DOCSLIB.ORG

Conference Non-Associative Algebras Related Topics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Conference Non-Associative Algebras Related Topics Program and Abstracts Conference Non-Associative Algebras and Related Topics 25-29 July 2011 Coimbra-Portugal Foreword We would like to welcome you to the NAART (Non-Associative Algebras and Related Topics) and wish you a pleasant stay in our town. The Theory of Non-Associative Algebras not only constitutes a well-developed and very active area of research but is also an intercurricular one, playing a cen- tral role in Mathematics and Physics. The aim of this conference is to bring together mathematicians and physicists interested in this field, with a central goal of increasing the quality of research, promoting interaction among researchers and discussing new directions for the future. The conference will last for five days and, besides a series of contributed sessions, there will also be five plenary talks and four courses. Contributions from all researchers in the area are welcome, including young researchers. The venue will be the historic University of Coimbra, founded in 1290 - the first university in Portugal and one of the oldest in Europe. The host of the event is the Department of Mathematics of the University of Coimbra (DMUC), drawing on the support of several other Portuguese research institutions. We are deeply indebted to the Centro de Matem´aticada Universidade de Coim- bra, to the Centro de Matem´atica da Universidade do Porto, to the Centro de Matem´aticada Universidade da Beira Interior, to the Funda¸c~aopara a Ci^enciae Tecnologia and also to the Banco Santander Totta for the material support. Helena Albuquerque Chair of the Organizing Committee Contents Contents i List of Participants 1 Schedule 3 Schedule of Contributed Talks 5 Mini Courses 9 Plenary Sessions 13 Contributed Talks 18 Posters 61 Index 73 i List of Participants 1 LIST OF PARTICIPANTS Marcia Aguiar (University of S~aoPaulo, Brazil) Helena Albuquerque (University of Coimbra, Portugal) Maribel Ram´ırezAlvarez (Universidad de Almer´ıa,Spain) Abu Zaid Ansari (AMU, Aligarh, India) Manuel Arenas (Universidad de Chile, Chile) Farkhad Arzikulov (I.M.I.T. , Andizhan State University, Uzbekistan) Raymond Aschheim (Polytopics, France) Imen Ayadi (Universit´ede Metz, ISGMP, France) Ignacio Bajo (Universidad de Vigo, Spain) Elisabete Barreiro (University of Coimbra, Portugal) Patr´ıciaBeites (University of Beira Interior, Portugal) Sa¨ıdBenayadi (Universit´ede Metz, ISGMP, France) Georgia Benkart (University of Wisconsin-Madison, USA) In^esBorges (ISCAC, Portugal) Said Boulmane (Faculty of Sciences Mekn`es,Morocco) Murray Bremner (University of Saskatchewan, Canada) Emmanuel Briand (University of Sevilla, Spain) Paula Carvalho (University of Porto, Portugal) Francesco Catino (University of Salento, Italia) Fl´avioU. Coelho (University of S~aoPaulo, Brazil) Isabel Cunha (University of Beira Interior, Portugal) Erik Darp¨o(University of Oxford, UK) Jos´eMar´ıaS´anchez Delgado (Universidad de C´adiz,Spain) S¨uleymanDemir (Anadolu University, Turkey) Geoffrey Dixon (USA) Cristina Draper (Universidad de M´alaga,Spain) Alberto Elduque (University of Zaragoza, Spain) Vyacheslav Futorny (University of S~aoPaulo, Brazil) Maxim Goncharov (Inst. of Math. SB RAS, Novosibirsk, Russia) Edgar