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 Ciˆenciae 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) InˆesBorges (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 and their automorphisms. Schedule of contributed talks 6

Tuesday, July 26th

Room 2.3 (Chair: Daniel Mondoc)

15:00 - 15:25: Irina Sviridova, Algebras with involution and their identities, 15:30 - 15:55: Juana S´anchez-Ortega, Leibniz triple systems.

Room 2.4 (Chair: Ignacio Bajo)

15:00 - 15:25: Ralph St¨ohr, On linear equations over free Lie algebras, 15:30 - 15:55: Vanesa Meinardi, Finite growth representations of infinite Lie conformal algebras. Room 2.5 (Chair: Richard Kerner)

15:00 - 15:25: Iryna Kashuba, Representation type of Jordan algebras, 15:30 - 15:55: Jes´usLaliena, The derived algebra of skew elements in a semiprime superalgebra with superinvolution. Schedule of contributed talks 7

Thursday, July 28th

Room 2.3 (Chair: Sa¨ıdBenayadi)

15:00 - 15:25: Elisabete Barreiro, Homogeneous symmetric antiassociative quasialgebras, 15:30 - 15:55: Gil Salgado, Heisenberg Lie superalgebras and its invariant superorthogonal and supersymplectic forms, 16:00 - 16:25: Manuel Avelino Insua Hermo, On (co)homology of Hom-Leibniz algebras. Room 2.4 (Chair: Ivan Shestakov)

15:00 - 15:25: Erik Darp¨o, Four-dimensional power-commutative real division algebras, 15:30 - 15:55: Henrique Guzzo Jr, Multiplicative mappings of alternative rings, 16:00 - 16:25: Sara Madariaga, Abelian groups in the slice category of bialgebras, 17:00 - 17:25: Valentin Ovsienko, A series of algebras generalizing the octonions and Hurwitz-Radon identity. Room 2.5 (Chair: Liudmila Sabinina)

15:00 - 15:25: Arkady Tsurkov, Strongly stable automorphisms of the category of free linear algebras, 15:30 - 15:55: Ana Paula Santana, Exact sequences in the Borel-Schur algebra, 16:00 - 16:25: Ivan Yudin, Normalized bar resolution and Schur algebra. Schedule of contributed talks 8

Friday, July 29th

Room 2.3 (Chair: Manuel Arenas)

10:30 - 10:55: Raymond Aschheim, Non associative quantum gravity, 11:00 - 11:25: Osamu Suzuki, Nonassociative algebra associated to genetics and Jordan algebra, 12:00 - 12:25: Sergei Sverchkov, Algebraic theory of DNA recombination, 12:30 - 12:55: Maribel Toc´on, A capacity 2 theorem for graded Jordan systems. Room 2.4 (Chair: Consuelo Mart´ınezLopez)

10:30 - 10:55: Abu Zaid Ansari, Lie ideals and generalized derivations in semiprime rings, 11:00 - 11:25: Geoffrey Dixon, Octonions, lattices, and preferred parentheses, 12:00 - 12:25: Daniel Mondoc, On generalized Jordan triple systems of second order and extended Dynkin diagrams, 12:30 - 12:55: Florin Panaite, Alternative twisted tensor products and Cayley algebras. Room 2.5 (Chair: Jes´usLaliena)

10:30 - 10:55: C´esarPolcino Milies, Alternative loop algebras over finite fields, 11:00 - 11:25: Pasha Zusmanovich, A commutative 2-cocycles approach to classification of simple Novikov algebras. Mini Courses 9

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Quadratic (resp. odd-quadratic) Lie superalgebras

Sa¨ıdBenayadi1

A quadratic (resp. odd-quadratic) Lie superalgebra is a Lie superalgebra with a non- degenerate supersymmetric, even (resp. odd), invariant bilinear form. Quadratic (resp. odd-quadratic) Lie superalgebras appear in particular in differential (super-)geometry and in physical models based on Lie superalgebras. The classification of quadratic (resp. odd-quadratic) classical simple Lie superalgebras had been obtained by V. Kac. But many solvable Lie superalgebras also belong to this class. In this mini course:

1. We will construct some non-trivial examples of quadratic (resp. odd-quadratic) Lie superalgbras.

2. Methods of double extension of quadratic (resp. odd-quadratic) Lie superalgebras will be developed.

3. We will give some inductive descriptions of quadratic (resp. odd-quadratic) Lie superalgebras.

4. We will introduce the quadratic dimension of a quadratic Lie superalgebra and we will establish some relations between this invariant and others invariants of quadratic Lie superalgebras.

5. At the end, we will discuss some open problems on quadratic (resp. odd-quadratic) Lie superalgebras.

1Laboratoire de Math´ematiqueset Applications de Metz, CNRS UMR 7122, Universit´ePaul Verlaine-Metz, Ile du Saulcy, F-57045 Metz cedex 1, France [email protected] Mini Courses 10

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Gradings on simple Lie algebras

Alberto Elduque1

A survey of some recent classification results for gradings over abelian groups on finite dimensional simple Lie algebras over algebraically closed fields will be given. This will require the classification of gradings on matrix algebras, on the algebra of octonions and on the Albert algebra (exceptional simple Jordan algebra). Any grading of a finite dimensional algebra over an abelian group is equivalent to a representation of a diagonalizable affine group scheme on the affine group scheme of automorphisms of the algebra. This point of view, necessary to deal with the modular case, will be stressed throughout.

1Departamento de Matem´aticas e Instituto Universitario de Matem´aticasy Apli- caciones, Universidad de Zaragoza, 50009 Zaragoza, Spain [email protected] Mini Courses 11

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Kac-Moody algebras, vertex operators and applications

Vyacheslav Futorny1

This mini-course will focus on vertex type constructions for certain classes of infinite- dimensional Lie algebras, including Affine Kac-Moody algebras and elliptic Affine alge- bras (the latter are particular cases of Krichever-Novikov algebras associated with elliptic curves). During the first lecture we will discuss some aspects of the representation theo- ry of Kac-Moody algebras based on free field realizations of Affine Lie algebras and theory of vertex algebras. Vertex algebras have origin in string theory, they provide a mathematical foundation of 2-dimensional conformal field theory. Vertex algebras have numerous applications in many areas of mathematics, in particular they are ubiquitous in representation theory of infinite-dimensional Lie algebras. Some generalizations of vertex constructions for Affine Lie algebras will be considered. In the second lecture vertex realizations of Affine Lie algebras will be applied to the study of representations of the Lie algebra of vector fields on N-dimensional torus (these are joint results with Y. Billig). Finally, we will discuss free field realizations of elliptic Lie algebras which were recently obtained in a joint work with B. Cox and A. Bueno.

1Instituto de Matem´aticae Estat´ıstica,Universidade de S˜aoPaulo, Caixa Postal 66281, S˜aoPaulo, CEP 05315-970, Brazil [email protected] Mini Courses 12

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Differentiably simple (super)algebras

Alexander Pozhidaev1

A superalgebra A is called differentiably simple if it lacks homogeneous ideals in- variant under Der(A). In this talk we discuss the theory of differentiably simple (su- per)algebras and some its applications.

1Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia [email protected] Plenary sessions 13

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Planar diagram algebras

Georgia Benkart1

Planar diagram algebras arise in the representation theory of low-rank Lie algebras, Lie superalgebras and quantum groups. They arise in statistical mechanics and have played a prominent role in Jones’ work on subfactors of von Neumann algebras and on invariants of knots and links. This talk will focus on planar diagram algebras, some introduced in our recent joint work with Halverson, and on their combinatorial connec- tions with various well-studied sequences of numbers such as the Catalan and Motzkin numbers.

1University of Wisconsin-Madison, USA [email protected] Plenary sessions 14

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Geometry of algebraic quasigroups

Shahn Majid1

Building on recent work we use a Hopf-algebra like theory of “Hopf coquasigroups” to study quasigroups such as S7. As well as tangent and cotangent bundles we now look at further aspects of the geometry such as metrics and other bundles, using algebraic methods previously developed for quantum groups.

1University of London, UK [email protected] Plenary sessions 15

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Cheng Kac superalgebras

Consuelo Mart´ınez1

Cheng-Kac superconformal algebra CK(6) was found by S. J. Cheng and V. Kac in 1997 and independently by Grozman, Leites and Shchepochkina in 2001. Its existence obliged to reformulated the previous conjecture by V. Kac and van de Leur about the structure of superconformal algebras. In 2001 they appear in a more general context in the classification of simple Jordan superalgebras in prime characteristic in the case of non-semisimple even part. We will explain what is know about those superalgebras and related structures. We will pay also special attention to the case of prime characteristic, explaining some results obtained jointly with E. Barreiro and A. Elduque.

1Departamento de Matem´aticas,Universidad de Oviedo, Oviedo, Spain [email protected] Plenary sessions 16

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

On speciality of Malcev algebras

Ivan Shestakov1

A Malcev algebra is called special if it can be embedded into a commutator algebra of a certain alternative algebra. We will give some new results related with the problem of speciality of Malcev algebras.

1University of S˜aoPaulo, Brazil [email protected] Plenary sessions 17

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Jordan superalgebras

Efim Zelmanov1

I will try to give a broad survey of the theory of Jordan superalgebras and their connections.

1University of California, San Diego, USA [email protected] Contributed Talks 18

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Lie ideals and generalized derivations in semiprime rings †

Nadeem Ur Rehman1, Abu Zaid Ansari 1

Let R be an associative ring with center Z(R). For each x, y ∈ R denote the commu- tator xy−yx by [x, y] and the anti-commutator xy+yx by x◦y. An additive subgroup L of R is said to be Lie ideal of R if [L, R] ⊆ L. An additive mapping F : R → R is called generalized inner derivation if F (x) = ax + xb for fixed a, b ∈ R. For such a mapping F , it is easy to see that F (xy) = F (x)y + xIb(y) for all x, y ∈ R. This observation leads to the following definition given in [Communication Algebra 26(1998), 1149-1166]. An additive mapping F : R → R is called generalized derivation with associated derivation d if F (xy) = F (x)y + xd(y) for all x, y ∈ R. In the present paper we shall show that L ⊆ Z(R) such that R is semiprime ring satisfying several conditions.

Keywords: Derivations, generalized derivations, prime rings, semiprime rings, Lie ideals

Mathematics Subject Classification 2010: 16W25, 16N60, 16U80

References

[1] Ashraf M., Ali A. and Ali S., Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math. 32(2) (2007),415-421.

[2] Ashraf M. Khan A., Lie ideals and generalized derivations of semiprime rings, (to appear).

[3] Ashraf M., Rehman N., On derivations and commutativity in prime rings, East west J. Math. 3(1) (2001) 87-91.

