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Generalized Bruhat Cells and Completeness of Hamiltonian Flows of Kogan-Zelevinsky Integrable Systems
Let 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 are complete in the sense that all
the integral curves of their Hamiltonian vector fields are defined on
. It follows that all the Kogan-Zelevinsky integrable systems on
have complete Hamiltonian flows, generalizing the result of Gekhtman and
Yakimov for the case of . 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 , 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
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