From almost (para)-complex structures to affine structures on Lie groups

Abstract

Let G = H ⋉ K denote a semidirect product Lie group with Lie algebra g = h⊕k, where k is an ideal and h is a subalgebra of the same dimension as k. There exist some natural split isomorphisms S with S 2 = ± Id on g: given any linear isomorphism j : h → k, we get the almost complex structure J(x, v) = (−j −1 v, jx) and the almost paracomplex structure E(x, v) = (j −1 v, jx). In this work we show that the integrability of the structures J and E above is equivalent to the existence of a left-invariant torsionfree connection ∇ on G such that ∇J = 0 = ∇E and also to the existence of an affine structure on H. Applications include complex, paracomplex and symplectic geometries.2010 Mathematics Subject Classification. 53C15, 53C55, 53D05, 22E25, 17B56.

Generalised fluxes, Yang-Baxter deformations and the O(d,d) structure of non-abelian T -duality

“…Theorem 3.2. of [92], adjusted to our case, states that these structures are integrable, iff the isomorphism θ is an 1-cocycle of (g, ad(g) g * ), meaning that…”

“…These (para)-complex structures I and J have not been applied yet to the geometric study of ordinary DFT (abelian bialgebra) or integrable deformations (quasi-Frobenius semi-abelian bialgebra case), where they might be useful. Many more details can be found in [92]. 20 The consideration in [92] are more general than the one we need.…”

“…Many more details can be found in [92]. 20 The consideration in [92] are more general than the one we need. They consider a Lie group, which is a semidirect product of two Lie groups of equal dimension d, Q = H K, so the corresponding Lie algebra is q = h ⊕ k, where h is a subalgebra and k is an ideal.…”

“…Following this definition, they have to consider general representations π : h → End(k) describing the action [h, k] ⊂ k. The study of Drinfel'd doubles fixes the choice of representation such that it is compatible with the Ad-invariant O(d, d)-metric. 21 In principle we could define the same structures for a generic Drinfel'd double, but only in the case of the semi-abelian double we can solve the integrability condition in a straightforward way and apply the results of [92].…”

“…Two other (families of) candidates for a splitting structure have been recently discussed in detail in [92] in a slightly different setting. 20 In the framework of Drinfel'd doubles, in which we are interested, they are only applicable for the semi-abelian Drinfel'd double D = T G with d = g ⊕ d (u(1)) d , where we can define two almost (para)-complex structures, 21 given a (vector space) isomorphism θ : g → g * = (u(1)) d…”