Goodaire (Memorial University, Canada) Valerio Guido (University of Salento, Italy) Malika Ait Ben Haddou (Faculty of Sciences Mekn`es,Morocco) Manuel Avelino Insua Hermo (Dpt. Matem´aticaAplicada I, University of Vigo, Spain) Radu Iordanescu (Institute of Mathematics of the Romanian Academy, Romania) Clara Jim´enez-Gestal (Universidad de La Rioja, Spain) Santos Gonz´alez Jim´enez(University of Oviedo, Spain) Henrique Guzzo Jr (University of S~aoPaulo, Brazil) Ji Hye Jung (Seoul National University, South Korea) Elamin Kaidi (Universidad de Almeria, Spain) Mustafa Emre Kansu (Dumlupinar University, Turkey) Iryna Kashuba (University of S~aoPaulo, Brazil) Richard Kerner (University of Paris 6, France) Daehong Kim (Seoul National University, South Korea) Miran Kim (Seoul National University, South Korea) Myungho Kim (Seoul National University, South Korea) List of Participants 2 Mikhail Kotchetov (Memorial University of Newfoundland, Canada) Jes´usLaliena (Universidad de La Rioja, Spain) Christian Lomp (University of Porto, Portugal) Samuel Lopes (University of Porto, Portugal) Sara Madariaga (University of Logro~no,Spain) Shahn Majid (University of London, UK) Consuelo Mart´ınez(University of Oviedo, Spain) Vanesa Meinardi (Universidad Nacional de C´ordoba,Argentina) C´esarPolcino Milies (University of S~aoPaulo, Brazil) Jos´eManuel Casas Mir´as(Dpt. Matem´aticaAplicada I, University of Vigo, Spain) Daniel Mondoc (Lund University, Sweden) Sophie Morier-Genoud (Universit´eParis 6, France) Alejandro Nicol´as(University of Cantabria, Spain) Valentin Ovsienko (CNRS, Universit´eLyon 1, Lyon, France) Florin Panaite (Institute of Mathematics of the Romanian Academy, Romania) Sandra Pinto (University of Coimbra, Portugal) Rosemary Miguel Pires (Universidade Federal Fluminense, Brasil) Alexander Pozhidaev (Sobolev Inst. of Math., SB RAS, Novosibirsk, Russia) Nat´aliaMaria Bessa Pacheco Rego (IPCA, Portugal) Roberto Rizzo (University of Salento, Italy) Rodrigo Rodrigues (University of S~aoPaulo, Brazil) Hansol Ryu (Seoul National University, South Korea) Liudmila Sabinina (Facultad de Ciencias, UAEM, M´exico) Gil Salgado (UASLP, M´exico) Juana S´anchez-Ortega (University of Malaga, Spain) Ana Paula Santana (University of Coimbra, Portugal) Paulo Saraiva (University of Coimbra, Portugal) Ivan Shestakov (University of S~aoPaulo, Brazil) Agata Smoktunowicz (University of Edinburgh, UK) Dragos Stefan (University of Bucharest) Ralph St¨ohr(University of Manchester, UK) Osamu Suzuki (Nihon University, Japan) Sergei Sverchkov (University of S~aoPaulo, Brazil) Irina Sviridova (University of Brasilia, Brazil) Mohammad Reza Molaei Taherabadi (University of Kerman (Shahid Bahonar), Iran) Murat Tanı¸slı(Anadolu University, Turkey) Maribel Toc´on(Universidad de C´ordoba, Spain) Arkady Tsurkov (University of S~aoPaulo, Brazil) Maria del Carmen Rodriguez Vallarte (UASLP, M´exico) Ivan Yudin (CMUC, University of Coimbra, Portugal) Efim Zelmanov (University of California, San Diego, USA) Pasha Zusmanovich (Tallinn University of Technology, Tallinn, Estonia) Schedule 3 SCHEDULE Monday, July 25th 9:00 - 9:50: Registration. 9:50 - 10:00: Opening ceremony. 10:00 - 11:00: Opening plenary session: Georgia Benkart, Planar diagram algebras. 11:00 - 11:30: Coffee break. 11:30 - 13:00: Mini course 1: Alberto Elduque, Gradings on simple Lie algebras. 