The 4th structure

Artikel i vetenskaplig tidskrift, 2006

According to conclusions made by F. Montaner and E. Zelmanov (unpublished), there exist four Lie bialgebra structures on g[u] up to classical twisting. Therefore we investigated four possible cases for the classical double and the corresponding Lie co-bracket on g[u]. Among them, a new case, where the Lie bialgebra structures are in an one-to-one correspondence with the first introduced here quasi-rational solutions of the CYBE. We also considered quasi-rational r-matrices in sl(n), particularly, we calculated all the quasi-rational r-matrices in sl(2)