The main goal of this poster, which is written in the form of a survey and tries to show some aspects of the research of authors on Lie algebras, is to pay homage to the memory of Pilar Pisón Casares, who was firstly teacher of some of them and later colleague of all of them during different stages of her stay as a member of the Departamento de Algebra, Computación, Geometría y Topología de la Universidad de Sevilla

Benjumea Acevedo, Juan Carlos

Falcón Ganfornina, Raúl Manuel

Fernández Ternero, Desamparados

Márquez García, María del Carmen

Núñez Valdés, Juan

English

idUS. Depósito de Investigación Universidad de Sevilla

References[1] J.M. Ancochea et M. Goze. Classificatiòn des algèbres de Lie filiformes de dimension 8 Archiv. Math. 50 (1988), 511-525.[2] J.M. Ancochea et M. Goze. Classificatiòn des algèbres de Lie nilpotentes de dimension 7 Archiv. Math. 52 (1989), 175-185.[3] Bruck, R. H. Some results in the theory of quasigroups. Transactions of the American Mathematical Society 55 (1944), pp. 19-54.[4] F.J. Echarte Reula, J.R. Gómez Mart́ın and J. Núñez Valdés. Les algébres de Lie filiformes complexes derivées d’autres algébres de Lie Collection Travaux en Cours: Lois d’algèbres et variétésalgébriques [Hermann Editeur, Paŕıs] 50 (1996), 45-55.[5] Echarte Reula, F.J., Márquez Garćıa M. C., Núñez Valdés, J. c-Graded fiiform Lie algebras. Bull Braz Math Soc, New Series 36(1) (2005) 59-77.[6] Echarte Reula, F.J., Márquez Garćıa M. C., Núñez Valdés, J. A constructive method to determine the variety of filiform Lie algebras. Czechoslovak Mathematical Journal 56:131 (2006) 1281-1299.[7] F.J. Echarte Reula, J. Núñez Valdés and F. Ramı́rez López. Description of some families of filiform Lie algebras Houston Journal of Mathematics. In press (to appear in 2008).[8] R. M. Falcón and J. Núñez-Valdés. Extension of Santilli’s isotopies to non-injective isoalgebras. Nonlinear Funct. Anal. Appl., 10(5) , 843-863, 2005.[9] R. M. Falcón and J. Núñez-Valdés. A particular case of extended isotopisms: Santilli’s isotopisms. Hadronic Journal, 29(3), 285-298, 2006.[10] R. M. Falcón Graphs related to principal autotopisms of Latin squares. Fifth Conference on Discrete Mathematics and Computer Science (Spanish), 279-286, Ciencias (Valladolid), 23, 2006.[11] R. M. Falcón Critical and forcing sets related to the autotopism group of a Latin square. International Congress of Mathematicians. Madrid, 2006.[12] D. Fernández-Ternero and J. Núñez-Valdés. Nilpotent Lie algebras of maximal rank and of Kac-Moody type: F(1)4 Comm. Algebra, 29(4):1551-1570, 2001.[13] D. Fernández-Ternero and J. Núñez-Valdés. Classification of metabelian Lie algebras of maximal rank C. R. Acad. Sci. Paris Sér. I Math., 332(11):969-974, 2001.[14] D. Fernández-Ternero and J. Núñez-Valdés. Nilpotent Lie algebras of maximal rank and of Kac-Moody type: E(1)6 J. Algebra, 250:664-695, 2002.[15] E. Dı́az, R. Fernández-Mateos, D. Fernández-Ternero and J. Núñez-Valdés. Graphs associated with nilpotent Lie algebras of maximal rank Rev. Mat. Iberoamericana, 19(2):325-338, 2003.[16] D. Fernández-Ternero. Nilpotent Lie algebras of maximal rank and of Kac-Moody type: D(3)4 J. Lie Theory, 15(1):249-260, 2005.[17] Khakimdjanov, Y.B. Variété des lois d’algèbres de Lie nilpotentes Geometriae Dedicata 40 (1991), 269-295 .[18] A. I. Malcev. Solvable Lie algebras Trans. Amer. Math. Soc. Transl. 9 (1962), 228-262.[19] L. J. Santharoubane. Kac-Moody Lie algebras and the classification of nilpotent Lie algebras of maximal rank Canad. J. Math., 34:1215-1239, 1982.[20] G.M. Tuynman. An elementary proof of Lie’s Third Theorem Publications de l’U.E.R. Mathématiques Pures et Appliquées, I.R.M.A. Univ. Lille 1 34 (1994), X1-X4.[21] M. Vergne. Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algebres de Lie nilpotentes Bull. Soc. Math. France 98 (1970), 81-116.6 Isotopic Lie algebrasBruck [3] defined a quasigroup as a nonempty set G endowed with a product ·, such thatif any two of the three symbols a, b, c in the equation a · b = c are given as elements ofG, the third one is uniquely determined as an element of G. Moreover, two quasigroups(G, ·) and (Ĝ, ·̂) are isotopic if there are three bijections α, β, γ from G to Ĝ, such thatγ(a · b) = α(a)̂·β(b), for all a, b ∈ G. The triple Θ = (α, β, γ) is called an isotopismfrom (G, ·) to (Ĝ, ·̂). Bruck’s isotopisms of quasigroups can be used to define some differentisotopies of Lie algebras. Specifically, starting from a Lie algebra of dimension n, we usedin [8, 9] a family of isotopisms, called Santilli’s isotopisms, in order to define Lie algebras ofdimension greater than n.Since the Caley’s table of a quasigroup of n elements is a Latin square of order n, that isto say, a n × n array L = (li,j) with elements chosen from the set [n] = {1, 2, ..., n}, suchthat each symbol occurs precisely once in each row and each column, the previous studyhas allowed us to