Abstract
The paper deals with linearization problem of Poisson-Lie structures on the \((1+1)\) Poincaré and \(2D\) Euclidean groups. We construct the explicit form of linearizing coordinates of all these Poisson-Lie structures. For this, we calculate all Poisson-Lie structures on these two groups mentioned above, through the correspondence with Lie Bialgebra structures on their Lie algebras which we first determine.
Highlights
Poisson-Lie structure on a Lie group G is a Poisson structure {. , .} on C∞(G), such that the multiplication μ : G × G −→ G is a Poisson map, namely{f ◦ μ, g ◦ μ}C∞(G×G)(x, y) = {f, g}C∞(G)(μ(x, y)), x, y ∈ G, f, g ∈ C∞(G).By Drinfel’d [5, 6], this is equivalent to giving an antisymmetric contravariant 2−tensor π on G such that the Schouten–Nijenhuis bracket [π, π] = 0 and satisfies the multiplicativity relation π(xy) = lx∗π(y) + ry∗π(x), ∀x, y ∈ G, where lx∗ and ry∗ are the left and right translations in G by x and y, respectively
The aim of this paper is the explicit construction of smooth linearizing coordinates for the Poisson-Lie structures on the 2D Euclidean group generated by the Lie algebra s3(0) and the (1 + 1) Poincare group generated by the Lie algebra τ3(−1)
In this work we present a Lie bialgebra structures on the Lie algebras s3(0) and τ3(−1) and we adopt the classification given in [9]
Summary
Poisson-Lie structure on a Lie group G is a Poisson structure {. , .} on C∞(G), such that the multiplication μ : G × G −→ G is a Poisson map, namely. The relation above shows that the Poisson-Lie structure π must vanishing at the identity e ∈ G, so that its derivative deπ : G → 2 G at e is well defined, where G is the Lie algebra of G This linear homomorphism turns out to be a 1-cocycle with respect to the adjoint action, and the dual homomorphism 2 G∗ → G∗ satisfies the Jacobi identity; i.e., the dual G∗ of G becomes a Lie algebra. The aim of this paper is the explicit construction of smooth linearizing coordinates for the Poisson-Lie structures on the 2D Euclidean group generated by the Lie algebra s3(0) and the (1 + 1) Poincare group generated by the Lie algebra τ3(−1).
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