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Lie and Jordan algebras, their representations and generalizations

Grant number: 05/60337-2
Support type:Research Projects - Thematic Grants
Duration: May 01, 2006 - April 30, 2010
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Algebra
Principal researcher:Ivan Chestakov
Grantee:Ivan Chestakov
Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
Principal researchers:Alexandre Grichkov ; Plamen Emilov Kochloukov ; Vyacheslav Futorny
Associated grant(s):10/50606-4 - Alexander A. Mikhalev | Moscow State University - Rússia, AV.EXT
10/50171-8 - Sergei Sverchkov | Siberian Branch of the Russian Academy of Agricultural Sciences - Rússia, AV.EXT
09/17533-6 - Representation theoretical aspect of conformal field theory, AV.EXT
+ associated grants 09/50981-2 - Alexandr Nikolaevich Zubkov | Omsk State Pedagogical University - Russia, AV.EXT
08/50142-8 - Alexander Pozhidaev | Sobolev Institute of Mathematics, SBRAS - Russia, AV.EXT
07/54820-8 - Ivan Penkov | Jacobs University Bremen - Germany, AV.EXT
07/50405-6 - Nikolay Romanovskiy | Sobolev Institute of Mathematics - Russia, AV.EXT
07/50712-6 - Yuri Bahturin | Memorial University of Newfoundland - Canada, AV.EXT
07/51589-3 - Natalia Zhukavets | Czech Technical University - República Tcheca, AV.EXT
06/60763-4 - Sergei Ovsienko | University of Kyiv - Ukraine, AV.EXT - associated grants
Associated scholarship(s):08/10688-1 - Galois orders, twisted generalized Weyl algebras and their representations, BP.PD
09/05887-8 - Characters and cohomology of modules for affine Kac-Moody algebras and generalizations, BP.DR
07/58048-8 - Binary-Lie algebras, BP.PD

Abstract

The main part of the project is devoted to Lie and Jordan algebras and superalgebras, which are the principal classes of non associative algebras. Besides, alternative and Malcev algebras and superalgebras will be considered, likewise Moufang loops and various generalizations of algebras mentioned above. The main directions of research will be the following: 1) structure and representations of algebras; 2) geometrical aspects of the theory of algebras; 3) combinatorial aspects of the theory of algebras; 4) applications and generalizations. The typical questions in the first direction are related with the study and classification of simple structures (basic atomic building blocks out of which all other structures are built): simple and prime algebras and superalgebras, irreducible and indecomposable modules, Verma modules, etc. Besides, the specialty problem will be studied for Jordan and Malcev algebras. In the second direction, the structure of certain algebraic varieties related with finite dimensional algebras and with various categories of representations will be investigated. The combinatorial part of the theory of algebras has many common features in associative and non-associative cases, both in the problems and in the methods of their solution. For example, the theory of PI-algebras played an important role in the classification of simple alternative and Jordan algebras. On the other hand, the algebras of Poisson brackets (which are non-associative) proved to be very useful in the study of automorphisms of polynomial algebras. Therefore, in this part of the research we consider both associative and non-associative algebras. We plan to study automorphisms and subalgebras of free algebras, polynomial identities of algebras and superalgebras from various classes. Besides, we will try to extend to non-associative algebras some results from the theory of Hopf algebras. Finally, in the part of applications and generalizations, we will study Moufang loops, which have strong relations with alternative algebras, and some generalizations of Jordan algebras. (AU)

Scientific publications (34)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
PETROGRADSKY, V. M.; SHESTAKOV, I. P. Self-similar associative algebras. Journal of Algebra, v. 390, p. 100-125, SEP 15 2013. Web of Science Citations: 4.
PETROGRADSKY, V. M.; SHESTAKOV, I. P. On Properties of a Fibonacci Restricted Lie Algebra. JOURNAL OF LIE THEORY, v. 23, n. 2, p. 407-431, 2013. Web of Science Citations: 1.