[4] Ashraf M., Rehman N., On commutativity of rings with derivations, Results Math. 42(2002), 3-8.

[5] Argac, N., On prime and semiprime rings with deivations, Algebra Colloq. 13(3) (2006), 371-380.

[6] Awtar R., Lie and Jordan structures in prime rings with derivations, Proc. Amer. Math. Soc. 41 (1973) 67-74.

[7] Awtar R., Lie structure in prime rings with derivations, Publ. Math. (Debrecen) 31 (1984) 209-215.

[8] Bell, H.E. and Daif, M.N. On derivations and commutativity in prime rings, Acta Math. Hungar. 66(4)(1995), 337-343.

[9] Bell, H.E. and Rehman, N., Generalized derivations with commutativity and anti- commu- tativity conditions, Math. J. Okayama Univ. 49(2007), 139-147.

†This research is supported by UGC, India, Grant No. 36-8/2008(SR). Contributed Talks 19

[10] Bergen, J., Herstein, I.N. and Kerr, J.W., Lie ideals and derivations of prime rings, J. Algebra 71(1981), 259-267.

[11] Bresar, M., On the distance of the compositions of two derivations to the generalized deriva- tions, Glasgow Math. J.33(1)(1991), 89-93.

[12] Daif, M.N. and Bell, H.E. Remarks on derivations on semiprime rings, Internat. J. Math. Math. Sci. 15 (1) (1991) 204-206.

[13] Deng, Q. and Ashraf, M., On strong commutativity preserving mappings, Results in Math. 30(1996), 259-263.

[14] Herstein, I.N., On the Lie structure of associative rings, J. Algebra 14(4) (1970) 561-571.

[15] Herstein, I.N., Topics in ring theory, Univ. Chicago Press, Chicago (1969).

[16] Herstein, I.N., Rings with invilotion, Univ. Chicago Press, Chicago and Londan (1976).

[17] Hongan, M., Rehman, N. and Al Omary, R., Lie ideals and Jordan triple derivations of rings, (to appear)

[18] Posner, E.C., Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100.

[19] Quadri, M.A. , Khan, M.S. and Rehman, N., Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math. 34(9)(2003), 1393-1396.

[20] Rehman, N., On commutativity of rings with generalized derivations, Math. J. Okayama Univ. 44(2002) 43-49.

1Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, India [email protected], [email protected] Contributed Talks 20

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011 On speciality of binary-Lie algebras

Manuel Arenas ‡1

In the present talk, I will present some results obtained in a joint work with Professor Ivan Shestakov. We proved that, for every assocyclic algebra A, the algebra A− is binary- Lie. We found a simple not Malcev binary-Lie superalgebra T that can not be embedded in A−s for any assocyclic superalgebra A. We used the Grassman envelope of T to prove the corresponding result for algebras. This solve negatively a problem by V.T. Filippov from [1]. Finally, we proved that the superalgebra T is isomorphic to the commutator superalgebra A−s for a simple binary (-1,1) superalgebra A.

Keywords: Assocyclic algebra, binary-Lie algebra, speciality problem, superalgebra, (-1,1)-algebra.

Mathematics Subject Classification 2010: 17D02

References

[1] The Dniestr notebook. Unsolved problems in the theory of rings and modules. (Dnes- trovskaya tetrad’. Nereshennye problemy teorii kolets i modulej). ed. by V. T. Filippov, V. K. Kharchenko, I. P. Shestakov, 4th ed. (Russian), Novosibirsk: Institut Matematiki SO RAN, 74 p. (1993). English transl. in Non-associative algebra and its applications, edited by L. V. Sabinin, L. Sbitneva, I. Shestakov, Proceedings of the 5th international conference, Oaxtep, Mexico, July 27 - August 2, 2003. Lecture Notes in Pure and Applied Mathematics 246, (2006).

[2] A. T. Gainov, Identical relations for binary-Lie rings, (Russian) Uspehi Mat. Nauk (N.S.) 12, 3 (75), 141-146, (1957).

[3] E. Kleinfeld and L. Widmer, Rings satisfying (x, y, z) = (y, z, x), Comm. in algebra, 17(11), 2683-2687, (1989).

[4] E. N. Kuzmin, Binary Lie Algebras of Small Dimentions, Algebra and Logic, vol. 37 (1998), N o 3, 181-186.

[5] I. P. Shestakov, Superalgebras and counter-examples, Sibirskii Matem. Zh., 32, N 6 (1991), 187–196; English transl.: Siberian Math.J. 32, N 6 (1991), 1052–1060.

[6] I. P. Shestakov, Simple (-1,1) superalgebras, (Russian) Algebra i Logika, 37, N 6 (1998), 721–739; English transl.: Algebra and Logic 37 (1998), no.6, 411–422.

1Departamento de Matem´aticas,Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Nu˜noa,Santiago,˜ Chile [email protected], [email protected]

‡supported by FONDECYT grant 11100092 Contributed Talks 21

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Non associative quantum gravity

Raymond Aschheim1

Non associative algebras have an emergent and fundamental place in quantum gra- vity. I will first present the infinion algebra, which is the limit for large n of algebras build by Cayley-Dickson process, and give explicit table for its restriction to 1024 dimensions and general construction; then review three possible applications in quantum gravity. Their imaginary part can replace su(2) in Regge Calculus for embedded spinfoams [1], and in Loop Quantum Gravity for abstract spinfoams [2], so that a non associative be- havior emerging in dimensions 5 to 8 extend naturally the spacetime to internal encoding of all fermions, with e8 roots as quantum numbers. This concept of matter as an effect of spacetime nonassociativity is also shown in the new framework of relative locality [3], where we propose to use logarithms of imaginary infinions as momentum space basis.

Keywords: division algebra, quantum gravity

Mathematics Subject Classification 2010: 17A35, 17B81

References

[1] E. Maglioro and C. Perini, Regge gravity from spinfoams, in Proceedings of LOOPS11 International Conference on Quantum Gravity, J. of Physics: Conference Series to ap- pears in (2011), available in arxiv:1105.0216v6 [gr-qc]. ,

[2] R. Aschheim, Spin Foam with topologically encoded tetrad on trivalent spin networks, in Proceedings of LOOPS11 International Conference on Quantum Gravity, J. of Physics: Conference Series to appears in (2011). ,

[3] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, The principle of relative locality, in Proceedings of LOOPS11 International Conference on Quantum Gravity, J. of Physics: Conference Series to appears in (2011), available in arxiv:1101.0931 [hep-th].

1POLYTOPICS, 8 villa Haussmann, 92130 Issy, France [email protected] Contributed Talks 22

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Indefinite K¨ahlermetrics on Lie algebras with abelian complex structure

Ignacio Bajo1

We study indefinite K¨ahlermetrics on Lie algebras endowed with abelian complex structures. It is seen that each of those Lie algebras is completely determined by an associative and commutative complex algebra admitting a special hermitian form. We propose several constructions and provide an inductive description of such Lie algebras. As a geometrical application, the curvatures of the pseudo-K¨ahlermetric are computed and sufficient conditions to ensure flatness or Ricci-flatness are given.

1Departamento de Matem´aticaAplicada II, E.T.S.I. Telecomunicaci´on,Univer- sidad de Vigo, 36280 Vigo, Spain [email protected] Contributed Talks 23

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Homogeneous symmetric antiassociative quasialgebras

Helena Albuquerque1, Elisabete Barreiro1, Sa¨ıdBenayadi2

This talk will focus on homogeneous symmetric antiassociative quasialgebras, mea- ning antiassociative quasialgebras equipped with a non-degenerate, supersymmetric, homogeneous (even or odd), associative bilinear form. We recall that any Z2-graded quasialgebra is either an associative superalgebra or an antiassociative quasialgebra. Symmetric homogeneous associative superalgebras were studied by I. Ayadi on her doc- toral work under S. Benayadi’s supervision [3, 4]. We show that any odd-symmetric antiassociative quasialgebra is, in particular, an associative superalgebra. On even- symmetric antiassociative quasialgebras we present some notions of generalized double extensions in order to give inductive descriptions of this class of superalgebras. Our goal is to complete the study of Z2-graded quasialgebras provided with homogeneous symmetric structures.

Keywords: antiassociative quasialgebras, homogeneous symmetric structures, genera- lized double extension, inductive description

References

[1] H. Albuquerque, A. Elduque, and J.M. P´erez-Izquierdo, Z2-quasialgebras, Comm. Algebra 30 (2002), 2161–2174.

[2] H. Albuquerque and Shahn Majid, Quasialgebra structure of the octonions, J. Algebra 220 (1999), 188–224.

[3] I. Ayadi, “Super-alg`ebresnon Associatives avec des Structures Homog`enes”,Doctoral The- sis, Universit´ePaul Verlaine-Metz, 2011.

[4] I. Ayadi and S. Benayadi, Symmetric Novikov superalgebras J. Math. Phys. 51 (2010), no. 2, 023501, 15 pp.

1CMUC, Departamento de Matem´atica, Universidade de Coimbra, 3001-454 Coimbra, Portugal [email protected], [email protected] 2Laboratoire LMAM, CNRS UMR 7122, Universit´ePaul Verlaine-Metz, Ile du Saulcy, F-57012 Metz cedex 01, France [email protected] Contributed Talks 24

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Malcev and Bol dialgebras

Murray R. Bremner1

We apply the algorithm of Kolesnikov to the defining identities for Malcev algebras to determine the defining identities for Malcev dialgebras. We then use computer algebra to show that these identities are equivalent to the identities of degree ≤ 4 satisfied by the dicommutator in every alternative dialgebra. We generalize Kolesnikov’s algorithm to the case of general multioperator algebras, and apply this to the defining identities for Bol algebras to determine the defining identities for Bol dialgebras. We then use computer algebra to show that these identities are equivalent to the identities of degree ≤ 5 satisfied by the dicommutator and the Jordan diassociator in every right alternative dialgebra. This is joint work with Alexander Pozhidaev, Luiz Peresi, and Juana Sanchez- Ortega.

1Department of Mathematics and Statistics, University of Saskatchewan 106 Wiggins Road (McLean Hall) Saskatoon, SK, S7N 5E6 Canada [email protected] Contributed Talks 25

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

The representation dimension of Artin algebras

Fl´avioUlhoa Coelho1

The representation dimension of an algebra was introduced by Maurice Auslander in the 70’s of the last century with the aim of measuring how far an algebra is to be representation-finite [1]. Recall that an algebra A is representation finite provided there are only finitely many non-isomorphic indecomposable finitely generated A-modules. However, more than 25 years passed before it was considered again in the mainframe of the Representation Theory of Algebras. We shall give a very quick survey on the development of such notion and end our talk with new results on two classes of algebras: (i) Iterated tilted algebras, jointly with D. Happel and L. Unger [3]; and (ii) Strongly simply connected algebras of polynomial growth, jointly with I. Assem and S. Trepode [2].

