Hom-Lie color algebra structures

This paper introduces the notion of Hom-Lie color algebra, which is a natural general- ization of Hom-Lie (super)algebras. Hom-Lie color algebras include also as special cases Lie (super) algebras and Lie color algebras. We study the homomorphism relation of Hom-Lie color algebras, and construct new algebras of such kind by a \sigma-twist. Hom-Lie color admissible algebras are also defined and investigated. They are finally classified via G-Hom-associative color algebras, where G is a subgroup of the symmetric group S_3.Comment: 16 page

On the structure of maximal solvable extensions and of Levi extensions of nilpotent algebras

We establish an improved upper estimate on dimension of any solvable algebra s with its nilradical isomorphic to a given nilpotent Lie algebra n. Next we consider Levi decomposable algebras with a given nilradical n and investigate restrictions on possible Levi factors originating from the structure of characteristic ideals of n. We present a new perspective on Turkowski's classification of Levi decomposable algebras up to dimension 9.Comment: 21 pages; major revision - one section added, another erased; author's version of the published pape

Contractions of Low-Dimensional Lie Algebras

Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of requisite invariant and semi-invariant quantities are calculated for wide classes of Lie algebras including all low-dimensional Lie algebras. An algorithm that allows one to handle one-parametric contractions is presented and applied to low-dimensional Lie algebras. As a result, all one-parametric continuous contractions for the both complex and real Lie algebras of dimensions not greater than four are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases. Levels and co-levels of low-dimensional Lie algebras are discussed in detail. Properties of multi-parametric and repeated contractions are also investigated.Comment: 47 pages, 4 figures, revised versio

On Deformations of n-Lie algebras

The aim of this paper is to review the deformation theory of $n$-Lie algebras. We summarize the 1-parameter formal deformation theory and provide a generalized approach using any unital commutative associative algebra as a deformation base. Moreover, we discuss degenerations and quantization of $n$-Lie algebras.Comment: Proceeding of the conference Dakar's Workshop in honor of Pr Amin Kaidi. arXiv admin note: text overlap with arXiv:hep-th/9602016 by other author

On post-Lie algebras, Lie--Butcher series and moving frames

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. They have been studied extensively in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan's method of moving frames. Lie--Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie--Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, are explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie--Butcher series are related to invariants of curves described by moving frames.Comment: added discussion of post-Lie algebroid

All solvable extensions of a class of nilpotent Lie algebras of dimension n and degree of nilpotency n-1

We construct all solvable Lie algebras with a specific n-dimensional nilradical n_(n,2) (of degree of nilpotency (n-1) and with an (n-2)-dimensional maximal Abelian ideal). We find that for given n such a solvable algebra is unique up to isomorphisms. Using the method of moving frames we construct a basis for the Casimir invariants of the nilradical n_(n,2). We also construct a basis for the generalized Casimir invariants of its solvable extension s_(n+1) consisting entirely of rational functions of the chosen invariants of the nilradical.Comment: 19 pages; added references, changes mainly in introduction and conclusions, typos corrected; submitted to J. Phys. A, version to be publishe

On Some Geometric Structures Associated to a k-Symplectic Manifold

A canonical connection is attached to any k-symplectic manifold. We study the properties of this connection and its geometric applications to k-symplectic manifolds. In particular we prove that, under some natural assumption, any ksymplectic manifold admits an Ehresmann connection, discussing some corollaries of this result, and we find vanishing theorems for characteristic classes on a k-symplectic manifold.Comment: To appear on J. Phys. A: Math. Theo

Some Results on Cubic and Higher Order Extensions of the Poincar\'e Algebra

In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincar\'e algebra. We first give some general properties of Lie superalgebras with some emphasis on the supersymmetric extension of the Poincar\'e algebra or Supersymmetry. Some general features on the so-called Wess-Zumino model (the simplest field theory invariant under Supersymmetry) are then given. We further introduce an additional algebraic structure called Lie algebras of order F, which naturally comprise the concepts of ordinary Lie algebras and superalgebras. This structure enables us to define various non-trivial extensions of the Poincar\'e algebra. These extensions are studied more precisely in two different contexts. The first algebra we are considering is shown to be an (infinite dimensional) higher order extension of the Poincar\'e algebra in $(1+2)-$dimensions and turns out to induce a symmetry which connects relativistic anyons. The second extension we are studying is related to a specific finite dimensional Lie algebra of order three, which is a cubic extension of the Poincar\'e algebra in $D-$space-time dimensions. Invariant Lagrangians are constructed.Comment: Mini course given at the Workshop higher symmetries in physics, Madrid, Spain, November 6-8, 200