15:00 - 16:30: Contributed talks. 16:30 - 17:00: Coffee break. 17:00 - 18:00: Contributed talks. Tuesday, July 26th 9:00 - 10:30: Mini course 2: Alexander Pozhidaev, Differentiably simple (super)algebras. 10:30 - 11:30: Mini course 1 (part two): Alberto Elduque, Gradings on simple Lie algebras. 11:30 - 12:00: Coffee break. 12:00 - 13:00: Plenary session: Ivan Shestakov, On speciality of Malcev algebras. 15:00 - 16:00: Contributed talks. 16:00 - 16:30: Coffee break. 16:30 - 18:00: Visit to the University of Coimbra. Wednesday, July 27th 9:00 - 10:30: Mini course 3: Sa¨ıdBenayadi, Quadratic (resp. odd-quadratic) Lie superalgebras. 10:30 - 11:30: Mini course 2 (part two): Alexander Pozhidaev, Differentiably simple (super)algebras. 11:30 - 12:00: Coffee break. 11:30 - 13:00: Plenary session: Shahn Majid, Geometry of algebraic quasigroups. 14:30 - 20:00: Tour of Montemor-o-Velho and Figueira da Foz. 20:00: Conference dinner. Schedule 4 Thursday, July 28th 9:30 - 10:30: Mini course 4: Vyacheslav Futorny, Kac-Moody algebras, vertex operators and applications. 10:30 - 11:30 : Mini course 3 (part two): Sa¨ıdBenayadi, Quadratic (resp. odd-quadratic) Lie superalgebras. 11:30 - 12:00: Coffee break. 12:00 - 13:00: Plenary session: Consuelo Mart´ınez, Cheng Kac superalgebras. 15:00 - 16:30: Contributed talks. 16:30 - 17:00: Coffee break. 17:00 - 17:30: Contributed talks. 17:30 - 18:30: Poster Session. Friday, July 29th 9:00 - 10:30: Mini course 4 (part two): Vyacheslav Futorny, Kac-Moody algebras, vertex operators and applications. 10:30 - 11:30: Contributed talks. 11:30 - 12:00: Coffee break. 12:00 - 13:00: Contributed talks. 15:00 - 16:00: Closing plenary session: Efim Zelmanov, Jordan superalgebras. Schedule of contributed talks 5 SCHEDULE OF CONTRIBUTED TALKS Monday, July 25th Room 2.3 (Chair: Florin Panaite) 15:00 - 15:25: Murat Tanı¸slı, Maxwell equations for magnetic sources and massive photon in octonion algebra, 15:30 - 15:55: Mustafa Emre Kansu, Electromagnetism with complex octonions, 16:00 - 16:25: Richard Kerner, Space-time symmetry groups derived from cubic and ternary algebras, 17:00 - 17:25: Rosemary Miguel Pires, Code loops: automorphisms and representations, 17:30 - 17:55: S¨uleymanDemir, Hyperbolic octonionic massive gravitoelectromagnetism with monopoles. Room 2.4 (Chair: Alberto Elduque) 15:00 - 15:25: Fl´avioU. Coelho, The representation dimension of Artin algebras, 15:30 - 15:55: Ignacio Bajo, Indefinite K¨ahlermetrics on Lie algebras with abelian complex structure, 16:00 - 16:25: Mikhail Kotchetov, Weyl groups of fine gradings on Lie algebras, 17:00 - 17:25: Mohammad Reza Molaei Taherabadi, A generalization of left invariant vector fields and Lie algebras generated by them. Room 2.5 (Chair: Henrique Guzzo Jr) 15:00 - 15:25: Murray Bremner, Malcev and Bol dialgebras, 15:30 - 15:55: Maxim Goncharov, Structures of Malcev Bialgebras on a simple non-Lie Malcev algebra, 16:00 - 16:25: Manuel Arenas, On speciality of binary-Lie algebras, 17:00 - 17:25: Liudmila Sabinina, Moufang loops
Load more

Recommended publications
  • Arxiv:1902.01706V2 [Math.RA] 1 Apr 2019 O Emti Eut E,Freape 6 ,1,1,1] He N Dimensional 14]