Keywords: representation dimension, Artin algebras

Mathematics Subject Classification 2010: 16G70, 16G20, 16E10

References

[1] M. Auslander, Representation dimension of artin algebras, Math. Notes, Queen Mary Col- lege, London (1971).

[2] I. Assem, F. U. Coelho and S. Trepode, The representation dimension of a class of tame algebras, Preprint, 2010.

[3] F. U. Coelho, D. Happel and L. Unger, Auslander generators for iterated tilted algebras, Proc. Amer. Math. Soc. 138 (2010), 1587-1593.

1Departamento de Matem´atica- IME, Universidade de S˜aoPaulo, Rua do Mat˜ao, 1010, Cidade Universit´aria,S˜aoPaulo, 05508-090, SP, Brasil [email protected] Contributed Talks 26

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Four-dimensional power-commutative real division algebras

Erik Darp¨o1

An algebra is said to be power-commutative if any subalgebra generated by a sin- gle element is commutative. While this is always true for associative algebras, there are many examples of non-associative algebras failing to satisfy the power-commutative condition, as well as power-commutative algebras that are not associative. However, most types of algebras that have been studied hitherto are power-commutative: al- ternative algebras, Lie algebras, non-commutative Jordan algebras, flexible algebras, power-associative algebras etc. In the talk, I shall present a classification of all four-dimensional power-commutative real division algebras, which naturally extends existing classifications in the power- associative and flexible cases.

1Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB, England [email protected] Contributed Talks 27

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Hyperbolic octonionic massive gravitoelectromagnetism with monopoles

S¨uleymanDemir1, Murat Tanı¸slı1

The non-associative hyperbolic octonions can be expressed in the sub-algebra of conic sedenions [1]. This type of octonions differs from classical octonions because of having both imaginary and hyperbolic basis. The gravitoelectromagnetism term states the theoretical analogy between the field equations in gravity and Maxwell’s equations in electromagnetism. In relevant literature, the possible relations between gravitation and electromagnetism have been formulated by using many hypercomplex number systems including biquaternions [2, 3], octonions [4], Clifford numbers [5] and sedenions [6]. In this study, after presenting the hyperbolic octonion formalism, a new formulation is proposed for the generalization of massive gravitoelectromagnetism with monopole terms. The gravitoelectromagnetic wave equation including Proca-type generalization and monopole terms is obtained in compact and elegant manner. Similarly, by using hyperbolic octonion formalism, the most generalized form of Klein-Gordon equation in gravity has been developed for the particle carrying gravitational masses.

Keywords: Hyperbolic octonion, gravitoelectromagnetism, Proca equation

Mathematics Subject Classification 2010: 78A25, 83C22, 83C50, 83E99

References

[1] K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions-further re- sults, Appl. Math. Comput. 84 (1997), 27–47.

[2] V. Majernik, Some astrophysical consequences of the extented Maxwell-like gravitational field equations, Astrophys. Space. Sci. 84 (1982), 191–204.

[3] S. Demir and M. Tanıs¸lı, Biquaternionic Proca-type generalization of gravity, Eur. Phys. J. Plus, 126 (2011), no. 5, 51–58.

[4] P. S. Bisht, S. Dangwal and O. P. S. Negi, Unified split octonion formulation of dyons, Int. J. Theor. Phys., 47 (2008) 2297–2313.

[5] S. Ulrych, Gravitoelectromagnetism in a complex Clifford algebra, Phys. Lett. B, 633, (2006) 631–635.

[6] J. Koplinger¨ , Gravity and electromagnetism on conic sedenions, Appl. Math. Comput. 188 (2007), no. 1, 948–953.

1Anadolu University, Science Faculty, Department of Physics, 26470, Eski¸sehir, Turkey [email protected], [email protected] Contributed Talks 28

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Octonions, lattices, and preferred parentheses

Geoffrey Dixon1

The octonions are the poster child for algebraic non-associativity, but although in general, x(yz) 6= (xy)z, and there seems to be no reason to prefer one placement of paren- theses over the other, a preference does arise when providing E8 lattices, represented over O, algebraic structures.

[email protected] Contributed Talks 29

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Structures of Malcev bialgebras on a simple non-Lie Malcev algebra§

Maxim Goncharov1

Lie bialgebras were introduced by Drinfeld [1] while studying the solutions to the classical Yang-Baxter equation. In [3], the definition of a bialgebra in the sense of Drinfeld (D-bialgebra) related with any variety of algebras was given. Malcev algebras were introduced by Malcev as tangent algebras for local analytic Moufang loops. The class of Malcev algebras generalizes the class of Lie algebras. In [2] some properties of Malcev bialgebras were studied. In particular, there were found conditions for a Malcev algebra with a comultiplication to be a Malcev bialgebra. In the present work we describe all Malcev bialgebra structures on a simple non-Lie Malcev algebra.

Keywords: Lie bialgebra, Malcev bialgebra, classical Yang-Baxter equation, nonas- sociative coalgebra, simple non-Lie Malcev algebra

Mathematics Subject Classification 2010: 17D10, 17B62,16T25

References

[1] V.G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equation, Sov, Math, Dokl. 27, (1983), 68–71.

[2] V.V. Vershinin, On Poisson-Malcev Structures, Acta Applicandae Mathematicae 75 (2003), 281–292.

[3] V.N. Zhelyabin, Jordan bialgebras and their relation to Lie bialgebras, Algebra and logic 36 (1997), 1-15.

1Sobolev Institute of Mathematics, Novosibirsk, Russia [email protected]

§The author was supported by Lavrent’ev Young Scientists Competition (No 43 on 04.02.2010), by the Program ”Development of scientific potential of the higher school” (project 2.1.1.10726), Russian Foundation for basic Research (Grant 09-01-00157-a, 11-01-00938-), the State Support Programs for the Leading Scientific Schools and the Young Scientists of the Russian Federation (Grands Nsh-3669.2010.1), and the Federal Targeting Programs (contracts 02.740.11.0429, 02.740.11.5191, 14.740.11.0346), the Integration Grant of the Siberian Division of the Russian Academy of Sciences (No. 97). Contributed Talks 30

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Multiplicative mappings of alternative rings

H. Guzzo Jr1, J. C. M. Ferreira2

Let R and S be arbitrary alternative rings. A mapping ϕ of R onto S is called a multiplicative isomorphism if ϕ is bijective and satisfies ϕ(xy) = ϕ(x)ϕ(y) for all x, y ∈ R. In this work we prove the following result, let R0 be an alternative ring containing a family Λ of idempotents. Suppose that R0 has a nonzero subring R which satisfies:

(i) For each e ∈ Λ, eR ⊆ R and Re ⊆ R;

(ii) If x ∈ R is such that xR = 0, then x = 0;

(iii) If x ∈ R is such that (eR)x = 0 (or e(Rx) = 0) for all e ∈ Λ, then x = 0 (and hence Rx = 0 implies x = 0);

(iv) For each e ∈ Λ and x ∈ R, if (exe)R(1 − e)
= 0 then exe = 0.

Then any multiplicative isomorphism ϕ of R onto an arbitrary alternative ring is addi- tive. As an application we generalize the famous Martindale’s result.

Keywords: Alternative rings, multiplicative mappings

Mathematics Subject Classification 2010: 17D05, 17A36

References

[1] F. Y. Lu and J. H. Xie, Multiplicative Mappings of Rings, Acta Mathematica Sinica 22 (2006), 1017–1020.

[2] W. S. Martindale III, When are multiplicative mappings additive?, Proc. Amer. Math. Soc. 21 (1969), 695–698.

[3] M. Slater, Prime alternative rings, I, Journal of Algebra 15 (1970), 229–243.

1Universidade de S˜aoPaulo, Departamento de Matem´atica,Rua do Mat˜ao,1010, 05508-090 - S˜aoPaulo, Brazil [email protected] 2Universidade Federal do ABC, Centro de Matem´atica,Computa¸c˜aoe Cogni¸c˜ao, Rua Santa Ad´elia,166, 09210-170 - Santo Andr´e,Brazil [email protected] Contributed Talks 31

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

On (co)homology of Hom-Leibniz algebras

Jos´eManuel Casas1 ¶, Manuel Avelino Insua1, Nat´aliaPacheco2

The Hom-Lie algebra structure was initially introduced in [2] motivated by examples of deformed Lie algebras coming from twisted discretizations of vector fields. Hom-Lie algebras are K-vector spaces endowed with a bilinear skew-symmetric bracket satisfying a Jacobi identity twisted by a map. When this map is the identity map, then the definition of Lie algebra is recovered. Following the generalization in [4] from Lie to Leibniz algebras, it is natural to describe same generalization in the framework of Hom-Lie algebras. In this way, the notion of (multiplicative) Hom-Leibniz algebra was firstly introduced in [3] as K-vector spaces L together with a linear map α : L → L, endowed with a bracket operation [−, −]: L ⊗ L → L which satisfies (α[x, y] = [α(x), α(y)]) the Hom-Leibniz identity

[α(x), [y, z]] = [[x, y], α(z)] − [[x, z], α(y)], for all x, y, z ∈ L (1)

Our goal in the present talk is to present the construction of (co)homology (co)chain complexes which allow us the computation of (co)homologies of multiplicative Hom- Leibniz algebras, its application to classify a particular type of abelian extensions and develop a theory of universal central extensions. We introduce the notions of representation and co-representation, which are the ade- quate coefficients for defining the co-chain and chain complexes from which we compute the cohomology and homology of a Hom-Leibniz algebra with coefficients. We establish the equivalence between the categories of representations and co-representations, we in- terpret low dimensional (co)homology vector spaces, in particular the second cohomology is the set of isomorphism classes of abelian α-extensions (the structure of representations induced by the extension doesn’t coincide with the previous action, but it coincides over the image of α), and we establish the relationships with Hom-Lie algebra (co)homology. We use the homology complex to construct universal central extensions of perfect Hom- Leibniz algebras. Our second homology appears as the kernel of the universal central extension. When we write the results obtained for the particular case α = Id, then we recover the corresponding results for Leibniz algebras in [4, 5].

Keywords: Hom-Leibniz algebra, (co)-representation, (co)homology, extensions

Mathematics Subject Classification 2010: 17A32, 16E40

References

[1] J. M. Casas, M. A. Insua and N. Pacheco, (Co)homology of Hom-Leibniz algebras, submitted to Linear Algebra and its Applications (2011).

¶(1) supported by MICINN, grant MTM2009-14464-C02-02 (European FEDER support in- cluded) and by Xunta de Galicia, grant Incite09 207 215 PR Contributed Talks 32

[2] J. T. Hartwing, D. Larson and S. D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2006), 314–361.