Co-Deletion of Chromosome 1p/19q and IDH1/2 Mutation in Glioma Subsets of Brain Tumors in Chinese Patients

OBJECTIVE: To characterize co-deletion of chromosome 1p/19q and IDH1/2 mutation in Chinese brain tumor patients and to assess their associations with clinical features. METHODS: In a series of 528 patients with gliomas, pathological and radiological materials were reviewed. Pathological constituents of tumor subsets, incidences of 1p/19q co-deletion and IDH1/2 mutation in gliomas by regions and sides in the brain were analyzed. RESULTS: Overall, 1p and 19q was detected in 339 patients by FISH method while the sequence of IDH1/2 was determined in 280 patients. Gliomas of frontal, temporal and insular origin had significantly different pathological constituents of tumor subsets (P<0.001). Gliomas of frontal origin had significantly higher incidence of 1p/19q co-deletion (50.4%) and IDH1/2 mutation (73.5%) than those of non-frontal origin (27.0% and 48.5%, respectively) (P<0.001), while gliomas of temporal origin had significantly lower incidence of 1p/19q co-deletion (23.9%) and IDH1/2 mutation (41.7%) than those of non-temporal origin (39.9% and 63.2%, respectively) (P = 0.013 and P = 0.003, respectively). Subgroup analysis confirmed these findings in oligoastrocytic and oligodendroglial tumors, respectively. Although the difference of 1p/19q co-deletion was not statistically significant in temporal oligodendroglial tumors, the trend was marginally significant (P = 0.082). However, gliomas from different sides of the brain did not show significant different pathological constituents, incidences of 1p/19q co-deletion or IDH1/2 mutation. CONCLUSION: Preferential distribution of pathological subsets, 1p/19q co-deletion and IDH1/2 mutation were confirmed in some brain regions in Chinese glioma patients, implying their distinctive tumor genesis and predictive value for prognosis

Normosmic Congenital Hypogonadotropic Hypogonadism Due to TAC3/TACR3 Mutations: Characterization of Neuroendocrine Phenotypes and Novel Mutations

CONTEXT: TAC3/TACR3 mutations have been reported in normosmic congenital hypogonadotropic hypogonadism (nCHH) (OMIM #146110). In the absence of animal models, studies of human neuroendocrine phenotypes associated with neurokinin B and NK3R receptor dysfunction can help to decipher the pathophysiology of this signaling pathway. OBJECTIVE: To evaluate the prevalence of TAC3/TACR3 mutations, characterize novel TACR3 mutations and to analyze neuroendocrine profiles in nCHH caused by deleterious TAC3/TACR3 biallelic mutations. RESULTS: From a cohort of 352 CHH, we selected 173 nCHH patients and identified nine patients carrying TAC3 or TACR3 variants (5.2%). We describe here 7 of these TACR3 variants (1 frameshift and 2 nonsense deleterious mutations and 4 missense variants) found in 5 subjects. Modeling and functional studies of the latter demonstrated the deleterious consequence of one missense mutation (Tyr267Asn) probably caused by the misfolding of the mutated NK3R protein. We found a statistically significant (p<0.0001) higher mean FSH/LH ratio in 11 nCHH patients with TAC3/TACR3 biallelic mutations than in 47 nCHH patients with either biallelic mutations in KISS1R, GNRHR, or with no identified mutations and than in 50 Kallmann patients with mutations in KAL1, FGFR1 or PROK2/PROKR2. Three patients with TAC3/TACR3 biallelic mutations had an apulsatile LH profile but low-frequency alpha-subunit pulses. Pulsatile GnRH administration increased alpha-subunit pulsatile frequency and reduced the FSH/LH ratio. CONCLUSION: The gonadotropin axis dysfunction associated with nCHH due to TAC3/TACR3 mutations is related to a low GnRH pulsatile frequency leading to a low frequency of alpha-subunit pulses and to an elevated FSH/LH ratio. This ratio might be useful for pre-screening nCHH patients for TAC3/TACR3 mutations