On multigraded generalizations of Kirillov-Reshetikhin modules

We study the category of Z^l-graded modules with finite-dimensional graded pieces for certain Z+^l-graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov-Reshetikhin modules and give a recursive formula for computing their graded characters

On minimal affinizations of representations of quantum groups

In this paper we study minimal affinizations of representations of quantum groups (generalizations of Kirillov-Reshetikhin modules of quantum affine algebras introduced by Chari). We prove that all minimal affinizations in types A, B, G are special in the sense of monomials. Although this property is not satisfied in general, we also prove an analog property for a large class of minimal affinization in types C, D, F. As an application, the Frenkel-Mukhin algorithm works for these modules. For minimal affinizations of type A, B we prove the thin property (the l-weight spaces are of dimension 1) and a conjecture of Nakai-Nakanishi (already known for type A). The proof of the special property is extended uniformly for more general quantum affinizations of quantum Kac-Moody algebras.Comment: 38 pages; references and additional results added. Accepted for publication in Communications in Mathematical Physic

The quantum bialgebra associated with the eight-vertex R-matrix

The quantum bialgebra related to the Baxter's eight-vertex R-matrix is found as a quantum deformation of the Lie algebra of sl(2)-valued automorphic functions on a complex torus.Comment: 4 page

Langlands duality for finite-dimensional representations of quantum affine algebras

We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of q-alg/9708006 and 0809.4453. We prove this duality for the Kirillov-Reshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct "interpolating (q,t)-characters" depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.Comment: 40 pages; several results and comments added. Accepted for publication in Letters in Mathematical Physic

Intraventricular haemorrhage and posthaemorrhagic ventricular dilatation: moving beyond CSF diversion

Advances in medical care have led to more premature babies surviving the neonatal period. In these babies, germinal matrix haemorrhage (GMH), intraventricular haemorrhage (IVH) and posthaemorrhagic ventricular dilatation (PHVD) are the most important determinants of long-term cognitive and developmental outcomes. In this review, we discuss current neurosurgical management of IVH and PHVD, including the importance of early diagnosis of PHVD, thresholds for intervention, options for early management through the use of temporising measures and subsequent definitive CSF diversion. We also discuss treatment options for the evolving paradigm to manage intraventricular blood and its breakdown products. We review the evidence for techniques such as drainage, irrigation, fibrinolytic therapy (DRIFT) and neuroendoscopic lavage in the context of optimising cognitive, neurodevelopmental and quality of life outcomes in these premature infants

Narrative review of epilepsy: getting the most out of your neuroimaging

Neuroimaging represents an important step in the evaluation of pediatric epilepsy. The crucial role of brain imaging in the diagnosis, follow-up and presurgical assessment of patients with epilepsy is noted and has to be familiar to all neuroradiologists and trainees approaching pediatric brain imaging. Morphological qualitative imaging shows the majority of cerebral lesions/alterations underlying focal epilepsy and can highlight some features which are useful in the differential diagnosis of the different types of epilepsy. Recent advances in MRI acquisitions including diffusion-weighted imaging (DWI), post-acquisition image processing techniques, and quantification of imaging data are increasing the accuracy of lesion detection during the last decades. Functional MRI (fMRI) can be really useful and helps to identify cortical eloquent areas that are essential for language, motor function, and memory, and diffusion tensor imaging (DTI) can reveal white matter tracts that are vital for these functions, thus reducing the risk of epilepsy surgery causing new morbidities. Also positron emission tomography (PET), single photon emission computed tomography (SPECT), simultaneous electroencephalogram (EEG) and fMRI, and electrical and magnetic source imaging can be used to assess the exact localization of epileptic foci and help in the design of intracranial EEG recording strategies. The main role of these “hybrid” techniques is to obtain quantitative and qualitative informations, a necessary step to evaluate and demonstrate the complex relationship between abnormal structural and functional data and to manage a “patient-tailored” surgical approach in epileptic patients

Universal Baxterization for Z\mathbb{Z}-graded Hopf algebras

We present a method for Baxterizing solutions of the constant Yang-Baxter equation associated with $\mathbb{Z}$-graded Hopf algebras. To demonstrate the approach, we provide examples for the Taft algebras and the quantum group $U_q[sl(2)]$.Comment: 8 page

A Generalized Q-operator for U_q(\hat(sl_2)) Vertex Models

In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of $U_q(\hat{sl}_2)$ over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator.Comment: 22 pages, Latex2e. This replacement is a revised version that includes a simple explicit expression for the Q matrix for the 6-vertex mode

On irreducibility of tensor products of evaluation modules for the quantum affine algebra

Every irreducible finite-dimensional representation of the quantized enveloping algebra U_q(gl_n) can be extended to the corresponding quantum affine algebra via the evaluation homomorphism. We give in explicit form the necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules.Comment: 22 pages. Some references are adde

A Jordanian quantum two-photon/Schrodinger algebra

A non-standard quantum deformation of the two-photon algebra $h_6$ is constructed, and its quantum universal R-matrix is given. Representations of this new quantum algebra are studied on the Fock space and translated into Fock-Bargmann realizations that provide a direct formalism for the definition of deformed states of light. Finally, the isomorphism between $h_6$ and the (1+1) Schr\"odinger algebra is used to introduce a new (non-standard) Hopf algebra deformation of this latter symmetry algebra.Comment: 12 pages, LaTeX, misprints correcte