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Abelian gradings in Lie algebras
[article]

2011
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arXiv
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pre-print

Given a Lie algebra L graded by a group G, if L is does not contain orthogonal graded ideals and G is generated by the support of L, then G is an abelian group.

arXiv:1105.2237v1
fatcat:lz7vknhyiraijfllxstc2ruxnu
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3868747.pdf

2019
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Figshare
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Revised Supplementary material

doi:10.6084/m9.figshare.7958582.v1
fatcat:qauctakwazeovfrztw3cm3pufy
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OWLing Clinical Data Repositories With the Ontology Web Language

2014
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JMIR Medical Informatics
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Conflicts of Interest Xavier Pastor and Raimundo

doi:10.2196/medinform.3023
pmid:25599697
pmcid:PMC4288111
fatcat:jfbdxixtmnd3vmyymlvfsd3zaq
*Lozano*-Rubí are employees of Hospital Clinic of Barcelona and University of Barcelona, which receive royalties from a third party by the commercialization ... ©Raimundo*Lozano*-Rubí, Xavier Pastor,*Esther**Lozano*. Originally published in JMIR Medical Informatics (http://medinform.jmir.org), 01.08.2014. ...##
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Decompositions of Matrices into Potent and Square-Zero Matrices
[article]

2021
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arXiv
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pre-print

In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such a matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most two when the Jacobson radical of the ring has zero-square. This somewhat extends results of ours in Lin. Multilin. Algebra (2021) established for matrices considered on arbitrary fields. Our main theorem also improves on recent results due to Abyzov et al. in

arXiv:2102.10460v1
fatcat:a2nqlp2aafhnnl3kom7oecznja
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... t. Zametki (2017), Šter in Lin. Algebra Appl. (2018) and Shitov in Indag. Math. (2019).##
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An Artinian theory for Lie algebras

2008
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Journal of Algebra
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Complemented Lie algebras are introduced in this paper (a notion similar to that studied by O. Loos and E. Neher in Jordan pairs). We prove that a Lie algebra is complemented if and only if it is a direct sum of simple nondegenerate Artinian Lie algebras. Moreover, we classify simple nondegenerate Artinian Lie algebras over a field of characteristic 0 or greater than 7, and describe the Lie inner ideal structure of simple Lie algebras arising from simple associative algebras with nonzero socle.

doi:10.1016/j.jalgebra.2007.10.038
fatcat:dpyfzfdr5fcp7bs6tgjf6fsxlq
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The Jordan algebras of a Lie algebra

2007
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Journal of Algebra
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We attach a Jordan algebra L x to any ad-nilpotent element x of index of nilpotence at most 3 in a Lie algebra L. This Jordan algebra has a behavior similar to that of the local algebra of a Jordan system at an element. Thus, L x inherits nice properties from L and keeps relevant information about the element x.

doi:10.1016/j.jalgebra.2006.02.035
fatcat:x53w7nj55zalvabszxpomk6tw4
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The Jordan socle and finitary Lie algebras

2004
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Journal of Algebra
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In this paper we introduce the notion of Jordan socle for nondegenerate Lie algebras, which extends the definition of socle given in [A. Fernández López et al., 3-Graded Lie algebras with Jordan finiteness conditions, Comm. Algebra, in press] for 3-graded Lie algebras. Any nondegenerate Lie algebra with essential Jordan socle is an essential subdirect product of strongly prime ones having nonzero Jordan socle. These last algebras are described, up to exceptional cases, in terms of simple Lie

doi:10.1016/j.jalgebra.2004.06.013
fatcat:3nzulra2bfehperekuyqeiudka
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... ebras of finite rank operators and their algebras of derivations. When working with Lie algebras which are infinite dimensional over an algebraically closed field of characteristic 0, the exceptions disappear and the algebras of derivations are computed. 2004 Elsevier Inc. All rights reserved.##
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Jordan centers and Martindale-like covers