[3] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008), no. 2, 51–64.

[4] J.-L. Loday, Une version non commutative des alg`ebresde Lie: les alg`ebresde Leibniz, L0Enseignement Math´ematique 39 (1993), 269–292.

[5] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), 139–158.

1Dpto. Matem´aticaAplicada I, Univ. de Vigo, E. I. Forestal, 36005 Pontevedra, Spain [email protected], [email protected] 2IPCA, Dpto. de Ciˆencias,Campus do IPCA, Lugar do Ald˜ao,4750-810 Vila Frescainha, S. Martinho, Barcelos, Portugal [email protected] Contributed Talks 33

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Electromagnetism with complex octonions

Mustafa Emre Kansu1, Murat Tanı¸slı2, S¨uleymanDemir2

Octonions, which have both non-commutative and non-associative algebraic pro- perties, generate alternative division algebra [1]. In this study, the general information about complex octonions are introduced by using Cayley-Dickson multiplication rules. Maxwell’s equations, which are very important for electromagnetism, are formed without electric and magnetic charge densities. Local energy conservation equation, which have previously produced in biquaternionic form [2], is rearranged by using complex octonions. By defining complex octonionic differential operator and field equation, local energy conservation equation is rewritten. So, the electromagnetic energy density and flow are obtained [3].

Keywords: Octonion, Maxwell’s equations, electromagnetic energy

Mathematics Subject Classification 2010: 08Axx, 35Q61

References

[1] S. Okubo, Introduction to Octonion and Other Non-associative Algebras in Physics, Cam- bridge University Press, Cambridge, UK (1995); J. C. Baez, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145–205.

[2] M. Tanıs¸lı, Gauge transformation and electromagnetism with biquaternions, Europhysics Letters 74 (2006), issue 4, 569–573.

[3] J. D. Jackson, Classical Electrodynamics (Third Edition), John Wiley - Sons Inc., New York, U.S.A. (1999).

1Dumlupınar University, Faculty of Art and Science, Department of Physics, 43100, K¨utahya, Turkey [email protected] 2Anadolu University, Science Faculty, Department of Physics, 26470, Eski¸sehir, Turkey [email protected], [email protected] Contributed Talks 34

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Representation type of Jordan algebras

Iryna Kashuba1

This talk is devoted to the problem of the classification of indecomposable Jordan bimodules over finite dimensional Jordan algebras when squared radical is zero. Recall, that to each Jordan algebra corresponds the Lie algebra TKK(J) called Tits-Kantor- Koecher construction. Moreover the category of one-sided Jordan (bi)modules over J is equivalent to some subcategory S 1 of Lie modules over TKK(J). We describe the quivers 2 of S 1 and consequently give a characterization of Jordan algebras of finite and tame type 2 with respect to their one-sided bimodules. This is joint results with V. Serganova.

1University of S˜aoPaulo, Brazil [email protected] Contributed Talks 35

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Space-time symmetry groups derived from cubic and ternary algebras

Richard Kerner1

We investigate certain Z3-graded associative algebras with cubic Z3 invariant cons- titutive relations, introduced by one of us some time ago. The invariant forms on finite algebras of this type are given in the lowest-dimensional cases with two and three gene- rators. Various types of non-associative ternary algebras of such forms are investigated. We show how the Lorentz symmetry represented by the SL(2,C) group can be intro- duced without any notion of metric, just as the symmetry of Z3-graded cubic algebra and its constitutive relations. Its representation is found in terms of the Pauli matrices. The relationship of such algebraic constructions with quark states is also considered.

1Laboratoire de Physique Th´eoriquede la Mati`ereCondens´ee,Universit´ePierre- et-Marie-Curie - CNRS UMR 7600 Tour 23, 5-`eme´etage,Boˆıte121, 4, Place Jussieu, 75005 Paris, France [email protected] Contributed Talks 36

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Weyl groups of fine gradings

Mikhail Kotchetov1

L Given a grading Γ : U = g∈G Ug on a nonassociative algebra U by an abelian group G, we have two subgroups of Aut(U): the automorphisms that stabilize each component Ug (as a subspace) and the automorphisms that permute the components. By the Weyl group of Γ we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the classical extended Weyl group. A grading is called fine if it cannot be refined. We compute the Weyl groups of all fine gradings on matrix algebras, octonions, the Albert algebra, and the simple Lie algebras of types A, B, C, D (except D4), F4 and G2 over an algebraically closed field. The characteristic is assumed to be different from 2 in the case of the Albert algebra and the simple Lie algebras (also different from 3 for type G2). This is joint work with A. Elduque.

1Memorial University of Newfoundland, Canada [email protected] Contributed Talks 37

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011 The derived algebra of skew elements in a semiprime superalgebra with superinvolution Jes´usLaliena1

The study of the relationship between the structure of an associative algebra A and that of the Lie algebra A− was started by I. N. Herstein and W. E. Baxter in the fifties. Regarding superalgebras, this line of research was motivated by the classification of the fi- nite dimensional simple Lie superalgebras given by V. Kac in 1977, particularly the types given from simple associative superalgebras and from simple associative superalgebras with superinvolution. The Lie structure of prime superalgebras and simple superalge- bras without superinvolution was studied by F. Montaner [4] and S. Montgomery [5], and later C. G´omez-Ambrosi and I. Shestakov [1] investigated the Lie structure of the Lie superalgebra of skew elements, K, and also of its derived algebra, [K,K], when we have a simple associative superalgebra with superinvolution over a field of characteristic not 2. These results were extended to prime associative superalgebras with superinvo- lution [2]. In the case of semiprime associative superalgebras, there have been described in [3] the Lie ideals of the superalgebra K. In this talk we will deal with the case of the derived superalgebra [K,K].

Keywords: semiprime associative superalgebras, superinvolutions, skewsymmetric elements, Lie structure, derived superalgebra

Mathematics Subject Classification 2010: MSC 17B60, 16W50

References

[1] C. Gomez-Ambrosi´ and I. P. Shestakov, On the Lie structure of the skew elements of a simple superalgebras with superinvolution, J. Algebra 208 (1998), 43–71. [2] C. Gomez-Ambrosi,´ J. Laliena and I. P. Shestakov, On the Lie structure of the Skew Elements of a Prime Superalgbra sith Superinvolution, Comm. Algebra 28 (7) (2000), 3277–3291. [3] J. Laliena and S. Sacristan´ , Lie structure in semiprime superalgebras with superinvo- lution, J. Algebra (2007), . [4] F. Montaner, On the Lie structure of associative superalgebras, Comm. Algebra 26 (7) (1998), 2337–2349.

[5] S. Montgomery, Constructing simple Lie superalgebras from associative graded algebras, J. Algebra 195 (1997), 558–579.

1C/ Luis de Ulloa s/n, Edificio Vives, Departamento de Matem´aticasy Com- putaci´on,26004 Logro˜no,Espa˜na [email protected] Contributed Talks 38

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Abelian groups in the slice category of bialgebras

Sara Madariaga1

A representation of a loop Q in a variety of loops V is an abelian group in the slice category V/Q [1]. In this talk we discuss the slice category of bialgebras in connection with the representation theory of formal loops and Sabinin algebras.

References

[1] A. Dharwadker and J.D.H. Smith, Split extensions and representations of Moufang loops, Communications in Algebra 23 (1995), 4245–4255.

[2] J. Mostovoy and J. M. Perez-Izquierdo´ , Formal multiplications, bialgebras of distri- butions and nonassociative Lie theory. Transform. Groups 15 (2010), 625–653.

1Departamento de Matem´aticas y Computaci´on, Universidad de La Rioja, Logro˜no,Spain [email protected] Contributed Talks 39

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011 Finite growth representations of infinite Lie conformal algebras

Carina Boyallian1, Vanesa Meinardi1

In the present paper we classify all finite growth representations of all infinite rank conformal subalgebras of gcN that contain a Virasoro subalgebra. This problem reduces to the study of finite growth representations on the corresponding extended annihilation algebras, which are certain subalgebras of DN (see Ref. [8]). The main tools used here are the recent results [2] on the classification of quasifinite highest weight modules over the central extension of DN and some of its important subalgebras (Refs.[4, 6]).

Keywords: representations, weight

Mathematics Subject Classification 2010: 06B15, 17B10

References

[1] C. Boyallian, V.G. Kac and J.I. Liberatti, On the classification of subalgebras of CendN and gcN , Journal of Algebra 260 (2003) 32-63. [2] C. Boyallian, V.G. Kac, J. Liberati and C. Yan, Quasifinite highest weight modules over the Lie algebra of matrix differential operators on the circle, Journal of Math. Phys. 39 (1998), 2910-2928.

[3] V. G. Kac, Infinite-dimensional Lie Algebras, 3rd edition, Cambridge University Press, Cam- bridge, 1990.

[4] C. Boyallian and V.B. Meinardi, QHWM of the orthogonal type Lie subalgebra of the Lie algebra of matrix differential operators on the circle. Journal of Mahtematical Physics. 51 online (2010).

N [5] C. Boyallian and V.B. Meinardi, Quasifinite highest weight modules over W∞ . Journal Physics A., Math. Theor. 44 235201 (2011).

N [6] C. Boyallian and V.B. Meinardi, Representations of a sympletic type subalgebra of W∞ . Journal of Mahtematical Physics. 54 online (14 de junio del 2011.)

[7] V. G. Kac and A. Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Comm. Math. Phys. 157 (1993), 429-457.

[8] V.G. Kac, Vertex Algebras for beginners, University Lecture Series, 10 (American Matema- tical Society, Providence, RI, 1996), Second edition 1998.

1FaMAF, Facultad de Matem´aticaAstronom´ıa y F´ısica, 5000 C´ordoba, Ar- gentina. [email protected], [email protected]. Contributed Talks 40

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Alternative loop algebras over finite fields

C´esarPolcino Milies1

We determine the structure of the semisimple group algebra of certain groups over those finite fields that produce the fewest number of simple components. We use this information to obtain similar information about the loop algebras of all indecomposable RA loops and observe that the isomorphism problem has a negative answer for alternative loop algebras over finite fields. This is joint work with R. Ferraz and E.G. Goodaire.

1Instituto de Matem´aticae Estat´ıstica,Universidade de S˜aoPaulo, Caixa Postal 66.281, CEP 05314-970, S˜aoPaulo SP, Brasil [email protected] Contributed Talks 41

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

On generalized Jordan triple systems of second order and extended Dynkin diagrams

Daniel Mondoc1

In this talk we give the correspondence between simple complex generalized Jor- dan triple systems of second order and extended Dynkin diagrams (joint work with N. Kamiya).