DAVYDOV, ALEXEI; FUTORNY, VYACHESLAV. COMMUTATIVE ALGEBRAS IN DRINFELD CATEGORIES OF ABELIAN LIE ALGEBRAS. PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, v. 55, n. 3, p. 613-633, OCT 2012. Web of Science Citations: 0.
FUTORNY, VYACHESLAV; HARTWIG, JONAS T. Multiparameter twisted Weyl algebras. Journal of Algebra, v. 357, p. 69-93, MAY 1 2012. Web of Science Citations: 10.
GONCALVES, DIMAS JOSE; KOSHLUKOV, PLAMEN. A-identities for upper triangular matrices: A question of Henke and Regev. Israel Journal of Mathematics, v. 186, n. 1, p. 407-426, NOV 2011. Web of Science Citations: 1.
COX, BEN; FUTORNY, VYACHESLAV. DJKM ALGEBRAS I: THEIR UNIVERSAL CENTRAL EXTENSION. Proceedings of the American Mathematical Society, v. 139, n. 10, p. 3451-3460, OCT 2011. Web of Science Citations: 4.
GIAMBRUNO, ANTONIO; SHESTAKOV, IVAN; ZAICEV, MIKHAIL. Finite-dimensional non-associative algebras and codimension growth. ADVANCES IN APPLIED MATHEMATICS, v. 47, n. 1, p. 125-139, JUL 2011. Web of Science Citations: 23.
ARENAS, MANUEL; SHESTAKOV, IVAN. ON SPECIALITY OF BINARY-LIE ALGEBRAS. JOURNAL OF ALGEBRA AND ITS APPLICATIONS, v. 10, n. 2, p. 257-268, APR 2011. Web of Science Citations: 0.
DI VINCENZO, ONOFRIO M.; KOSHLUKOV, PLAMEN. On the {*}-polynomial identities of M-1,M-1(E). Journal of Pure and Applied Algebra, v. 215, n. 3, p. 262-275, MAR 2011. Web of Science Citations: 3.
KOSHLUKOV, PLAMEN; SILVA, DIOGO DINIZ P. S. 2-Graded polynomial identities for the Jordan algebra of the symmetric matrices of order two. Journal of Algebra, v. 327, n. 1, p. 236-250, FEB 1 2011. Web of Science Citations: 3.
BAHTURIN, YURI; BRESAR, MATEJ; SHESTAKOV, IVAN. Jordan Gradings on Associative Algebras. ALGEBRAS AND REPRESENTATION THEORY, v. 14, n. 1, p. 113-129, FEB 2011. Web of Science Citations: 3.
KASHUBA, IRYNA; OVSIENKO, SERGE; SHESTAKOV, IVAN. Representation type of Jordan algebras. ADVANCES IN MATHEMATICS, v. 226, n. 1, p. 385-418, JAN 15 2011. Web of Science Citations: 5.
GUTIERREZ FERNANDEZ, JUAN C.; GRISHKOV, A.; MONTOYA, MARY L. R.; MURAKAMI, LUCIA S. I. COMMUTATIVE POWER-ASSOCIATIVE ALGEBRAS OF NILINDEX FOUR. COMMUNICATIONS IN ALGEBRA, v. 39, n. 9, p. 3151-3165, 2011. Web of Science Citations: 1.
FUTORNY, VYACHESLAV; OVSIENKO, SERGE. Galois orders in skew monoid rings. Journal of Algebra, v. 324, n. 4, p. 598-630, AUG 15 2010. Web of Science Citations: 12.
KOSHLUKOV, PLAMEN; ZAICEV, MIKHAIL. Identities and isomorphisms of graded simple algebras. Linear Algebra and its Applications, v. 432, n. 12, p. 3141-3148, JUL 1 2010. Web of Science Citations: 15.
POZHIDAEV, ALEXANDER P.; SHESTAKOV, IVAN P. Structurable superalgebras of Cartan type. Journal of Algebra, v. 323, n. 12, p. 3230-3251, JUN 15 2010. Web of Science Citations: 0.
