A ⊗ C satisfyingcertain compatibility conditions with the multiplication and unit of A as well as withthe comultiplication and counit of C. Brzezinski and the second author of the paperunder review proved that one can define a Yang-Baxter system on A and C if, andonly if, A is entwined with C [Commun. Algebra 33, No. 4, 1083-1093 (2005; Zbl.1085.16028)]. The purpose of this paper is to show that braidings and entwinings ofvarious algebraic structures can be constructed from a braid system, or equivalently,from a Yang-Baxter system.