3-Lie bialgebras and 3-Lie classical Yang–Baxter equations in low dimensions. Issue 8 (3rd August 2018)

• Record Type: Journal Article
• Authors: Du, Chengyu; Bai, Chengming; Guo, Li
• Is Part Of: Linear & multilinear algebra. Volume 66:Issue 8(2018)
• Page Start: 1633
• Page End: 1658
• Publication Date: 2018-08-03
• Subjects: Bialgebra -- 3-Lie algebra -- 3-Lie bialgebra -- classical Yang–Baxter equation -- Manin triple; 15A75 -- 16T10 -- 16T25 -- 17A30 -- 17B62 -- 81T30; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2017.1366413
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract: In this paper, we give low-dimensional examples of local cocycle 3-Lie bialgebras and double construction 3-Lie bialgebras which were introduced in the study of the classical Yang–Baxter equation and Manin triples for 3-Lie algebras. We give an explicit and practical formula to compute the skew-symmetric solutions of the 3-Lie classical Yang–Baxter equation (CYBE). As an illustration, we obtain all skew-symmetric solutions of the 3-Lie CYBE in complex 3-Lie algebras of dimensions 3 and 4 and then the induced local cocycle 3-Lie bialgebras. On the other hand, we classify the double construction 3-Lie bialgebras for complex 3-Lie algebras in dimensions 3 and 4 and then give the corresponding eight-dimensional pseudo-metric 3-Lie algebras.