1Centre for Mathematical Sciences, Lund University, Sweden [email protected] Contributed Talks 42

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

A series of algebras generalizing the octonions and Hurwitz-Radon identity

Sophie Morier-Genoud1, Valentin Ovsienko2

We study non-associative twisted group algebras with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their proper- ties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present an application of the constructed algebras to the celebrated Hurwitz problem on square identities.

Keywords: Graded commutative algebras, non-associative algebras, Clifford alge- bras, octonions, square identities, Hurwitz-Radon function, code loop, Parker loop

References

[1] H. Albuquerque and S. Majid, Quasialgebra structure of the octonions, J. Algebra 220 (1999), no. 1, 188–224.

[2] A. Elduque, Gradings on octonions, J. Algebra 207 (1998), no. 1, 342–354.

[3] S. Morier-Genoud and V. Ovsienko, Simple graded commutative algebras, J. Algebra 323 (2010), no. 6, 1649–1664.

1Universit´eParis Diderot Paris 7, UFR de math´ematiquescase 7012, 75205 Paris Cedex 13, France [email protected] 2CNRS, Institut Camille Jordan, Universit´eClaude Bernard Lyon 1, 43 boule- vard du 11 novembre 1918, 69622 Villeurbanne cedex, France [email protected] Contributed Talks 43

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Alternative twisted tensor products and Cayley algebras

Helena Albuquerque1, Florin Panaite2

We present a construction from [1] called alternative twisted tensor products for not necessarily associative algebras, which arose as a common generalization of several different constructions: the Cayley-Dickson process, the Clifford process and the twisted tensor product of two associative algebras, one of them being commutative. Some very basic facts concerning the Cayley-Dickson process (the equivalence between the two different formulations of it and the lifting of the involution) are particular cases of general results about alternative twisted tensor products of algebras. As a class of examples of alternative twisted tensor products, we present a tripling process for an algebra endowed with a strong involution, containing the Cayley-Dickson doubling as a subalgebra and sharing some of its basic properties.

Keywords: Cayley algebras, twisted tensor products

Mathematics Subject Classification 2010: 17A01

References

[1] H. Albuquerque and F. Panaite, Alternative twisted tensor products and Cayley al- gebras, Comm. Algebra 39 (2011), no. 2, 686–700.

1CMUC, Departamento de Matem´atica, Universidade de Coimbra, 3001-454 Coimbra, Portugal [email protected] 2Institute of Mathematics of the Romanian Academy, P.O.Box 1-764, RO- 014700, Bucharest, Romania [email protected] Contributed Talks 44

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011 Code loops: automorphisms and representations

Rosemary Miguel Pires1, Alexandre Grichkov2

We present a introduction about code loops based on the paper of Robert L. Griess, [2]. We characterized code loops of rank n as homomorphic image of certain free Moufang loops with n generators and we introduce the concept of characteristic vectors associated with code loops. With the results of the theory studied, we classify all the code loops of rank 3, we find all the groups of outer automorphisms of these loops and finally, we determine all their basic representations. This work was obtained as part of the PhD thesis under the supervisor Alexandre Grichkov.

Keywords: Code Loops, even codes, characteristic vectors, outer automorphisms, representations.

Mathematics Subject Classification 2010: MSC20-02, MSC20N05.

References

[1] R. M. Pires, Loops de c´odigo:automorfismos e representa¸c˜oes, IME-USP (2011) - Tese de Doutorado.

[2] R. L. Griess Jr., Code loops, J.Algebra 100 (1986), 224–234.

[3] R. H. Bruck, A survey of binary systems, Springer-Verlag (1958).

[4] O. Chein and E. G. Goodaire, Moufang Loops with a Unique Nonidentity Commutator (Associator, Square), J. Algebra 130 (1990), 369–384.

[5] H. O. Pflugfelder, Quasigroups and Loops: An Introduction, Berlin:Heldermann (1990).

[6] O. Chein and H. O. Pflugfelder, Quasigroups and Loops: Theory and Applications, Berlin:Heldermann (1990).

[7] E. G. Goodaire, E. Jaspers and C. Polcino, Alternative Loop Rings, North Holland Math 184 (1996).

1Instituto de CiˆenciasExatas (ICEx), Departamento de Matem´atica,Universi- dade Federal Fluminense (UFF), Volta Redonda-RJ, Brasil [email protected], [email protected] 2Instituto de Matem´aticae Estat´ıstica (IME), Universidade de S˜aoPaulo (USP), S˜aoPaulo-SP, Brasil [email protected], [email protected] Contributed Talks 45

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Moufang loops and their automorphisms

Liudmila Sabinina1

I would like to speak about Moufang loops in which inner mappings are automor- phisms. Main result of the talk is a theorem proved jointly with A. Grishkov and P. Plaumann. Let Fn be the free group with a basis X = {x1, . . . , xn} and let Cn be the free commutative Moufang loop with a basis Y = {y1, . . . , yn}. Put zi = (xi, yi) ∈ Fn × Cn for 1 ≤ i ≤ n and denote by An the subloop of Fn × Cn generated by Z = {z1, . . . , zn}.

Theorem. The loop An is a free automorphic Moufang loop of rank n.

Some corollaries of the theorem will be given.

Keywords: Automorphic Moufang loops

Mathematics Subject Classification 2010: 20N05

1Facultad de Ciencias, Universidad Aut´onomadel Estado de Morelos, Avenida Universidad 1001, 62209 Cuernavaca, Morelos, M´exico [email protected] Contributed Talks 46

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Heisenberg Lie superalgebras and its invariant superorthogonal and supersymplectic forms

M.C. Rodr´ıguez-Vallarte1, G. Salgado1, O.A. S´anchez-Valenzuela2

Finite-dimensional complex Lie superalgebras of Heisenberg type obtained from a given Z2-homogeneous supersymplectic form defined on a vector superspace, are classi- fied up to isomorphism. Those arising from even supersymplectic forms, have an ordinary Heisenberg Lie algebra as its underlying even subspace, whereas those arising from odd supersymplectic forms get based on an abelian Lie algebras. The question of whether this sort of Heisenberg Lie superalgebras do or not do support a given invariant super- geometric structure is addressed, and it is found that none of them do. It is proved, however, that 1-dimensional extensions by appropriate Z2-homogeneous derivations do. Such “appropriate” derivations are characterized, and their invariant supergeometric structures are fully described [1].

Keywords: Heisenberg Lie superalgebras, ad-invariant supersymmetric bilinear forms

Mathematics Subject Classification 2010: 17B30, 17B70, 81R05

References

[1] M.C. Rofr´ıguez-Vallarte, G. Salgado and O.A. Sanchez-Valenzuela´ , Heisen- berg Lie superalgebras and its invariant superorthogonal and supersymplectic forms, Jour- nal Algebra 332 (2011), no. 1, 71–86.

1Facultad de Ciencias, UASLP, Av. Salvador Nava s/n, Zona Universitaria, C.P. 78290, SLP, M´exico [email protected], [email protected], [email protected] 2CIMAT, Apdo. Postal 402, C.P. 36000, Guanjuato, Gto., M´exico [email protected], [email protected] Contributed Talks 47

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Leibniz triple systems

Murray R. Bremner1, Juana S´anchez-Ortega2

We introduce a new variety of triple systems, which are related to Leibniz algebras in the same way that Lie algebras are connected with Lie triple systems. More precisely, we show that the defining identities for these structures are satisfied by the iterated bracket in a Leibniz algebra and the (permuted) associator in a Jordan dialgebra. Thereby, we call them “Leibniz triple systems”, and we construct their universal Leibniz envelopes.

1Department of Mathematics and Statistics, University of Saskatchewan 106 Wiggins Road (McLean Hall) Saskatoon, SK, S7N 5E6 Canada [email protected] 2Department of Algebra, Geometry and Topology, University of M´alaga,Spain [email protected] Contributed Talks 48

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Exact sequences in the Borel-Schur algebra

Ana Paula Santana1

Schur algebras are fundamental tools in the representation theory of the general linear group and of the symmetric group. In this talk I explain how we can use the induction functor from the Borel-Schur algebra to the Schur algebra, over a commutative ring, to construct projective resolutions of Weyl modules. This is joint work with I. Yudin.

Keywords: Schur algebras, projective resolutions

Mathematics Subject Classification 2010: 20G43, 16E05

References

[1] A. P. Santana and I. Yudin, Characteristic free resolutions for Weyl and Specht modules, http://arxiv.org/abs/1104.1959

1CMUC, Departamento de Matem´atica, Universidade de Coimbra, 3001-454 Coimbra, Portugal [email protected] Contributed Talks 49

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

On linear equations over free Lie algebras

Ralph St¨ohr1

We investigate equations of the form

[x1, u1] + [x2, u2] + ··· + [xk, uk] = 0 over a free Lie algebra L. In the case where the coefficients u1, u2, . . . , uk are free gene- rators of L, we generalize a number of earlier results on equations with two variables obtained in [1] to equations with an arbitrary number of indeterminates. Our main result is a complete description of the multilinear fine homogeneous component of the solution space. This is joint work with Alaa Altassan.

References

[1] V.N. Remeslennikov and Ralph Stohr¨ , The equation [x, u] + [y, v] = 0 in free Lie algebras, Internat. J. Algebra Comput. 17 (2007), no. 5/6, 1165–1187.

1University of Manchester, UK [email protected], web:http://www.maths.manchester.ac.uk/ rs/ Contributed Talks 50

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Non associative algebra associated to genetics and Jordan algebra

Osamu Suzuki1

A non associative algebra associated to genetics is introduced and it is proved that it is a Jordan algebra. By this fact we see that Jordan algebra is important not only in physics but in biology. Following Prof. A. Michali, we introduce a gametic algebra and Mendel algebra as a special case. The Mendel algebra is defined on a n-dimensional vector space R[S1,S2, ..., Sn] by the following product: 1 S ∗ S = (S + S ) i j 2 i j Then we obtain a commutative and non associative algebra. We may understand the product is an algebraic description of separation law in Mendel’s law. Then we can prove the following theorem:

Theorem A Mendel algebra is a Jordan algebra.

Proof is done by a direct calculation. From the fact that specialization of Mendel algebra and the tensor product is Mendel algebra, we see that the family of Jordan algebras above obtained are closed under these operations. We notice that the specialization and tensor product are mathematical description of mating process and independent law in genetics.

Remark (1)It is proved that a Mendel algebra is at the same time flexible algebra; (2)We can obtain a family of non associative algebras from Mendel algebras and we can discuss a hierarchy structure of these algebras.

Keywords: Mendelian genetics, Mendel algebra, Jordan algebra

References

[1] A. Micali and Ph. Revoy, Sur la algebres gametiques , Proc. Edinburgh Math. Soc. 43 (1986), no. 1,2, 187–197.

