Exceptional Lie Algebras, SU(3) and Jordan Pairs

A simple unifying view of the exceptional Lie algebras is presented. The underlying Jordan pair content and role are exhibited. Each algebra contains three Jordan pairs sharing the same Lie algebra of automorphisms and the same external su(3) symmetry. Eventual physical applications and implications of the theory are outlined.Comment: 24 pages, 6 figure

Neuropathic pain: the scope of the problem

No abstract availabl

Current and innovative pharmacological options to treat typical and atypical trigeminal neuralgia

Trigeminal neuralgia is a representative neuropathic facial pain condition, characterised by unilateral paroxysmal pain in the distribution territory of one or more divisions of the trigeminal nerve, triggered by innocuous stimuli. A subgroup of patients with trigeminal neuralgia [TN (previously defined as atypical TN)] also suffer from concomitant continuous pain, i.e. a background pain between the paroxysmal attacks. The aim of this review is to provide current, evidence-based, knowledge about the pharmacological treatment of typical and atypical TN, with a specific focus on drugs in development. We searched for relevant papers within PubMed, EMBASE, the Cochrane Database of Systematic Reviews and the Clinical Trials database (ClinicalTrials.gov), taking into account publications up to February 2018. Two authors independently selected studies for inclusions, data extraction, and bias assessment. Carbamazepine and oxcarbazepine are the first-choice drugs for paroxysmal pain. When sodium channel blockers cannot reach full dosage because of side effects, an add-on treatment with lamotrigine or baclofen should be considered. In patients with atypical TN, both gabapentin and antidepressants are expected to be efficacious and should be tried as an add-on to oxcarbazepine or carbamazepine. Although carbamazepine and oxcarbazepine are effective in virtually the totality of patients, they are responsible for side effects causing withdrawal from treatment in an important percentage of cases. A new, better tolerated, Nav1.7 selective state-dependent, sodium channel blocker (vixotrigine) is under development. Future trials testing the effect of combination therapy in patients with TN are needed, especially in patients with concomitant continuous pain and in TN secondary to multiple sclerosis

Representation Theory Approach to the Polynomial Solutions of q - Difference Equations : U_q(sl(3)) and Beyond,

A new approach to the theory of polynomial solutions of q - difference equations is proposed. The approach is based on the representation theory of simple Lie algebras and their q - deformations and is presented here for U_q(sl(n)). First a q - difference realization of U_q(sl(n)) in terms of n(n-1)/2 commuting variables and depending on n-1 complex representation parameters r_i, is constructed. From this realization lowest weight modules (LWM) are obtained which are studied in detail for the case n=3 (the well known n=2 case is also recovered). All reducible LWM are found and the polynomial bases of their invariant irreducible subrepresentations are explicitly given. This also gives a classification of the quasi-exactly solvable operators in the present setting. The invariant subspaces are obtained as solutions of certain invariant q - difference equations, i.e., these are kernels of invariant q - difference operators, which are also explicitly given. Such operators were not used until now in the theory of polynomial solutions. Finally the states in all subrepresentations are depicted graphically via the so called Newton diagrams.Comment: uuencoded Z-compressed .tar file containing two ps files