Some results on L-dendriform algebras

We introduce a notion of L-dendriform algebra due to several different motivations. L-dendriform algebras are regarded as the underlying algebraic structures of pseudo-Hessian structures on Lie groups and the algebraic structures behind the $\mathcal O$-operators of pre-Lie algebras and the related $S$-equation. As a direct consequence, they provide some explicit solutions of $S$-equation in certain pre-Lie algebras constructed from L-dendriform algebras. They also fit into a bigger framework as Lie algebraic analogues of dendriform algebras. Moreover, we introduce a notion of $\mathcal O$-operator of an L-dendriform algebra which gives an algebraic equation regarded as an analogue of the classical Yang-Baxter equation in a Lie algebra.Comment: 15 page

Spin gap behavior and charge ordering in \alpha^{\prime}-NaV_2O_5 probed by light scattering

We present a detailed analysis of light scattering experiments performed on the quarter-filled spin ladder compound $\alpha^\prime$-NaV$_{2}$O$_{5}$ for the temperature range 5 K$\le$T$\le$300 K. This system undergoes a phase transition into a singlet ground state at T=34 K accompanied by the formation of a super structure. For T$\leq$34 K several new modes were detected. Three of these modes are identified as magnetic bound states. Experimental evidence for charge ordering on the V sites is detected as an anomalous shift and splitting of a V-O vibration at 422 cm$^{-1}$ for temperatures above 34 K. The smooth and crossover-like onset of this ordering at T$_{\rm CO}$= 80 K is accompanied by pretransitional fluctuations both in magnetic and phononic Raman scattering. It resembles the effect of stripe order on the super structure intensities in La$_2$NiO$_{4+\delta}$.Comment: 36 pages, 11 figures, accepted for publication in PRB (sept.99