Generalized Lie bialgebras and Jacobi structures on Lie groups

Abstract

We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras. Lie bialgebras are examples of generalized Lie bialgebras. Moreover, we prove that the last ones can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structures. We also propose a method to obtain generalized Lie bialgebras. It is a generalization of the YangBaxter equation method. Finally, we describe the structure of a compact generalized Lie bialgebra.Mathematics Subject Classification (2000): 17B62, 22Exx, 53D05, 53D10, 53D17.

Jacobi–Lie symmetry and Jacobi–Lie T-dual sigma models on group manifolds

“…where Λ(g) is the Jacobi structure on the Lie group G [19,22]. By substituting (45) in (44) and using the following relation (from (39)) [15] ∂ µ a(g) i j = L l µ a(g) i k f kl j , (46) one can see that Λ ij (g) must be satisfied in the following condition…”

Jacobi groupoids and generalized Lie bialgebroids

“…Now, using Theorem 2.6 in [30] and since G is α-connected and the 2-vector L ∂ ∂tΛ +Λ is affine, we deduce thatΛ is homogeneous if and only if: Let ((g, φ 0 )(g * , X 0 )) be a generalized Lie bialgebra, that is, a generalized Lie bialgebroid over a single point, and G be a connected simply connected Lie group with Lie algebra g. Then, using (5.9), Proposition 4.4 and Theorem 5.9 we deduce the following facts: a) there exists a unique multiplicative function σ : G → R and a unique σ-multiplicative 2-vector Λ on G such that (δσ)(e) = φ 0 and the intrinsic derivative of Λ at e is −d * X 0 , d * X 0 being the X 0 -differential of the Lie algebra g * ; b) # Λ (δσ) = − → X 0 − e −σ ← − X 0 and c) the pair (Λ, E) is a Jacobi structure on G, where E = − − → X 0 . These results were proved in [17] (see Theorem 3.10 in [17]). …”

“…On the other hand, if G is a Lie group with identity element e and P is a σ-affine multivector field on G such that P (e) = 0, then P is a σ-multiplicative multivector field in the sense of [17].…”

“…If G is a connected Lie group with identity element e, σ : G → R is a multiplicative function and Λ is a σ-multiplicative 2-vector such that the intrinsic derivative of Λ at e is zero then Λ identically vanishes (see [17]). …”

“…In particular, Jacobi groupoids (G ⇒ M, Λ, E, σ) where M is a single point are just the Lie groups studied in [17], whose infinitesimal invariants are generalized Lie bialgebras.…”