1Department of Computer Sciences and System Analysis, Setagaya-ku, Sakura- jjousui 3-25-40,Tokyo, Japan [email protected] Contributed Talks 51

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Algebraic theory of DNA recombination

Sergei R. Sverchkov1

In this presentation we give a brief introduction of the algebraic theory for DNA recombination within of intersection of two areas: theoretical genetics and theory of non- associative algebras. We investigate the structure and representation of n-ary algebras arising from DNA recombination, where n is a number of DNA segments participating in recombination. For the algebraic formalization of DNA recombination we will use the basic idea of the formalization of algebra of observables in quantum mechanics, which was first realized by Pascual Jordan. We proved that every identity satisfied by n-ary DNA recombination, with no restriction on the degree, is consequence of n-ary commutativity and a single n-ary identity of the degree 3n-2. It solves the well- known open problem in the theory of n-ary intermolecular recombination.

Keywords: Jordan algebras, DNA recombination, theoretical genetics, theory of n- ary algebras.

1Novosibirsk State University, Russia [email protected] Contributed Talks 52

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Algebras with involution and their identitiesk

Irina Sviridova1

We will consider associative non-commutative algebras over a field F of zero characte- ristic. Anti-automorphism of order 2 of an algebra is called involution of this algebra. If we will fix an involution ∗ on an associative F -algebra A then it can be considered as a new unary operation on A. We can study identities of the algebra A in this new signature. These identities are called identities with involution of A (∗-identities). It is well-known (see for example [1]) that all ∗-identities of an F -algebra with involution form two-side verbal ideal of the free associative algebra with involution closed under the action of involution. If this ideal is non-zero then the algebra A is also PI-algebra (it satisfies the non-trivial ordinary polynomial identity without involution). It is well-known also that any algebra A with involution ∗ as a vector space over F is the direct sum A = S + T of the subspace S = {a ∈ A|a∗ = a} of symmetric elements (satisfying equality a∗ = a) and of the subspace T = {a ∈ A|a∗ = −a} of skew- symmetric elements (satisfying a∗ = −a). Then the vector subspace S with operation x◦y = xy +yx forms Jordan algebra, and the subspace T with operation [x, y] = xy −yx forms Lie algebra. The next relation between finitely generated associative algebras with involution and finite dimensional algebras with involution over a field of zero characteristic is obtained. Theorem. For any finitely generated associative algebra A with involution over a field F of zero characteristic there exists a finite dimensional over F algebra C with involution which has exactly the same ∗-identities as the algebra A.

Keywords: Algebras with involution, identities with involution

Mathematics Subject Classification 2010: 16R50, 16W10

References

[1] A. Giambruno and M. Zaicev, Polynomial identities and asymptotic methods. Mathe- matical Surveys and Monographs 122 American Mathematical Society, Providence, RI, 2005.

1Departamento de Matem´atica,Universidade de Bras´ılia, 70910-900 Bras´ılia, DF, Brazil [email protected]

kSupported by CNPq, and by CNPq-FAPDF PRONEX grant 2009/00091-0 (193.000.580/2009) Contributed Talks 53

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

A generalization of left invariant vector fields and Lie algebras generated by them

M.R. Molaei Taherabadi1

In this talk we consider top spaces [1] as a class of completely regular semigroups with the C∞ structures which are compatible with their product and their inverse mappings. We will present a non-associative algebra by using of a top space with a finite number of identities [2].

Keywords: Top space, left invariant vector field

Mathematics Subject Classification 2010: 17B45, 17B66

References

[1] M.R. Molaei, Top spaces, Journal of Interdisciplinary Mathematics 7 (2004), no. 2, 173– 181.

[2] M.R. Molaei and M.R. Farhangdoost, Lie algebras of a class of top spaces, Balkan Journal of Geometry and Its Applications 14 (2009), no. 1, 46–51.

1Mahani Mathematical Research Center, and Department of Mathematics, Uni- versity of Kerman (Shahid Bahonar), Kerman, Iran [email protected] Contributed Talks 54

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Maxwell equations for magnetic sources and massive photon in octonion algebra

Murat Tanı¸slı1, Bernard Jancewicz2

In this study, the octonion algebra and its general properties are defined by the Cayley-Dickson’s multiplication rules for octonion units. The field equations, potential and Maxwell equations for electromagnetism are investigated with the octonionic equa- tions and these equations can be compared with their vectorial representations. Also, the potential and wave equations for fields with sources are given. Then, using Maxwell equations, a Lorenz like condition is suggested for electromagnetism. Because the exis- ting equations give the well known Lorenz condition for magnetic monopole and source. These equations include the photon mass.

Keywords: complex octonion, Lorenz condition, electromagnetism, magnetic monopole, photon mass, Proca-Maxwell equations

Mathematics Subject Classification 2010: 35Q61, 08Axx

References

[1] V. Majern´ık, Gen. Rel. Grav. 35, 2003, 1833.

[2] V. Majern´ıkand M. Nagy, Lett. Nuovo Cimento 16, 1976, 265.

[3] V. Majern´ık, Adv. Appl. Clifford Algebras 9, 1999, 119.

[4] A. Gamba, Il Nuovo Cimento 111A, 1998, 293.

[5] M. Tanı¸slıand G. Ozg¨ur,¨ Acta Phys. Slovaca, 53,2003,243.

[6] M. Tanı¸slı, Europhysics Letters, 74(4), 2006, 569.

[7] T. Tolan, K. Ozda¸sand¨ M. Tanı¸slı, Il Nuovo Cimento B, 121(1), 2006, 43-55.

[8] N. Candemir, M. Tanı¸slı,K. Ozda¸sand¨ S. Demir, Zeitschrift f¨urNaturforschung A, 63(a), 2008, 15-18.

[9] F. Toppan, hep-th/0503210v1

[10] B.A. Bernevig, J.P. Hu, N. Toumbas and S.C. Zhang, Phys. Rev. Lett. 91, 2003, 23.

[11] K.J. Vlaenderen and A. Waser, Electrodynamics with the Scalar Field, http://www.info.global-scaling-verein.de/Documents/Electrodynamics-WithThe ScalarField03.pdf

[12] O.P.S. Negi, S. Bisht and P.S. Bisht, Nuovo Cimento B, 113,1998,1449.

[13] B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics, (World Scientific), 1988. Contributed Talks 55

[14] S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, (Oxford University Press, New York), 1995.

[15] J. Daboul and R. Delbourgo, J. Math. Phys. 40, 1999,4134.

[16] Y. Tian, Adv. Appl. Clifford Algebras 10, 2000, 61.

[17] Okubo S., Introduction to Octonion and Other Non-Associative Algebras in Physics, (Cam- bridge University Press, Cambridge), 1995.

[18] H.L. Carrion, M. Rojas and F. Toppan, hep-th/0302113v1.

[19] J.D. Jackson, Classical Electrodynamics, (John Wiley and Sons, New York), 1999.

1Science Faculty, Department of Physics, Anadolu University, 26470 Eski¸sehir, Turkey [email protected] 2Intitute of Theoretical Physics, University of Wroclaw, Wroclaw, Poland [email protected] Contributed Talks 56

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

A capacity 2 theorem for graded Jordan systems

Maribel Toc´on1, Erhard Neher2

The Osborn Capacity 2 Theorem for Jordan algebras tackles the hardest case in the classification of Jordan algebras with finite capacity, namely that of capacity two, i.e., the identity element is a sum of two orthogonal division idempotents. Removing the division condition on the diagonal Peirce spaces leads to the definition of triangulated Jordan algebras: The Jordan algebra has two complementary orthogonal idempotents which are connected by an invertible element in the off-diagonal Peirce space. The extension of this notion to triangulated Jordan systems (Jordan triple systems and Jordan pairs) is immediate. A Jordan system is called graded-simple-triangulated if it is triangulated and graded-simple with respect to a Λ-grading compatible with its triangularization, for Λ an abelian group. These are the Jordan systems which, via the Tits-Kantor-Koecher construction, correspond to B2-graded Lie algebras that are graded-simple with respect to a second compatible grading by Λ, a case of interest to the theory of extended affine Lie algebras. In this talk, we present a classification of graded-simple-triangulated Jordan systems with division-graded diagonal Peirce spaces, based on the classification of graded-simple- triangulated Jordan systems ([NT]), with the restriction that Λ be torsion-free. As a consequence we extend the Osborn Capacity 2 Theorem to the graded setting.

Keywords: Jordan structures covered by a triangle, coordinatization

Mathematics Subject Classification 2010: 17C10, 17C27

References

[1] E. Neher and M. Tocon´ , Lie tori of type B2 and graded-simple Jordan structures covered by a triangle, preprint available at http://homepage.uibk.ac.at./ ∼c70202/jordan/index.html.

1Departamento de Estad´ıstica e Investigaci´on Operativa, Universidad de C´ordoba,Puerta Nueva s/n, C´ordoba, 14071, Spain [email protected] 2Department of Mathematics and Statistics, University of Ottawa, Ottawa, On- tario K1N 6N5, Canada [email protected] Contributed Talks 57

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011 Strongly stable automorphisms of the category of free linear algebras Arkadt Tsurkov1

This research is motivated by universal algebraic geometry. We consider in universal algebraic geometry the some variety of universal algebras Θ and algebras H ∈ Θ from this variety. We take a “set of equations” T ⊂ F × F in some finitely generated free algebra F of the arbitrary variety of universal algebras Θ and we “resolve” these equations in Hom (F,H), where H ∈ Θ. The set Hom (F,H) serves as an “affine space over the 0 algebra H”. Denote by TH the set {µ ∈ Hom (F,H) | T ⊂ ker µ}. This is the set of all solutions of the set of equations T . For every set of “points” R of the affine space 0 T Hom (F,H) we consider a congruence of equations defined by this set: RH = ker µ. µ∈R 00 For every set of equations T we consider its algebraic closure TH in respect to the algebra 00 H. A set T ⊂ F × F is called H-closed if T = TH . One of the central questions of the theory is the following: When do two algebras have the same geometry? What does it mean that the two algebras have the same geometry? The notion of geometric equivalence of algebras gives a sort of answer to this question. Algebras H1 and H2 are called geometrically equivalent if and only if the H1-closed sets coincide with the H2-closed sets. The notion of automorphic equivalence is a generalization of the first notion. Algebras H1 and H2 are called automorphicaly equivalent if and only if the H1-closed sets coincide with the H2-closed sets after some “changing of coordinates”. We can detect the difference between geometric and automorphic equivalence of algebras of the variety Θ by researching of the automorphisms of the category Θ0 of the finitely generated free algebras of the variety Θ. By [?] the automorphic equivalence of algebras provided by inner automorphism degenerated to the geometric equivalence. So the various differences between geometric and automorphic equivalence of algebras can be found in the variety Θ if the factor group A/Y is big. Hear A is the group of all automorphisms of the category Θ0, Y is a normal subgroup of all inner automorphisms of the category Θ0. We have [1, Theorem 2] this decomposition A = YS. Hear S is a group of strongly stable automorphisms of the category Θ0. So A/Y =∼ S/S ∩ Y. In [1] the variety of all Lie algebras and the variety of all associative algebras over the field k were studied. The group A/Y in both these case is small, in particular, if the field k has not nontrivial automorphisms. We consider in this paper the variety of all linear algebras over the infinite field k which has not nontrivial automorphisms. The ∼ ∗ group A/Y = S/S ∩ Y is isomorphic to the group G/k I2, where G is the group of the  a b  regular 2 × 2 matrices over field k, which have a form where a, b ∈ k. So the b a group A/Y is enough rich in this case.

