Linearization of some Poisson-Lie tensor (Q1377239)

Linearization problems for Poisson-Lie (P.-L.) tensors (Weinstein, 1983; Conn, 1984; Dufour, 1990; Molinier, 1993) are considered. The existing bijection between P.-L. structures on a connected, simply-connected Lie group \(G\) and the bialgebra structures \((g,g^*)\) on the Lie algebra \(g\) and its dual mapping \(g^*\) of \(G\) (Drinfeld, 1983) is used as starting point. It is shown that any P.-L. tensor \(P\) on \(G\) such that the linear approximation \(g^*\) is reductive is analytically reducible. Some results on P.-L. structures whose linear approximations are direct sums with a semisimple factor are also presented. Finally, some examples of linearizable and nonlinearizable exact P.-L. structures on some nilpotent groups are given.