2006
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Journal of Algebra
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In this paper we show that the scalar center of a nondegenerate quadratic Jordan algebra is contained in the scalar center of any of its Martindale-like covers. any of its Martindale-like algebras of quotients [1, 4.1]. Our aim in this paper is showing that the general (quadratic) version of (ii) holds. Indeed we will work at the slightly more general setting of what we call "Martindale-like covers," defined in terms of natural "ideal absorption properties." This result is basic in our

doi:10.1016/j.jalgebra.2006.06.047
fatcat:kvmgt34bifcwtmbffx5pvgswim
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... ng paper on polynomial identities and speciality of quadratic Martindale-like quotients, as well as we expect it to be useful in the description of Martindale-like quotients of strongly prime quadratic Jordan algebras satisfying a polynomial identity. The proof of our main result is purely combinatorial, based on the fact that 2J + Ker 2 Id J is an essential ideal of any nondegenerate Jordan algebra J , which, with the use of annihilators, allows to split the problem into the 2-torsion free and the characteristic 2 cases. The paper is divided into four sections. Section 0 is devoted to recalling basic facts and notions, including the essentialness of 2J + Ker 2 Id J , mentioned above, and the definition of the scalar center. In Section 1 we study characteristic 2 phenomena needed in the sequel, and their natural extensions to arbitrary Jordan algebras in terms of the annihilator Ann J (Ker 2 Id J ) of Ker 2 Id J . In the next section we establish the fundamental properties of Martindale-like covers. Finally, in Section 3, we prove our main theorem asserting the inheritance of the scalar center by Martindale-like covers of nondegenerate Jordan algebras. It turns out that for a central element z of J , and a cover Q of J , V z is in the centroid of Q as soon as Q satisfies the natural outer ideal absorption properties, while for the fact that z is indeed central in Q, the inner ideal absorption property must be assumed too. 0. Preliminaries 0.1. We will deal with Jordan algebras over a ring of scalars Φ. The reader is referred to [5, 7, 11] for definitions and basic properties not explicitly mentioned or proved in this section. Given a Jordan algebra J , its products will be denoted x 2 , U x y, for x, y ∈ J . They are quadratic in x and linear in y and have linearizations denoted V x y = x • y, U x,z y = {x, y, z} = V x,y z, respectively. A Jordan algebra J is said to be unital if there is an element 1 ∈ J satisfying U 1 = Id J and U x 1 = x 2 , for any x ∈ J (such an element can be shown to be unique and it is called the unit of J ). Every Jordan algebra J embeds in a unital Jordan algebraĴ = J ⊕ Φ1 called its ( free) unitization [11, 0.6]. A Jordan algebra J is said to be nondegenerate if zero is the only absolute zero divisor, i.e., zero is the only x ∈ J such that U x = 0.##
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Jordan systems of Martindale-like quotients

2004
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Journal of Pure and Applied Algebra
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In this paper we introduce the notion of Jordan system (algebra, pair or triple system) of Martindale-like quotients with respect to a ÿlter of ideals as that whose elements are absorbed into the original system by ideals of the ÿlter, and prove that it inherits regularity conditions such as (semi)primeness and nondegeneracy. When we consider power ÿlters of sturdy ideals, the notions of Jordan systems of Martindale-like quotients and Lie algebras of quotients are related through the

doi:10.1016/j.jpaa.2004.04.001
fatcat:6jdqbejgkzcd5axnf7i6mr4jiu
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... -Koecher construction, and that allows us to give constructions of the maximal systems of quotients when the original systems are nondegenerate. The theory of rings of quotients has its origins between 1930 and 1940, in the works of Ore and Osano on the construction of the total ring of fractions. In that decade Ore proved that a necessary and su cient condition for a ring R to have a (left) classic ring of quotients is that for any regular element a in R, and any b ∈ R there exist a regular c ∈ R and d ∈ R such that cb = da (left Ore condition).##
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Center, Centroid, Extended Centroid, and Quotients of Jordan Systems

