Generalized Bruhat Cells and Completeness of Hamiltonian Flows of Kogan-Zelevinsky Integrable Systems

Let $G$ be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on every double Bruhat cell of $G$ are complete in the sense that all the integral curves of their Hamiltonian vector fields are defined on ${\mathbb{C}}$. It follows that all the Kogan-Zelevinsky integrable systems on $G$ have complete Hamiltonian flows, generalizing the result of Gekhtman and Yakimov for the case of $SL(n, {\mathbb{C}})$. We in fact construct a class of integrable systems with complete Hamiltonian flows associated to {\it generalized Bruhat cells} which are defined using arbitrary sequences of elements in the Weyl group of $G$, and we obtain the results for double Bruhat cells through the so-called open {\it Fomin-Zelevinsky embeddings} of (reduced) double Bruhat cells in generalized Bruhat cells. The Fomin-Zelevinsky embeddings are proved to be Poisson, and they provide global coordinates on double Bruhat cells, called {\it Bott-Samelson coordinates}, in which all the Fomin-Zelevinsky minors become polynomials and the Poisson structure can be computed explicitly.Comment: Title slightly changed; Section 1.3 expanded; some typos correcte

Transition metal oxides for high performance sodium ion battery anodes

Sodium-ion batteries (SIBs) are attracting considerable attention with expectation of replacing lithium-ion batteries (LIBs) in large-scale energy storage systems (ESSs). To explore high performance anode materials for SIBs is highly desired subject to the current anode research mainly limited to carbonaceous materials. In this study, a series of transition metal oxides (TMOs) is successfully demonstrated as anodes for SIBs for the first time. The sodium uptake/extract is confirmed in the way of reversible conversion reaction. The pseudocapacitance-type behavior is also observed in the contribution of sodium capacity. For Fe2O3anode, a reversible capacity of 386 mAh g-1at 100 mA g-1 is achieved over 200 cycles; as high as 233 mAhg-1is sustained even cycling at a large current-density of 5 A g-1