Examinando por Autor "Vera de Salas, Guillermo"

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A Description of Ad-nilpotent Elements in Semiprime Rings with Involution
(Springer, 2021-02-01) Brox, Jose; García, Esther; Gómez Lozano, Miguel; Muñoz Alcázar, Rubén; Vera de Salas, Guillermo
Ad-nilpotent elements in algebras and superalgebras
(Universidad Rey Juan Carlos, 2021) Vera de Salas, Guillermo
In thisthesiswewilldealwithad-nilpotentelementsinassociativealgebrasand superalgebraswithinvolutionandsuperinvolution,andad-nilpotentelementsinLie superalgebras.The rstaimofthiswork tswithHerstein'sbranchoftheorythat studies nilpotentinnerderivationsinalgebras.Therearemanystudiesonthisarea, highlightingforourworkthearticlesofW.S.MartindaleandC.R.Miers[55], [56] and T.K.Lee[54]. Later,inthesecondpart,westudyhowtoassociatesomeJordan structures toaLiesuperalgebra,motivatedbytheworkofA.Fern andez,E.Garc a and M.G omezLozano[24]. Objectives Three objectivesareaddressedthroughoutthisthesis.Inthe rstinstance,weseekto describeindetailthead-nilpotentelementsinsemiprimeassociativealgebraswithin- volution.Thesecondaimofthisthesisistocarryoverthedescriptionsofad-nilpotent elementsinsemiprimeassociativealgebrastoprimeassociativesuperalgebras,that is, togiveadetaileddescriptionofhomogeneousad-nilpotentelementsbelongingto prime associativesuperalgebras.Finally,motivatedbytheworkofA.Fern andez,E. Garc aandM.G omezLozanoin[24], toassociateaJordansuperstructuretoaLie superalgebrawithanad-nilpotentelementofacertainindex. Methodology Todevelopthe rsttwogoalswehaveworkedwithintheframeworkofsemiprime algebras withinvolutionandprimeassociativesuperalgebraswithsuperinvolution. V Moreover,theextendedcentroidwillbeanimportanttoolinthisthesis.Forthe last objective,wehaveworkedwithnonassociativesuperstructuressuchasLieand Jordan superalgebras,de nedbytheGrassmannenvelope,andJordansuperpairs. Wecanhighlightthehighcombinatorialcontentthroughouttheentirethesis. Results Wehavesuccessfullycoveredthethreeinitialgoals.First,wehavedescribedindetail ad-nilpotentelementsbelongingtoasemiprimeassociativealgebra.Moreover,we havesucceededinreducingthetorsionintheclassi cationofad-nilpotentelements in semiprimeassociativealgebraswithinvolutionduetothenewconceptofapure ad-nilpotentelement,introducedinthisthesisinChapter2.Theconditionsonthe scalar ringshasbeenweakenedtobefreeof 􀀀n s and s torsion with s := [n+1 2 ] instead of beingfreeof n! torsion.Ontheotherhand,fortheskew-symmetricad-nilpotent elementsofasemiprimeassociativealgebra R with involution , wehavegivena description thatdependsontheirad-nilpotentindexmodulo4.Inthisdescription wecanemphasize:Ifaskew-symmetricelement a is ad-nilpotentsuchthatitsindex of ad-nilpotenceof K := Skew(R; ) and R do notcoincide,thatis,adn aK = 0but adn aR 6= 0,(itcanonlyoccurforad-nilpotentindicesof K congruentto0or3modulo 4) thenacertaincornerof R satis es aPI,hence R holds aGPI.Theseresultshave beendevelopedthroughoutChapter2whichhaveoriginatedanarticlethathasbeen published inthejournal BulletinoftheMalaysianMathematicalSciencesSociety ([12]). Thesecondaim,todescribeinprimeassociativesuperalgebraswithsuperin- volutionnilpotentinnerderivations,hasalsobeenpositivelysolvedduringChapter 3. Thisdescriptiondependsontheparityofthehomogeneouselement:iftheelement is even,whathasbeendevelopedinthepreviouschapterinalgebrasettings([12]), is largelyrescued.However,iftheelementisodd,wehaveworkedonitssquare, whichisanevenad-nilpotentelement,andwehaveappliedthedescriptionsforeven ad-nilpotentelementsstudiedabove.Theseresultshasbeenpublishedinthejournal LinearandMultilinearAlgebra ([28]). DuringChapter4,wehavegivenexamples for eachofthecasesappearinginthedescriptionsoftheelementsinbothalgebras VI and superalgebras,thusshowingthatthesedescriptionsarenottrivial.Finally,in Chapter 5,wehaveassociatedaJordansuperstructuretoaLiesuperalgebra L with a homogeneous ad-nilpotentelement a of index3or4,accordingtoitsparity.Further- more, theJordansuperpairwehaveconstructedfollowingthespiritofthepaperofA. Fern andez,E.Garc aandM.G omezLozano[24], coincideswiththesubquotientof a Liesuperalgebraassociatedwithanabelianinnerideal[a; [a; L]]. Thislastchapter has beenpublishedandcanbeconsultedinthejournal CommunicationsinAlgebra ([30]).
Ad-Nilpotent Elements of Skew Index in Semiprime Rings with Involution
(Springer, 2021-11-11) Brox, Jose; García, Esther; Gómez Lozano, Miguel; Muñoz Alcázar, Rubén; Vera de Salas, Guillermo
Gradings induced by nilpotent elements
(Elsevier, 2022) García, Esther; Gómez Lozano, Miguel; Muñoz Alcázar, Rubén; Vera de Salas, Guillermo
An element a is nilpotent last-regular if it is nilpotent and its last nonzero power is von Neumann regular. In this paper we show that any nilpotent last-regular element a in an associative algebra R over a ring of scalars Φ gives rise to a complete system of orthogonal idempotents that induces a finite Z-grading on R; we also show that such element gives rise to an sl2-triple in R with semisimple adjoint map adh, and that the grading of R with respect to the complete system of orthogonal idempotents is a refinement of the Φgrading induced by the eigenspaces of adh. These results can be adapted to nilpotent elements a with all their powers von Neumann regular, in which case the element a can be completed to an sl2-triple and a is homogeneous of degree 2 both in the Z-grading of R and in the Φ-grading given by the eigenspaces of adh.
Nilpotent superderivations in prime superalgebras
(Taylor&Francis, 2021-04-28) García, Esther; Gómez Lozano, Miguel; Vera de Salas, Guillermo
In this paper we give an in-deph analysis of the nilpotency index of nilpotent homogeneous inner superderivations in associative prime superalgebras with and without superinvolution. We also present examples of all the different cases that our analysis exhibits for the nilpotency indices of the inner superderivations. En este artículo se realiza un análisis en profundidad del índice de nilpotencia de las superderivaciones internas homogéneas nilpotentes en una superálgebra asociativa prima con o sin superinvolución. También damos ejemplos de todos los diferentes casos que aparecen.
On classical orthogonal polynomials and the Cholesky factorization of a class of Hankel matrices
(World Scientific, 2023-04-11) Marriaga, Misael E.; Vera de Salas, Guillermo; Latorre, Marta; Muñóz Alcázar, Rubén
Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second-order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second-order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials.