"Doubled" generalized Landau-Lifshiz hierarchies and special quasigraded Lie algebras

Using special quasigraded Lie algebras we obtain new hierarchies of integrable nonlinear vector equations admitting zero-curvature representations. Among them the most interesting is extension of the generalized Landau-Lifshitz hierarchy which we call "doubled" generalized Landau-Lifshiz hierarchy. This hierarchy can be also interpreted as an anisotropic vector generalization of "modified" Sine-Gordon hierarchy or as a very special vector generalization of so(3) anisotropic chiral field hierarchy.Comment: 16 pages, no figures, submitted to Journal of Physics

Compatible Poisson brackets, quadratic Poisson algebras and classical r-matrices

We show that for a general quadratic Poisson bracket it is possible to define a lot of associated linear Poisson brackets: linearizations of the initial bracket in the neighborhood of special points. We prove that the constructed linear Poisson brackets are always compatible with the initial quadratic Poisson bracket. We apply the obtained results to the cases of the standard quadratic r-matrix bracket and to classical “twisted reflection algebra” brackets. In the first case we obtain that there exists only one non-equivalent linearization: the standard linear r-matrix bracket and recover well-known result that the standard quadratic and linear r-matrix brackets are compatible.We show that there are a lot of non-equivalent linearizations of the classical twisted Reflection Equation Algebra bracket and all of them are compatible with the initial quadratic bracket

Compatible Lie brackets related to elliptic curve

For the direct sum of several copies of sl_n, a family of Lie brackets compatible with the initial one is constructed. The structure constants of these brackets are expressed in terms of theta-functions associated with an elliptic curve. The structure of Casimir elements for these brackets is investigated. A generalization of this construction to the case of vector-valued theta-functions is presented. The brackets define a multi-hamiltonian structure for the elliptic sl_n-Gaudin model. A different procedure for constructing compatible Lie brackets based on the argument shift method for quadratic Poisson brackets is discussed.Comment: 18 pages, Late

From AKNS to derivative NLS hierarchies via deformations of associative products

Using deformations of associative products, derivative nonlinear Schrodinger (DNLS) hierarchies are recovered as AKNS-type hierarchies. Since the latter can also be formulated as Gelfand-Dickey-type Lax hierarchies, a recently developed method to obtain 'functional representations' can be applied. We actually consider hierarchies with dependent variables in any (possibly noncommutative) associative algebra, e.g., an algebra of matrices of functions. This also covers the case of hierarchies of coupled DNLS equations.Comment: 22 pages, 2nd version: title changed and material organized in a different way, 3rd version: introduction and first part of section 2 rewritten, taking account of previously overlooked references. To appear in J. Physics A: Math. Ge

Quality of life evaluation in acute leukemia patients receiving induction chemotherapy

Background: Over the past decades, special attention has been paid to study of quality of life (QoL) indicators in hematological patients receiving chemotherapy (CT). Nowadays QoL is conceptually viewed as an important complement to traditional objective evaluation measures. Aim. To assess QoL in patients with acute leukemia (AL) depending on the presence of concomitant ischemic heart disease (1HD) during the induction CT. Methods: Our study involved 83 patients with newly diagnosed AL, of which 19 were lymphoblastic, 64-myeloid leukemia, aged 16-72,43 (51.8%) men, 40 (48.2%) women, according to ECOG l-II. Patients received standard induction CT. According to concomitant IFID patients were divided into groups: I (n — 47) - AL without cardio­ logical diseases; II (n = 36) - AL with concomitant IHD. Patients were evaluated using SF-36 questionnaire to calculate physical and mental health components before treatment and after 2 induction courses of CT reaching remission

Cyclotomic Gaudin models: construction and Bethe ansatz

This is a pre-copyedited author produced PDF of an article accepted for publication in Communications in Mathematical Physics, Benoit, V and Young, C, 'Cyclotomic Gaudin models: construction and Bethe ansatz', Commun. Math. Phys. (2016) 343:971, first published on line March 24, 2016. The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-016-2601-3 © Springer-Verlag Berlin Heidelberg 2016To any simple Lie algebra $\mathfrak g$ and automorphism $\sigma:\mathfrak g\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case $\sigma = \text{id}$. We go on to construct joint eigenvectors and their eigenvalues for this hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain consisting of a tensor product of Verma modules. To do so we generalize an approach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involving vertex algebras and the Wakimoto construction. As part of this construction, we make use of a theorem concerning cyclotomic coinvariants, which we prove in a companion paper. As a byproduct, we obtain a cyclotomic generalization of the Schechtman-Varchenko formula for the weight function.Peer reviewe

Classical R-matrix theory for bi-Hamiltonian field systems

The R-matrix formalism for the construction of integrable systems with infinitely many degrees of freedom is reviewed. Its application to Poisson, noncommutative and loop algebras as well as central extension procedure are presented. The theory is developed for (1+1)-dimensional case where the space variable belongs either to R or to various discrete sets. Then, the extension onto (2+1)-dimensional case is made, when the second space variable belongs to R. The formalism presented contains many proofs and important details to make it self-contained and complete. The general theory is applied to several infinite dimensional Lie algebras in order to construct both dispersionless and dispersive (soliton) integrable field systems.Comment: review article, 39 page

Cyclotomic Gaudin models, Miura opers and flag varieties

Let g be a semisimple Lie algebra over C. Let ν∈Autg be a diagram automorphism whose order divides T∈Z≥1. We define cyclotomic g-opers over the Riemann sphere P1 as gauge equivalence classes of g-valued connections of a certain form, equivariant under actions of the cyclic group Z/TZ on g and P1. It reduces to the usual notion of g-opers when T=1. We also extend the notion of Miura g-opers to the cyclotomic setting. To any cyclotomic Miura g-oper ∇ we associate a corresponding cyclotomic g-oper. Let ∇ have residue at the origin given by a ν-invariant rational dominant coweight λˇ0 and be monodromy-free on a cover of P1. We prove that the subset of all cyclotomic Miura g-opers associated with the same cyclotomic g-oper as ∇ is isomorphic to the ϑ-invariant subset of the full flag variety of the adjoint group G of g, where the automorphism ϑ depends on ν, T and λˇ0. The big cell of the latter is isomorphic to Nϑ, the ϑ-invariant subgroup of the unipotent subgroup N⊂G, which we identify with those cyclotomic Miura g-opers whose residue at the origin is the same as that of ∇. In particular, the cyclotomic generation procedure recently introduced in [arXiv:1505.07582] is interpreted as taking ∇ to other cyclotomic Miura g-opers corresponding to elements of Nϑ associated with simple root generators. We motivate the introduction of cyclotomic g-opers by formulating two conjectures which relate them to the cyclotomic Gaudin model of [arXiv:1409.6937]