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- Title
A New Approach to Jordan D-Bialgebras via Jordan–Manin Triples.
- Authors
Hou, Dongping
- Abstract
Jordan D-bialgebras were introduced by Zhelyabin. In this paper, we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan–Manin triples called double constructions of pseudo-euclidean Jordan algebras. We also show that a Jordan D-bialgebra leads to the Jordan Yang–Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang–Baxter equation corresponds to an antisymmetric bilinear form, which we call a Jordan symplectic form on Jordan algebras. Furthermore, there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form.
- Subjects
JORDAN algebras; YANG-Baxter equation; BILINEAR forms; REPRESENTATIONS of algebras; ALGEBRA
- Publication
Algebra Colloquium, 2023, Vol 30, Issue 1, p73
- ISSN
1005-3867
- Publication type
Article
- DOI
10.1142/S100538672300007X