    The algebraic and geometric classification of nilpotent binary Lie algebras 1,2 Hani Abdelwahaba, Antonio Jesus´ Calderon´ b & Ivan Kaygorodovc a Department of Mathematics, Faculty of Sciences, Mansoura University, Mansoura, Egypt b Department of Mathematics, Faculty of Sciences, University of Cadiz, Cadiz, Spain c CMCC, Universidade Federal do ABC. Santo Andr´e, Brasil E-mail addresses: Hani Abdelwahab ([email protected]) Antonio Jes´us Calder´on ([email protected]) Ivan Kaygorodov ([email protected]) Abstract: We give a complete algebraic classification of nilpotent binary Lie algebras of dimension at most 6 over an arbitrary base field of char- acteristic not 2 and a complete geometric classification of nilpotent binary Lie algebras of dimension 6 over C. As an application, we give an algebraic and geometric classification of nilpotent anticommutative CD-algebras of dimension at most 6. Keywords: Nilpotent algebras, binary Lie algebras, Malcev algebras, CD- algebras, Lie algebras, algebraic classification, geometric classification, degeneration. MSC2010: 17D10, 17D30. INTRODUCTION There are many results related to both the algebraic and geometric classi- fication of small dimensional algebras in the varieties of Jordan, Lie, Leib- niz, Zinbiel algebras; for algebraic results see, for example, [2,5,8,14,16]; arXiv:1902.01706v2 [math.RA] 1 Apr 2019 for geometric results see, for example, [6,9,11,12,14]. Here we give an algebraic and geometric classification of low dimensional nilpotent binary Lie algebras. Malcev defined binary Lie algebras as algebras such that every two- generated subalgebra is a Lie algebra [17]. Identities of the variety of bi- nary Lie algebras were described by Gainov [4].
    [Show full text]
  • Malcev Algebras
    MALCEVALGEBRAS BY ARTHUR A. SAGLEO) 1. Introduction. This paper is an investigation of a class of nonassociative algebras which generalizes the class of Lie algebras. These algebras satisfy certain identities that were suggested to Malcev [7] when he used the com- mutator of two elements as a new multiplicative operation for an alternative algebra. As a means of establishing some notation for the present paper, a brief sketch of this development will be given here. If ^4 is any algebra (associative or not) over a field P, the original product of two elements will be denoted by juxtaposition, xy, and the following nota- tion will be adopted for elements x, y, z of A : (1.1) Commutator, xoy = (x, y) = xy — yx; (1.2) Associator, (x, y, z) = (xy)z — x(yz) ; (1.3) Jacobian, J(x, y, z) = (xy)z + (yz)x + (zx)y. An alternative algebra A is a nonassociative algebra such that for any elements xi, x2, x3 oí A, the associator (¡Ci,x2, x3) "alternates" ; that is, (xi, x2, xs) = t(xiv xh, xh) for any permutation i%,i2, i%of 1, 2, 3 where « is 1 in case the permutation is even, — 1 in case the permutation is odd. If we introduce a new product into an alternative algebra A by means of a commutator x o y, we obtain for any x, y, z of A J(x, y, z)o = (x o y) o z + (y o z) o x + (z o y) o x = 6(x, y, z). The new algebra thus obtained will be denoted by A(~K Using the preceding identity with the known identities of an alternative algebra [2] and the fact that the Jacobian is a skew-symmetric function in /l(-), we see that /1(_) satis- fies the identities (1.4) ïoï = 0, (1.5) (x o y) o (x o z) = ((x o y) o z) o x + ((y o z) o x) o x + ((z o x) o x) o y Presented to the Society, August 31, 1960; received by the editors March 14, 1961.