Keywords: Linear algebras, automorphisms of categories, universal algebraic geome- try Contributed Talks 58

Mathematics Subject Classification 2010: 17A50, 18A22

References

[1] B. Plotkin and G. Zhitomirski, On automorphisms of categories of free algebras of some varieties, Journal of Algebra 306 (2006), no. 2, 344 – 367.

1Institute of Mathematics and Statistics, University of S˜aoPaulo, Rua do Mat˜ao, 1010, 05508-090, Brazil [email protected] Contributed Talks 59

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Normalized bar resolution and Schur algebra

Ivan Yudin1

One of the basic results in Representation Theory of Schur algebras and of symmetric groups are Weyl character formula and Frobenius determinant formula, respectively. In this talk I will explain how normalized bar resolution for Borel-Schur algebra can be used to obtain resolutions of Weyl and Specht modules that “realize” the above mentioned formulae. This is joint work with A. P. Santana.

Keywords: Schur algebras, projective resolutions

Mathematics Subject Classification 2010: 20G43, 16E05

References

[1] A. P. Santana, I. Yudin, Characteristic free resolutions for Weyl and Specht modules, http://arxiv.org/abs/1104.1959

1CMUC, Departamento de Matem´atica, Universidade de Coimbra, 3001-454 Coimbra, Portugal [email protected] Contributed Talks 60

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

A commutative 2-cocycles approach to classification of simple Novikov algebras

Pasha Zusmanovich1

Finite-dimensional simple Novikov algebras were classified by Zelmanov in charac- teristic 0 and by Osborn and Xu in characteristic p. We will outline an alternative approach to these classifications, based on the notion of commutative 2-cocycles of Lie algebras – bilinear maps satisfying the usual cocycle equation, only being symmetric instead of skew-symmetric.

1Depart. of Mathematics, Tallinn University of Technology, Tallinn, Estonia [email protected] Posters 61

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

On finite-dimensional absolute-valued algebras satisfying (xp, xq, xr) = 0

M. I. Ram´ırezAlvarez1

By means of principal isotopes lH(a, b) of the algebra lH [Ra 99] we give an exhaustive and not repetitive description of all 4-dimensional absolute-valued algebras satisfying (xp, xq, xr) = 0 for fixed integers p, q, r ∈ {1, 2}. For such an algebras the number N(p, q, r) of isomorphism classes is either 2 or 3, or is infinite. Concretely

1. N(1, 1, 1) = N(1, 1, 2) = N(1, 2, 1) = N(2, 1, 1) = 2,

2. N(1, 2, 2) = N(2, 2, 1) = 3,

3. N(2, 1, 2) = N(2, 2, 2) = ∞.

Besides, each one of the above algebras contains 2-dimensional subalgebras. However, the problem in dimension 8 is far from being completely solved. In fact, there are 8-dimensional absolute-valued algebras, containing no 4-dimensional subalgebras, satis- fying (x2, x, x2) = (x2, x2, x2) = 0.

1Universidad de Almer´ıa,Spain [email protected] Posters 62

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

On purely real algebras Lie on a Hilbert space

Farkhad Arzikulov1

In the given work we investigated a real von Neumann algebra R on a Hilbert space such that R ∩ iR = {0} and an ultraweakly closed real algebra Lie L of sky-adjoint operators on a Hilbert space such that R(L) ∩ iR(L) = {0} for the real von Neumann algebra R(L), generated by L in B(H). These algebras were called a purely real von Neumann algebra and a ultraweakly closed purely real algebra Lie. It is proved that every ¯ ¯ maximal purely real algebra Lie on a Hilbert space H is isomorphic to B(HR)k ⊕B(HH)k for some Hilbert spaces HR and HH on R and H (quaternions) correspondingly such that ∗ B(H) = B(HF ) + iB(HF ), where F = R, H, B(HF )k = {x ∈ B(HF ): x = −x}, and ¯ ¯ ¯ ¯ ⊥ ¯ ¯ ⊥ for such Hilbert subspaces HR and HH, that HR = HR ⊕ H , HH = HH ⊕ H , the ¯ ¯ R H identity elements of the algebras B(HR) and B(HH) are mutually orthogonal and their sum is equal to the identity element of B(H). Also it was proved that any ultraweakly closed purely real abelian algebra Lie L of sky-adjoint operators on a Hilbert space is isomorphic to the algebra Lie of all continuous  0 1  functions f : X → on a hyperstonean compact X. −1 0 R

Keywords: real von Neumann algebra, Algebra Li

Mathematics Subject Classification 2010: 17B10, 46L10

1Institute of Mathematics and Information technologies, Tashkent, Uzbekistan [email protected], [email protected] Posters 63

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

On homogeneous symmetric associative superalgebras

Imen Ayadi1, Sa¨ıdBenayadi1

The main purpose of this work is to study associative superalgebras endowed with symmetric, even or odd, non-degenerate associative bilinear forms, i.e the so called symmetric homogeneous structures. First we show that any associative superalgebra with a non null product cannot admit simultaneously even symmetric and odd symmetric structures. Next, we prove that, unlike the Lie case, all simple associative superalgebras admit either even symmetric or odd symmetric structures by giving explicitly in every case the homogeneous symmetric structures. Second, we introduce some notions of generalized double extensions in order to give inductive descriptions of even symmetric associative superalgebras and odd symmetric associative superalgebras. Finally, we give an other interesting description of odd symmetric associative superalgebras whose even parts are semi-simple bimodules with no use of doubles extensions.

Keywords: Simple associative superalgebras, associative superalgebras, homogeneous symmetric structures, generalized double extension, inductive description

1Laboratoire de Math´ematiqueset Applications de Metz, CNRS UMR 7122, Universit´ePaul Verlaine-Metz, Ile du Saulcy, F-57045 Metz cedex 1, France [email protected], [email protected] Posters 64

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

On some Hurwitz reduced algebras of a ternary quaternion algebra

Patr´ıciaD. Beites1, Alejandro P. Nicol´as2

The definition of n-Lie algebra (n ≥ 2) was introduced in [2], being the most rele- vant and natural known generalization of the Lie algebra concept. Following the actual tendency, we use the term n-ary Filippov algebra instead of n-Lie algebra. Let F be a field such that ch(F ) 6= 2. Consider the 4-dimensional ternary Filippov algebra A4 over F equipped with a bilinear, symmetric and non-degenerate form (· , ·), and the canonical basis {e1, e2, e3, e4}. The multiplication table of A4 is the subsequent one

i [e1,..., eˆi, . . . , e4] = (−1) ei, i ∈ {1,..., 4}, (1) wheree ˆi means that ei is omitted. The remaining products are zero or obtained from (1) by anticommutativity. We define a new multiplication on the underlying vector space of A4,

{x, y, z} := −(y, z)x + (x, z)y − (x, y)z + [x, y, z], and denote the obtained ternary quaternion algebra by A, as in [1]. In this reference, simplicity, identities of levels 1 and 2, and derivations of A were studied. Furthermore, envelopes for ternary Filippov algebras were obtained. In the present work we consider some reduced algebras of A and its identities of level less or equal to 2. We focus on those that are Hurwitz algebras, namely (A, •2,1) where x •2,1 y = {x, e1, y}. Moreover, being a, b 6= 0, conditions for the existence of non-zero solutions of the linear equation

a •2,1 x = x •2,1 b are established. Under these, using matrix representations inspired by some results in [3], the general solution is determined.

Keywords: Filippov algebra, ternary quaternion algebra, reduced algebra, composi- tion algebra, matrix representation, linear equation

Mathematics Subject Classification 2010: 17A32, 17A75

References

[1] P. D. Beites, A. P. Nicolas,´ A. P. Pozhidaev and P. Saraiva, On identities of a ternary quaternion algebra, Comm. Algebra 39 (2011), no. 3, 830–842.

[2] V. T. Filippov, n-Lie Algebras, Siberian Math. J. 26 (1985), no. 6, 879–891. Posters 65

[3] Y. Tian, Matrix representations of octonions and their applications, Adv. Appl. Clifford Algebr. 10 (2000), no. 1, 61–90.

1Centro de Matem´aticaand Departamento de Matem´atica,Universidade da Beira Interior, Rua Marquˆesd’Avila´ e Bolama, 6201-001 Covilh˜a,Portugal [email protected] 2Departamento de Matem´aticaAplicada, Universidad de Valladolid, Campus Duques de Soria s/n, 42004 Soria, Espa˜na [email protected] Posters 66

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

On split Lie superalgebras

Antonio J. Calder´onMart´ın1, Jos´eM. S´anchez Delgado1

We study the structure of arbitrary split Lie superalgebras. We show that any of such P superalgebras L is of the form L = U + Ij with U a subspace of the abelian (graded) j subalgebra H and any Ij a well described (graded) ideal of L satisfying [Ij,Ik] = 0 if j 6= k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal (graded) ideals, each one being a simple split Lie superalgebra.

Keywords: Infinite dimensional Lie superalgebras, structure theory

Mathematics Subject Classification 2010: 17B65, 17B05, 17B20

References

[1] C. Boyallian and V. Meinardi, Quasifinite representations of the Lie superalgebra of quantum pseudodifferential operators. J. Math. Phys. 49 (2008), no. 2, 023505, 13 pp.

[2] A.J. Calderon´ , On split Lie algebras with symmetric root systems. Proc. Indian. Acad. Sci, Math. Sci. 118 (2008), 351-356.

[3] A.J. Calderon´ , On simple split Lie triple systems. Algebr. Represent. Theory 12 (2009), 401-415.

[4] K. Iohara and Y. Koga, Note on spin modules associated to Z-graded Lie superalgebras. J. Math. Phys. 50 (2009), no. 10, 103508, 9 pp.

[5] E. Poletaeva, Embedding of the Lie superalgebra D(2, 1; α) into the Lie superalgebra of pseudodifferential symbols on S1|2. J. Math. Phys. 48 (2007), no. 10, 103504, 17 pp.

