Crystals for Demazure Modules of Classical Affine Lie Algebras

We study, in the path realization, crystals for Demazure modules of affine Lie algebras of types $A^{(1)}_n,B^{(1)}_n,C^{(1)}_n,D^{(1)}_n, A^{(2)}_{2n-1},A^{(2)}_{2n}, and D^{(2)}_{n+1}$. We find a special sequence of affine Weyl group elements for the selected perfect crystal, and show if the highest weight is l\La_0, the Demazure crystal has a remarkably simple structure.Comment: Latex, 28 page

Factorization, reduction and embedding in integrable cellular automata

Factorized dynamics in soliton cellular automata with quantum group symmetry is identified with a motion of particles and anti-particles exhibiting pair creation and annihilation. An embedding scheme is presented showing that the D^{(1)}_n-automaton contains, as certain subsectors, the box-ball systems and all the other automata associated with the crystal bases of non-exceptional affine Lie algebras. The results extend the earlier ones to higher representations by a certain reduction and to a wider class of boundary conditions.Comment: LaTeX2e, 20 page

Expression and function of the Delta-1/Notch-2/Hes-1 pathway during experimental acute kidney injury

The Notch signaling pathway consists of several receptors and their ligands Delta and Jagged and is important for embryogenesis, cellular differentiation and proliferation. Activation of Notch receptors causes their cleavage yielding cytoplastic domains that translocate into the nucleus to induce target proteins such as the basic-loop-helix proteins Hes and Hey. Here we sought to clarify the significance of the Notch signaling pathway in acute kidney injury using a rat ischemia-reperfusion injury model and cultured NRK-52E cells. Analysis of the whole kidney after injury showed increased expression of Delta-1 and Hes-1 mRNA and protein along with processed Notch-2. Confocal microscopy, using specific antibodies, showed that Delta-1, cleaved Notch-2 and Hes-1 colocalized in the same segments of the injured renal proximal tubules. Recombinant Delta-1 significantly stimulated NRK-52E cell proliferation. Our study suggests that the Delta-1/Notch-2/Hes-1 signaling pathway may regulate the regeneration and proliferation of renal tubules during acute kidney injury

Generalised Perk--Schultz models: solutions of the Yang-Baxter equation associated with quantised orthosymplectic superalgebras

The Perk--Schultz model may be expressed in terms of the solution of the Yang--Baxter equation associated with the fundamental representation of the untwisted affine extension of the general linear quantum superalgebra $U_q[sl(m|n)]$, with a multiparametric co-product action as given by Reshetikhin. Here we present analogous explicit expressions for solutions of the Yang-Baxter equation associated with the fundamental representations of the twisted and untwisted affine extensions of the orthosymplectic quantum superalgebras $U_q[osp(m|n)]$. In this manner we obtain generalisations of the Perk--Schultz model.Comment: 10 pages, 2 figure

Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry

The box-ball system is an integrable cellular automaton on one dimensional lattice. It arises from either quantum or classical integrable systems by the procedures called crystallization and ultradiscretization, respectively. The double origin of the integrability has endowed the box-ball system with a variety of aspects related to Yang-Baxter integrable models in statistical mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz, geometric crystals, classical theory of solitons, tau functions, inverse scattering method, action-angle variables and invariant tori in completely integrable systems, spectral curves, tropical geometry and so forth. In this review article, we demonstrate these integrable structures of the box-ball system and its generalizations based on the developments in the last two decades.Comment: 73 page

Crystal Interpretation of Kerov-Kirillov-Reshetikhin Bijection II. Proof for sl_n Case

In proving the Fermionic formulae, combinatorial bijection called the Kerov--Kirillov--Reshetikhin (KKR) bijection plays the central role. It is a bijection between the set of highest paths and the set of rigged configurations. In this paper, we give a proof of crystal theoretic reformulation of the KKR bijection. It is the main claim of Part I (math.QA/0601630) written by A. Kuniba, M. Okado, T. Takagi, Y. Yamada, and the author. The proof is given by introducing a structure of affine combinatorial $R$ matrices on rigged configurations.Comment: 45 pages, version for publication. Introduction revised, more explanations added to the main tex