    [Show full text]
  • Split Octonions
    Split Octonions Benjamin Prather Florida State University Department of Mathematics November 15, 2017 Group Algebras Split Octonions Let R be a ring (with unity). Prather Let G be a group. Loop Algebras Octonions Denition Moufang Loops An element of group algebra R[G] is the formal sum: Split-Octonions Analysis X Malcev Algebras rngn Summary gn2G Addition is component wise (as a free module). Multiplication follows from the products in R and G, distributivity and commutativity between R and G. Note: If G is innite only nitely many ri are non-zero. Group Algebras Split Octonions Prather Loop Algebras A group algebra is itself a ring. Octonions In particular, a group under multiplication. Moufang Loops Split-Octonions Analysis A set M with a binary operation is called a magma. Malcev Algebras Summary This process can be generalized to magmas. The resulting algebra often inherit the properties of M. Divisibility is a notable exception. Magmas Split Octonions Prather Loop Algebras Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary Loops Split Octonions Prather Loop Algebras In particular, loops are not associative. Octonions It is useful to dene some weaker properties. Moufang Loops power associative: the sub-algebra generated by Split-Octonions Analysis any one element is associative. Malcev Algebras diassociative: the sub-algebra generated by any Summary two elements is associative. associative: the sub-algebra generated by any three elements is associative. Loops Split Octonions Prather Loop Algebras Octonions Power associativity gives us (xx)x = x(xx). Moufang Loops This allows x n to be well dened. Split-Octonions Analysis −1 Malcev Algebras Diassociative loops have two sided inverses, x .
    [Show full text]
  • Universal Enveloping Algebras for Simple Malcev Algebras
    Pr´e-Publica¸c˜oes do Departamento de Matem´atica Universidade de Coimbra Preprint Number 10–32 UNIVERSAL ENVELOPING ALGEBRAS FOR SIMPLE MALCEV ALGEBRAS ELISABETE BARREIRO AND SARA MADARIAGA Abstract: It is our goal to present an universal enveloping algebra for the simple (non-Lie) 7-dimensional Malcev algebra M(α, β, γ) in order to obtain an universal enveloping algebra for simple Malcev algebras and to study its properties. Keywords: Simple Malcev algebra; universal enveloping algebra. Introduction The universal enveloping algebra of an algebra is a basic tool in the study of its representation theory. Given a algebra A, let us denote by A− the algebra obtained from A by replacing the product xy by the commutator [x, y]= xy yx, for elements x, y A. It is known that if A is an associative − ∈ algebra one obtains a Lie algebra A− and, conversely, the Poincar´e-Birkhoff- Witt Theorem establishes that any Lie algebra is isomorphic to a subalgebra of A− for some associative algebra A ([1], [2]). If we start with an alternative algebra A (alternative means that x(xy)= 2 2 x y and (yx)x = yx , for x, y A) then A− is a Malcev algebra. Since any associative algebra is in particular∈ an alternative algebra, then the Mal- cev algebra A− is a natural generalization of the Lie algebra A−. In [4], P´erez-Izquierdo and Shestakov presented an enveloping algebra of Malcev algebras (constructed in a more general way), generalizing the classical no- tion of universal enveloping algebra for Lie algebras. They proved that for every Malcev algebra M there exist an algebra U(M) and an injective Malcev algebra homomorphism ι : M U(M)− such that the image of M lies into the alternative nucleus of U(M−→), being U(M) a universal object with respect to such homomorphism.
    [Show full text]
  • Enveloping Algebra for Simple Malcev Algebras
    Motivation Simple Malcev algebras Enveloping algebra of Lie algebras Enveloping algebra of simple Malcev algebras Enveloping algebra for simple Malcev algebras Elisabete Barreiro (joint work with Sara Merino) Centre for Mathematics - University of Coimbra Portugal International Conference on Algebras and Lattices Prague, June 2010 Elisabete Barreiro (Univ. Coimbra, Portugal) Enveloping for simple Malcev algebras ICAL Prague 2010 1 / 16 Motivation Simple Malcev algebras Enveloping algebra of Lie algebras Enveloping algebra of simple Malcev algebras .A algebra finite dimensional over a field K .A − algebra: replacing the product xy in A by the commutator [x; y] = xy − yx, x; y 2 A. Ak(A) algebra: provided with commutator [x; y] = xy − yx and associator A(x; y; z) = (xy)z − x(yz), x; y; z 2 A. A associative =) A− Lie algebra alternative =) A− Malcev algebra x(xy) = x2y (yx)x = yx2; x; y 2 A not necessarily associative =) Ak(A) Akivis algebra Lie Malcev Akivis Elisabete Barreiro (Univ. Coimbra, Portugal) Enveloping for simple Malcev algebras ICAL Prague 2010 2 / 16 Motivation Simple Malcev algebras Enveloping algebra of Lie algebras Enveloping algebra of simple Malcev algebras CONVERSELY Theorem (Poincare-Birkhoff-Witt)´ Any Lie algebra is isomorphic to a subalgebra of A− for a suitable associative algebra A. Theorem An arbitrary Akivis algebra can be isomorphically embedded into an Akivis algebra Ak(A) for an algebra A. Reference: I. P. Shestakov, Every Akivis algebra is linear, Geometriae Dedicata 77 (1999), 215–223. Lie X Malcev ? Akivis X Elisabete Barreiro (Univ. Coimbra, Portugal) Enveloping for simple Malcev algebras ICAL Prague 2010 3 / 16 Motivation Simple Malcev algebras Enveloping algebra of Lie algebras Enveloping algebra of simple Malcev algebras QUESTION: Is any Malcev algebra isomorphic to a subalgebra of A−, for some alternative algebra A? .