[6] N. Stumme, The structure of Locally Finite Split Lie Algebras, J. Algebra. 220 (1999), 664-693.

1Departamento de Matem´aticas, Universidad de C´adiz, 11510 Puerto Real, C´adiz,Spain [email protected], [email protected] Posters 67

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Gradings on e6

Cristina Draper1

A model of each fine grading on e6 is given. There are 14 of such fine gradings, 5 of them are inner. Not every outer fine grading comes from extending to e6 a grading either on c4 or on f4, there is one fine grading produced by a quasitorus with outer automorphisms but without order two outer automorphisms.

1Universidad de M´alaga,Spain [email protected] Posters 68

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux

Dimitar Grantcharov1, Ji Hye Jung2, Seok-Jin Kang2, Masaki Kashiwara3, Myungho Kim2

We give an explicit combinatorial realization of the crystal B(λ) for an irreducible highest weight Uq(q(n))-module V (λ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algo- rithms of decomposing the tensor product of Uq(q(n))-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.

Keywords: Quantum queer superalgebras, crystal bases, odd Kashiwara operators, semistandard decomposition tableaux, shifted Littlewood-Richardson coefficients

Mathematics Subject Classification 2010: 17B37, 81R50

1Department of Mathematics University of Texas at Arlington, Arlington, TX 76021, USA [email protected] 2Department of Mathematical Sciences, Seoul National University, Seoul 151- 747, Korea [email protected], [email protected], [email protected] 3Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606- 8502, Japan & Department of Mathematical Sciences, Seoul National University, Seoul 151- 747, Korea [email protected] Posters 69

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Crystal bases for the quantum queer superalgebra

Dimitar Grantcharov1, Ji Hye Jung2, Seok-Jin Kang2, Masaki Kashiwara3, Myungho Kim2

We develop the crystal basis theory for the quantum queer superalgebra Uq(q(n)). We define the notion of crystal bases and prove the tensor product rule for Uq(q(n))- ≥0 modules in the category Oint. Our main theorem shows that every Uq(q(n))-module in ≥0 the category Oint has a unique crystal basis.

Keywords: quantum queer superalgebras, crystal bases, odd Kashiwara operators.

Mathematics Subject Classification 2010: 17B37, 81R50

1Department of Mathematics University of Texas at Arlington, Arlington, TX 76021, USA [email protected] 2Department of Mathematical Sciences, Seoul National University, Seoul 151- 747, Korea [email protected], [email protected], [email protected] 3Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606- 8502, Japan & Department of Mathematical Sciences, Seoul National University, Seoul 151- 747, Korea [email protected] Posters 70

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011

Universal central extensions of Hom-Leibniz algebras

Jos´eManuel Casas1 ∗∗, Manuel Avelino Insua1, Nat´aliaPacheco2

The Hom-Lie algebra structure was initially introduced in [2] motivated by examples of deformed Lie algebras coming from twisted discretizations of vector fields. In [3] is defined a (multiplicative) Hom-Leibniz algebra as a triple (L, [−, −], α) con- sisting of a K-vector space L, a bilinear map [−, −]: L × L → L and a K-linear map α : L → L (which preserves the bracket) satisfying:

[α(x), [y, z]] = [[x, y], α(z)] − [[x, z], α(y)] (Hom − Leibniz identity) (2) for all x, y, z ∈ L. If we consider α = id, then the Hom-Leibniz identity becomes to Leibniz identity, so Leibniz algebras are a particular instance of (multiplicative) Hom-Leibniz algebras. Our goal in the present note is the study of universal central extensions of multiplica- tive Hom-Leibniz algebras. A new notion of centrality for an extension π :(K, αK )  (L, αL) is necessary. An extension π is called α-central if [α(Ker (π)),K] = 0 = [K, α(Ker (π))], and is called central if [Ker (π),K] = 0 = [K, Ker (π)]. Clearly central extension implies α-central extension and both notions coincide in case α = Id. Also two notions of universality are needed: a central extension π :(K, αk)  (L, αL) is universal if for every central extension τ :(A, αA)  (L, αL), there exists a unique homomorphism ϕ :(K, αk) → (A, αA) such that τϕ = π. If this property holds only for α-central extensions τ :(A, αA)  (L, αL), then π is called universal α-central. Clearly universal α-central implies universal central and both notions coincide when α = Id. After that, we can generalize some classical properties on universal central extensions of Leibniz algebras to the multiplicative Hom-Leibniz algebras setting, in particular, we show that a multiplicative Hom-Leibniz algebra (L, αL) admits universal central extension if and only if (L, αL) is perfect; in this case, the kernel of the universal central α extension is canonically isomorphic to HL2 (L) [1], the second homology with trivial coefficients of the multiplicative Hom-Leibniz algebra L. Nevertheless other results are weaker than the classical ones. For instance, the composition of two central extensions i π is not a central extension, but it is an α-central extension; if 0 → (M, αM ) → (K, αK ) → α α (L, αL) → 0 is a universal α-central extension, then HL1 (K) = HL2 (K) = 0, but the converse doesn’t hold.

Keywords: Hom-Leibniz algebra, universal α-central extension

Mathematics Subject Classification 2010: 17A32, 17A30

∗∗(1) supported by MICINN, grant MTM2009-14464-C02-02 (European FEDER support in- cluded) and by Xunta de Galicia, grant Incite09 207 215 PR Posters 71

References

[1] J. M. Casas, M. A. Insua and N. Pacheco, (Co)homology of Hom-Leibniz algebras, submitted to Linear Algebra and its Applications (2011).

[2] J. T. Hartwing, D. Larson and S. D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2006), 314–361.

[3] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008), no. 2, 51–64.

1Dpto. Matem´aticaAplicada I, Univ. de Vigo, E. I. Forestal, 36005 Pontevedra, Spain [email protected], [email protected] 2IPCA, Dpto. de Ciˆencias,Campus do IPCA, Lugar do Ald˜ao,4750-810 Vila Frescainha, S. Martinho, Barcelos, Portugal [email protected] Posters 72

Non-Associative Algebras and Related Topics Coimbra, July 25–29, 2011 The nil-radical and the bar-radical for alternative baric algebras Henrique Guzzo J´unior1, Rodrigo Lucas Rodrigues2

Let F be a field and let A be an algebra over F , not necessarily associative, neither commutative nor finite dimensional. If ω : A → F is a nonzero homomorphism, then the ordered pair (A, ω) is called a baric algebra over F and ω is its weight function. The kernel of ω is denoted by bar(A), and an ideal I is baric if I ⊆ bar(A). An alternative algebra A over F is an algebra satisfying the equalities x2y = x(xy) and yx2 = (yx)x, for all x, y ∈ A. We assume during this work that A is a finite dimensional algebra and F is a field of characteristic distinct of two, and we study two radicals. We denote by R(A) the nil-radical, which is the maximal nilpotent ideal of A, and by Rb(A) the bar-radical of A, defined as the intersection of maximal baric subalgebras. It is known that a baric alternative algebra possess an idempotent of weight 1, and in this case the bar-radical of A is the intersection of all maximal baric ideals. Our objective is investigate relations between these two radicals. We discovery that 2 these radicals are not necessarily equals and we can prove Rb(A) = RA ∩ (bar(A)) .

Keywords: bar-radical, baric algebra, nil-radical.

Mathematics Subject Classification 2010: MSC17A65, MSC17D05, MSC17D92.

References

[1] H. Guzzo Jr. and M. A. Couto, The radical in alternative baric algebras, Arch. Math. (Basel) 75 (2000), no. 3, 178–187.

[2] H. Guzzo Jr, The bar-radical of baric algebras, Arch. Math. (Basel) 67 (1996), no. 2, 106–118.

[3] H. Guzzo Jr., Some properties of bar-radical of baric algebras, Comm. Algebra 30 (2010), no. 10, 4827–4835.

[4] H. Guzzo Jr., The structure of baric algebras, Groups, rings, and Group rings, Lec. Notes Pure Appl. Math. Chapman and Hall/CRC, Boca Raton, FL (2006), 233–242.

1Instituto de Matem´aticae Estat´ıstica(IME), Departamento de Matem´atica, Universidade de S˜aoPaulo, S˜aoPaulo (USP), Brasil [email protected], [email protected] 2Instituto de Matem´aticae Estat´ıstica(IME), Departamento de Matem´atica, Universidade de S˜aoPaulo, S˜aoPaulo (USP), Brasil [email protected], [email protected] Index Alvarez, M. I. Ram´ırez,61 Nicol´as,Alejandro P., 64 Ansari, Abu Zaid, 18 Arenas, Manuel, 20 Ovsienko, Valentin, 42 Arzikulov, Farkhad, 62 Pacheco, Nat´alia,70 Aschheim, Raymond, 21 Panaite, Florin, 43 Ayadi, Imen, 63 Pires, Rosemary Miguel, 44 Bajo, Ignacio, 22 Pozhidaev, Alexander, 12 Barreiro, Elisabete, 23 Rodrigues, Rodrigo Lucas, 72 Beites, Patr´ıciaD., 64 Benayadi, Sa¨ıd,9 Sabinina, Liudmila, 45 Benkart, Georgia, 13 Salgado, Gil, 46 Bremner, Murray R., 24 S´anchez-Ortega, Juana, 47 Santana, Ana Paula, 48 Casas, Jos´eManuel, 70 Shestakov, Ivan, 16 Coelho, Fl´avioUlhoa, 25 St¨ohr,Ralph, 49 Darp¨o,Erik, 26 Suzuki, Osamu, 50 Delgado, Jos´eM. S´anchez, 66 Sverchkov, Sergei R., 51 Demir, S¨uleyman,27 Sviridova, Irina, 52 Dixon, Geoffrey, 28 Taherabadi, M.R. Molaei, 53 Draper, Cristina, 67 Tanı¸slı,Murat, 54 Elduque, Alberto, 10 Toc´on, Maribel, 56 Tsurkov, Arkadt, 57 Futorny, Vyacheslav, 11 Yudin, Ivan, 59 Goncharov, Maxim, 29 Guzzo Jr, H., 30 Zelmanov, Efim, 17 Zusmanovich, Pasha, 60 Insua, Manuel Avelino, 31

Jung, Ji Hye, 68

Kansu, Mustafa Emre, 33 Kashuba, Iryna, 34 Kerner, Richard, 35 Kim, Myungho, 69 Kotchetov, Mikhail, 36

Laliena, Jes´us,37

Madariaga, Sara, 38 Majid, Shahn, 14 Mart´ınez,Consuelo, 15 Meinardi, Vanesa, 39 Milies, C´esarPolcino, 40 Mondoc, Daniel, 41

73