research and develop new results on design theory. So, for example, weestablished a bijective relation between the graphs relating the different autotopisms of theautotopy group of a Latin square [10]. We also gave an upper bound of the size of thesmallest set of triples equivalent to Θ [11].5 Nilpotent Lie algebras of maximal rankLet L be a finite-dimensional nilpotent Lie algebra, DerL its derivation algebra, AutL itsautomorphism group. A torus on L is a commutative subalgebra of DerL whose elementsare semi-simple. The rank r of L is the dimension of a maximal torus. The Lie algebra Lis of maximal rank if r = L/[L,L].In [19] Santharoubane associated canonically a Kac-Moody algebra g(A) with each NLA Lof maximal rank, where A is a generalized Cartan matrix. The LA L is isomorphic to aquotient of the positive part n+ of the Kac-Moody algebra of type A.This classification problem is reduced to finding some ideals of the positive part n+ of theKac-Moody algebra g(A), up to the action of the automorphism group G of the Dynkindiagram of A.There are 3 families of Kac-Moody algebras: finite, affine and indefinite, and the secondfamily is divided in two subfamilies: non-twisted affine and twisted affine.The study of NLA of maximal rank associated with the finite Kac-Moody algebras (i.e. thefinite-dimensional simple Lie algebras) was already done.In [15] we used the graphs as a tool to study NLA. If A is affine, we can associate n+ with adirected graph (in fact, a digraph) and we can also associate a subgraph of this digraph withevery isomorphism class of NLA of maximal rank and of type A. We proposed an algorithmwhich computes these subgraphs and also groups them in isomorphism classes.I5 = h0; 1i110 4 81 56 910 I6 = h0; 2i7 110 4 82 56 910 I7 = h0; 6i110 4 856 910 I8 = h0; 7i7 110 4 85 910 I9 = h0; 11i110 4 85 910 I10 = h4; 5i4 85 910 I11 = h4; 7i7 114 8910 I12 = h4; 10i4 8910 I13 = h4; 11i114 89 At present, several authors are studying the NLA of maximalrank associated with non-twisted affine Kac-Moody algebras. Westudied the NLA of maximal rank and types F(1)4 and E(1)6 (see[12] and [14]). The main results are that up to isomorphismthere are: 1095 (respectively 2087) infinite series with discreteparameters and 28 (respectively, 126) infinite series with contin-uous parameters of nilpotent Lie algebras of maximal rank andof Kac-Moody type F(1)4 (respectively, E(1)6 ).We studied in [16] the NLA associated with the twisted affine Kac-Moody algebra g(D(3)4 ).The main result is that there are 88 infinite series (up to isomorphism) with discrete param-eters and 1 infinite series with continuous parameter for this type.In [13] 2-step nilpotent Lie algebras of maximal rank and of type A are classified, where Ais of finite or affine type. We proved that L is isomorphic to n+/C3n+, so that a metabelianLie algebra of maximal rank is uniquely determined by its type.In our research we have improved Tuynman’s work in a particular case, although vastenough: when dealing with filiform LA. To do this we give an explicit construction of asimply connected Lie group whose Lie algebra falls into a given FLA, say g.By using the group of automorphisms of FLA of a finite dimension n, particularly theunipotent automorphisms, we find a kind of basis with respect to which these automorphismscan be represented by triangular matrices. The dimension of the associated algebra, whichis a subalgebra of Der g, is, at most, 2n− 3.Starting from the model FLA of each dimension, its algebra of derivations contains assubalgebras those FLA such that [g, g] is abelian. Note that one of them is the own initialmodel algebra. These subalgebras determine, in a unique way, the corresponding subgroupsfrom the initial group of automorphisms which constitute, as it is pretended, a matrixrepresentation of the simply connected Lie group corresponding to each of such algebras.4 Lie groups of filiform Lie algebrasLie’s Third Theorem sets a uniquely correspondence between simply connected Liegroups and their associated Lie algebras. The accustomed proof in the literatureis based on Ado’s Theorem, which sets that given any LA g, there is a linear LAisomorphic to it.Recently (see [20]), Tuynman has given an elementary proof of Lie’s Third Theoremby using the correspondence between Lie subalgebras and Lie subgroups and the factthat, for a simply connected Lie group G, one has H2(G) = 0.3 c-Graded filiform Lie algebrasBy considering the usual gradation of the second space of Chevalley-Eilemberg’scohomology of the model FLA of dimension n, Pn, given by H2(Pn, Pn) =⊕c≥0H2c+1(Pn, Pn) and by taking into account that every FLA is isomorphic to(Pn)ψ, where ψ ∈ H2(Pn, Pn), it is possible to consider c-graded LA as algebrasisomorphic to (Pn)ψc, with ψc ∈ H2c+1(Pn, Pn) (see [17]).In [5] we obtain general results related c-graded FLA particularized to establishthe classification of 3-graded FLA. The