    [Show full text]
  • Ideals in Non-Associative Universal Enveloping Algebras of Lie Triple Systems
    IDEALS IN NON-ASSOCIATIVE UNIVERSAL ENVELOPING ALGEBRAS OF LIE TRIPLE SYSTEMS J.MOSTOVOY AND J.M. PEREZ-IZQUIERDO´ Abstract. The notion of a non-associative universal enveloping algebra for a Lie triple system arises when Lie triple systems are considered as Bol algebras (more generally, Sabinin algebras). In this paper a new construction for these universal enveloping algebras is given, and their properties are studied. It is shown that universal enveloping algebras of Lie triple systems have surprisingly few ideals. It is conjectured, and the conjecture is verified on several examples, that the only proper ideal of the universal enveloping algebra of a simple Lie triple system is the augmentation ideal. 1. Introduction. Given a smooth manifold M and a point e M, a local multiplication on M at e is a smooth map U U M where U is some neighbourhood of e 2and the point e is a two-sided unit, that is, xe = ex = x for all×x !U. If x is sufficiently close to e, both left and right multiplications by x are one-to-one. Therefore, there2always exists a neighbourhood V U where the operations of left and right division are defined by the identities a (ab) = b and (ab)=b = a resp⊂ectively. Two local multiplications at the same point e of a manifold M are consideredn to be equivalent if they coincide when restricted to some neighbourhood of (e; e) in M M. Equivalence classes of local multiplications are called infinitesimal loops. (Sometimes infinitesimal loops×are also called local loops.) The importance of infinitesimal loops lies in the fact that they are closely related to affine connections on manifolds.
    [Show full text]
  • On Malcev Algebras Nilpotent by Lie Center and Corresponding Analytic Moufang Loops
    On Malcev algebras nilpotent by Lie center and corresponding analytic Moufang loops. Alexander Grishkov, Marina Rasskazova, Liudmila Sabinina, Mohamed Salim May, 2020 Abstract In this note we describe the structure of finite dimensional Malcev al- gebras over a field of real numbers R, which are nilpotent module its Lie center. It is proved that the corresponding analytic global Moufang loops are nilpotent module their nucleus. Key words: Malcev algebras, Moufang loops, Global Moufang loops. 2010 Mathematics Subject Classification: 17D10, 20N05. 1 Introduction The theory of analytic loops started with the work of A.I. Malcev in [Ma1]. In this article, the correspondence between the analytic local diassociative loops and the binary Lie algebras was established. A loop is a set M, endowed with a binary operation M × M → M, with the neutral element e ∈ M and the condition that the equations ax = b,ya = b for all a,b ∈ M have a unique solution. A loop is called diassociative, if every two elements of this loop generate a subgroup. Malcev showed that for an analytic loop, with the Moufang identity: (xy)(zx) = x(yz)x the corresponding tangent algebra satisfies the following identities: 2 arXiv:2005.07331v1 [math.RA] 15 May 2020 x = [J(x,y,z), x] − J(x, y, [x, z]) = 0, where J(x,y,z) = [[x, y],z] + [[x, z], x] + [[z, x],y] (See [1]). Algebras with these defining identities are currently called Malcev algebras. The more difficult ques- tion, if every finite dimensional Malcev R-algebra is the tangent algebra of some local analytic Moufang loop was solved positively by E.Kuzmin in 1969 in [Kuzm2].