2006
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Communications in Algebra
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In this paper we prove that the extended centroid of a nondegenerate Jordan system is isomorphic to the centroid (and to the center in the case of Jordan algebras) of its maximal Martindale-like system of quotients with respect to the filter of all essential ideals.

doi:10.1080/00927870600878066
fatcat:4ki3frbjc5fnhhrqxhs7qi6ugi
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A Construction of Gradings of Lie Algebras

2007
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International mathematics research notices
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Gómez

doi:10.1093/imrn/rnm051
fatcat:hikona4i2ncznansg5dhe5iqmu
*Lozano*at the University of Ottawa, following an invitation of E. Neher. ...##
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An elemental characterization of strong primeness in Lie algebras

2007
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Journal of Algebra
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In this paper we prove that a Lie algebra L is strongly prime if and only if [x, [y, L]] = 0 for every nonzero elements x, y ∈ L. As a consequence, we give an elementary proof, without the classification theorem of strongly prime Jordan algebras, that a linear Jordan algebra or Jordan pair T is strongly prime if and only if {x, T, y} = 0 for every x, y ∈ T . Moreover, we prove that the Jordan algebras at nonzero Jordan elements of strongly prime Lie algebras are strongly prime.

doi:10.1016/j.jalgebra.2006.11.003
fatcat:ougqqtntyjdqxb7qbbuhvnkfgm
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Human CD6 Down-Modulation following T-Cell Activation Compromises Lymphocyte Survival and Proliferative Responses

2017
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Frontiers in Immunology
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Available evidence indicates that the CD6 lymphocyte surface receptor is involved in T-cell developmental and activation processes, by facilitating cell-to-cell adhesive contacts with antigen-presenting cells and likely modulating T-cell receptor (TCR) signaling. Here, we show that in vitro activation of human T cells under different TCR-ligation conditions leads to surface downregulation of CD6 expression. This phenomenon was (i) concomitant to increased levels of soluble CD6 (sCD6) in culture

doi:10.3389/fimmu.2017.00769
pmid:28713387
pmcid:PMC5492662
fatcat:hwzh3yvngvep3p7urjkrnu5zom
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... supernatants, (ii) partially reverted by protease inhibitors, (iii) not associated to CD6 mRNA downregulation, and (iv) reversible by stimulus removal. CD6 down-modulation inversely correlated with the upregulation of CD25 in both FoxP3 − (Tact) and FoxP3 + (Treg) T-cell subsets. Furthermore, ex vivo analysis of peripheral CD4 + and CD8 + T cells with activated (CD25 + ) or effector memory (effector memory T cell, CD45RA − CCR7 − ) phenotype present lower CD6 levels than their naïve or central memory (central memory T cell, CD45RA − CCR7 + ) counterparts. CD6 lo/− T cells resulting from in vitro T-cell activation show higher apoptosis and lower proliferation levels than CD6 hi T cells, supporting the relevance of CD6 in the induction of proper T-cell proliferative responses and resistance to apoptosis. Accordingly, CD6 transfectants also showed higher viability when exposed to TCR-independent apoptosis-inducing conditions in comparison with untransfected cells. Taken together, these results provide insight into the origin of sCD6 and the previously reported circulating CD6-negative T-cell subset in humans, as well as into the functional consequences of CD6 down-modulation on ongoing T-cell responses, which includes sensitization to apoptotic events and attenuation of T-cell proliferative responses.##
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A characterization of the Kostrikin radical of a Lie algebra

2011
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Journal of Algebra
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The socle of a nondegenerate Lie algebra

2008
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Journal of Algebra
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We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an

doi:10.1016/j.jalgebra.2007.10.042
fatcat:s7tcccqn3rfplaqac24xuwwcmu
## more »

... inite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element.
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