classification of c-graded LA allows, for onething, to settle the classification of FLA of the type (Pn)ψc+ψk and, for another,to progress in the knowledge of the structure of (Pn)ψ, because by considering thegraduation, ψ = ψc + ψc+1 + · · · + ψk then ψc and ψk have to be cocycles suchthat (Pn)ψc and (Pn)ψk are c-graded LA. Finally, we obtain that fixed c ≥ 2, thereexists m = m(c) ∈ N such that every c-graded LA of dimension n ≥ m is a derivedalgebra of a rigid SLA of dimension n + 1.For each dimension, it is possible to associate the space H2(Pn, Pn) with an affinealgebraic variety. Polynomials defining this variety are preserved when the dimensionincreases, although the number of them certainly increases. In fact, all polynomialsdefining these varieties are homogeneous of degree two and when increasing the degreeby one, the only polynomials which are added are linear in the new variables. Thissuggests that each component of the variety of FLA of dimension n + 1 may be abundle over the corresponding component of the variety of dimension n with affinefibres (see [17]).In [6], we describe a constructive way to find a small set of polynomials which definethis variety. It allows to improve previous results related with the cardinal of thisset. We also describe an algorithm to determine in a computational way the varietyof these LA. It is suitable for being used in the case of bigger dimensions and canbe easily implemented in any symbolic computational package. Concretely, we haveused MAPLE in our study.2 Filiform Lie algebrasAt the best of our knowledge, the problem of classifying all the different typesof Lie algebras is still unsolved. As it is already well-known, there exist threedifferent types of LA: the semi-simple, the solvable and those which are neithersemi-simple nor solvable. So, to know the classification of LA, in general, isequivalent to know the classification of each of these three types.However, by the Levi-Maltsev theorem [18] in 1945, any finite-dimensionalLA over a field of characteristic zero can be expressed as a semidirect sum ofa semi-simple subalgebra and its radical (which is solvable). It reduces thetask of classifying all LA to obtain the classification of semi-simple and ofSLA.The classification of semi-simple LA is completely solved, due to the worksby Killing, Cartan and others (last decade of the 19th century).With respect to the SLA, NLA and FLA, it is only known the explicit classi-fication of them up to dimensions 12, 6 and 7, respectively. It was really M.Vergne, in 1966, who gave a great impulse to the study of filiform LA in herPh. D. Thesis, later published in [21]. Indeed, Vergne showed that the vari-ety of NLA of given dimension n has an irreducible component of dimensionexceeding n2, consisting entirely of LA of maximal class.By continuing the works by Ancochea and Goze, who classified 8-dimensionalFLA and 7-dimensional NLA, (see [1, 2], respectively), one of the first achieve-ments of one of the authors was to introduce two invariants of FLA whichallowed him to classify this type of algebras in dimensions 10, 11 and 12.The main properties of such invariants, denoted for the first time by i and j,respectively, can be checked in [4], later extended and improved in [7].1 IntroductionThis poster, dedicated to the memory of Pilar Pisón Casares, shows theachievements obtained by authors during the time in which they coincidewith her at the Faculty of Mathematics in the University of Seville, whenthey did research on Lie algebras in general and on filiform Lie algebras, inparticular. Such results, firstly got in the Ph. D. Thesis of each of them, werelater lengthened and improved to be published in International MathematicalReviews.From now on, Lie algebras, solvable Lie algebras, nilpotent Lie algebras andfiliform Lie algebras will be denoted by LA, SLA, NLA and FLA, respectively.Abstract: The main goal of this poster, which is written in the form of a survey and tries toshow some aspects of the research of authors on Lie algebras, is to pay homage to the memory of PilarPisón Casares, who was firstly teacher of some of them and later colleague of all of them duringdifferent stages of her stay as a member of the Departamento de Álgebra, Computación, Geometŕıa yTopoloǵıa de la Universidad de Sevilla.J. C. Benjumea1, R. M. Falcón2, D. Fernández-Ternero1, M. C. Márquez1 and J. Núñez1jcbenjumea@us.es rafalgan@us.es desamfer@us.es cmgarcia@us.es jnvaldes@us.es1 Department of Geometry and Topology 2 Department of Applied Mathematic IUniversity of Seville (Spain).Some advances in the research on Lie algebrasConferencia Internacional sobre Álgebra Conmutativa, Combinatoria y Computacional (Sevilla, 11-16 Febrero)

Some advances in the research on Lie algebras

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Some advances in the research on Lie algebras

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