    [Show full text]
  • Lie Algebras with Triality
    Lie algebras with triality. Alexander Grishkov1 Rua do Mat˜ao,1010,IME, University of S˜aoPaulo, CEP 005598-300, S˜ao Paulo, Brazil. e-mail: [email protected] In memorian Valerii Filippov 1supported fy FAPESP and CNPq (Brazil) 1 1 ABSTRACT By analogy with the definition of group with triality we introduce Lie algebra with triality as Lie algebra L wich admits the group of automorphisms S3 = {σ, ρ|σ2 = ρ3 = 1, σρσ = ρ2} such that for any x ∈ L we have (xσ − x) + (xσ − x)ρ + (xσ − x)ρ2 = 0. We describe the structure of finite dimensional Lie algebra with triality over a field of characteristic 0 and give applications of Lie algebras with triality to the theory of Malcev algebras. 2 2 Introduction. Glauberman [12], [13] noted that the multiplication group G(M) of a given Moufang loop M admits a certain dihedral group S3 of automorphisms, with 2 3 2 |S3| = 6. For fixed generators σ, ρ ∈ S3 with σ = ρ = 1, σρσ = ρ , the following equation holds for any x in G(M): 2 (xσx−1)(xσx−1)ρ(xσx−1)ρ = 1. (1) S.Doro [3] called groups with such automorphisms groups with triality, since the most striking example is D4(q) with its graph automorphisms. In this paper we introduce the notion of a Lie algebra with triality such that the simple Lie algebra of type D4 is a Lie algebra with triality and a Lie algebra that corresponds to an algebraic or Lie group with triality is a Lie algebra with triality.
    [Show full text]
  • ON the RADICAL of a FREE MALCEV ALGEBRA an Algebra M
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 9, September 2012, Pages 3049–3054 S 0002-9939(2012)11163-3 Article electronically published on January 31, 2012 ON THE RADICAL OF A FREE MALCEV ALGEBRA I. P. SHESTAKOV AND A. I. KORNEV (Communicated by Kailash C. Misra) Abstract. We prove that the prime radical rad M of the free Malcev algebra M of rank more than two over a field of characteristic = 2 coincides with the set of all universally Engelian elements of M.Moreover,letT (M)bethe ideal of M consisting of all stable identities of the split simple 7-dimensional Malcev algebra M over F .Itisprovedthatrad M = J(M) ∩ T (M), where J(M) is the Jacobian ideal of M. Similar results were proved by I. Shestakov and E. Zelmanov for free alternative and free Jordan algebras. An algebra M is called a Malcev algebra if it satisfies the identities x2 =0, J(x, xy, z)=J(x, y, z)x, where J(x, y, z)=(xy)z +(zx)y +(yz)x is the Jacobian of the elements x, y, z [7, 9, 5]. Since for a Lie algebra the Jacobian of any three elements vanishes, Lie algebras fall into the variety of Malcev algebras. Among the non–Lie Malcev algebras, the traceless elements of the octonion algebra with the product given by the commutator [x, y]=xy − yx is one of the most important examples [9, 5, 6]. In 1977 I. P. Shestakov [11] proved that the free Malcev algebra Mn on n ≥ 9 free generators is not semiprime; that is, Mn contains nonzero nilpotent ideals.
