Title: 3-Lie bialgebras and 3-Lie classical Yang-Baxter equations in low dimensions,
Abstract: In this paper, we give some 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 dimension 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 8-dimensional pseudo-metric 3-Lie algebras.