Classification of real three-dimensional Poisson–Lie groups

All real three-dimensional Poisson–Lie (PL) groups are explicitly constructed and fully classified under group automorphisms by making use of their one-to-one correspondence with the complete classification of real three-dimensional Lie bialgebras given in Gomez (2000 J. Math. Phys. 41 4939). Many of these 3D PL groups are non-coboundary structures, whose Poisson brackets are given here for the first time. Casimir functions for all three-dimensional PL groups are given, and some features of several PL structures are commented. (paper)