    [Show full text]
  • On Malcev Algebras
    Pacific Journal of Mathematics ON MALCEV ALGEBRAS T. S. RAVISANKAR Vol. 42, No. 1 January 1972 PACIFIC JOURNAL OF MATHEMATICS Vol. 42, No. 1, 1972 ON MALCEV ALGEBRAS T. S. RAVISANKAR A Malcev algebra is an anticommutative algebra satisfying the identity (xy)(xz) = {{xy)z)x + ((yz)x)x + ((zx)x)y. Various notions of solvability have been introduced for a Malcev algebra mainly with the aim of proving the following result or one of its corollaries for the related notion of semisimplicity. THEOREM A. A semisimple Malcev algebra over a field of characteristic zero is a direct sum of ideals which are simple as algebras. In the present paper all these notions of solvability are shown to be equivalent. This fact enables simplifications of proofs of some known results. The Killing form of a Malcev algebra is considered and is shown to possess some of the nice properties of the classical Cartan-Killing form of a Lie algebra. Finally, the structure of the Malcev algebra is studied in relation to that of the derivation algebra of an associated Lie triple system. The equivalence of the notions of solvability is proved in §1 of the paper (Theorem 1.1). This result is used in §2 to give an alter- native proof of Theorem A. This equivalence is further used to simplify the proofs of certain results of Loos [6] and to throw some light on a conjecture of Sagle [7]. In §§3, 4 we consider the Killing form of a Malcev algebra over a field of characteristic prime to six.
    [Show full text]
  • An Envelope for Malcev Algebras
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 272 (2004) 379–393 www.elsevier.com/locate/jalgebra An envelope for Malcev algebras José M. Pérez-Izquierdo a,1 andIvanP.Shestakovb,c,∗,2 a Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain b Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, São Paulo, SP 05311-970, Brazil c Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia Received 9 October 2002 Communicated by Alexander Premet Abstract We prove that for every Malcev algebra M there exist an algebra U(M) and a monomorphism − − ι: M → U(M) of M into the commutator algebra U(M) such that the image of M lies into the alternative center of U(M),andU(M) is a universal object with respect to such homomorphisms. The algebra U(M), in general, is not alternative, but it has a basis of Poincaré–Birkhoff–Witt type over M and inherits some good properties of universal enveloping algebras of Lie algebras. In particular, the elements of M can be characterized as the primitive elements of the algebra U(M) with respect to the diagonal homomorphism ∆ : U(M) → U(M) ⊗ U(M). An extension of Ado– Iwasawa theorem to Malcev algebras is also proved. 2004 Published by Elsevier Inc. 1. Introduction An anticommutative algebra (M, [ , ]) is said to be a Malcev algebra if it satisfies the identity [J(x,y,z),x]=J(x,y,[x,z]),whereJ(x,y,z) =[[x,y],z]−[[x,z],y]− [x,[y,z]] is the jacobian of x,y,z [4,6,8].
    [Show full text]
  • Structure and Representations of Finite Dimensional Malcev Algebras 1
    Quasigroups and Related Systems 22 (2014), 97 − 132 Structure and representations of nite dimensional Malcev algebras E. N. Kuzmin Abstract. The paper [6] is devoted to the study of the basic structure theory of nite dimensional Malcev algebras. Similarly to the structure of nite dimensional Lie algebras, this theory has attracted a lot of attention and stimulated further research in this area. However, for the sake of brevity, detailed proofs of some results were omitted. Some authors experienced some diculty owing to the lack of detailed proofs (see, for example [14]). The present work mostly follows the outline of [6] and lls the gaps in the literature. Editors' Preface This is an English translation of Structure and representations of nite dimensional Malcev algebras by E. N. Kuzmin, originally published in Akademiya Nauk SSSR, Sibirskoe Otdelenie, Trudy Instituta Matematiki (Novosibirsk), Issledovaniya po Teorii Kolets i Algebr 16 (1989), 75 101. The translation by Marina Tvalavadze was edited by Murray Bremner and Sara Madariaga. A brief survey of recent developments is included at the end of the paper. 1. Representations of nilpotent Malcev algebras. Cartan subalgebras 1.1. Malcev algebras were rst introduced in [10] as Moufang-Lie algebras. They are dened by the identities, x2 = 0, (1) J(x, y, xz) = J(x, y, z)x, (2) where J(x, y, z) is the so-called Jacobian of x, y, z: J(x, y, z) = (xy)z + (yz)x + (zx)y. In any anticommutative algebra, the Jacobian J(x, y, z) is a skew-symmetric func- tion of its arguments. Expanding the Jacobian, the Malcev identity (2) can be 2010 Mathematics Subject Classication: Primary 17D10.
    [Show full text]