POZHIDAEV, A. P.; SHESTAKOV, I. P. Structurable Superalgebras of Cartan Type. DOKLADY MATHEMATICS, v. 81, n. 3, p. 373-378, JUN 2010. Web of Science Citations: 1.
MARTINEZ, CONSUELO; SHESTAKOV, IVAN; ZELMANOV, EFIM. JORDAN BIMODULES OVER THE SUPERALGEBRAS P(n) AND Q(n). TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v. 362, n. 4, p. 2037-2051, APR 2010. Web of Science Citations: 3.
POZHIDAEV, A. P.; SHESTAKOV, I. P. Noncommutative Jordan superalgebras of degree n a parts per thousand yen 2. DOKLADY MATHEMATICS, v. 81, n. 2, p. 204-208, APR 2010. Web of Science Citations: 0.
POZHIDAEV, A. P.; SHESTAKOV, P. Noncommutative Jordan superalgebras of degree n > 2. Algebra and Logic, v. 49, n. 1, p. 18-42, MAR 2010. Web of Science Citations: 3.
FUTORNY, VYACHESLAV; MOLEV, ALEXANDER; OVSIENKO, SERGE. The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras. ADVANCES IN MATHEMATICS, v. 223, n. 3, p. 773-796, FEB 15 2010. Web of Science Citations: 16.
DI VINCENZO, ONOFRIO MARIO; KOSHLUKOV, PLAMEN; SANTULO, JR., EDNEI APARECIDO. Graded identities for tensor products of matrix (super)algebras over the Grassmann algebra. Linear Algebra and its Applications, v. 432, n. 2-3, p. 780-795, JAN 15 2010. Web of Science Citations: 1.
PETROGRADSKY, VICTOR M.; SHESTAKOVAND, IVAN P.; ZELMANOV, EFIM. Nil graded self-similar algebras. Groups Geometry and Dynamics, v. 4, n. 4, p. 873-900, 2010. Web of Science Citations: 8.
BEKKERT, VIKTOR; DROZD, YURIY; FUTORNY, VYACHESLAV. Derived tame local and two-point algebras. Journal of Algebra, v. 322, n. 7, p. 2433-2448, OCT 1 2009. Web of Science Citations: 1.
RAO, SENAPATHI ESWARA; FUTORNY, VYACHESLAV. INTEGRABLE MODULES FOR AFFINE LIE SUPERALGEBRAS. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v. 361, n. 10, p. 5435-5455, OCT 2009. Web of Science Citations: 3.
BUENO, ANDRE; COX, BEN; FUTORNY, VYACHESLAV. Free field realizations of the elliptic affine Lie algebra sl(2, R) circle plus (Omega(R)/dR). JOURNAL OF GEOMETRY AND PHYSICS, v. 59, n. 9, p. 1258-1270, SEP 2009. Web of Science Citations: 8.
BRANDAO, JR., ANTONIO PEREIRA; KOSHLUKOV, PLAMEN; KRASILNIKOV, ALEXEI. Graded central polynomials for the matrix algebra of order two. MONATSHEFTE FUR MATHEMATIK, v. 157, n. 3, p. 247-256, JUL 2009. Web of Science Citations: 6.
SHESTAKOV, IVAN; ZHUKAVETS, NATALIA. Skew-symmetric identities of octonions. Journal of Pure and Applied Algebra, v. 213, n. 4, p. 479-492, APR 2009. Web of Science Citations: 6.
FREITAS, JOSE A.; KOSHLUKOV, PLAMEN. Polynomial identities for graded tensor products of algebras. Journal of Algebra, v. 321, n. 2, p. 667-681, JAN 15 2009. Web of Science Citations: 2.
ALVES, SERGIO M.; BRANDAO, JR., ANTONIO P.; KOSHLUKOV, PLAMEN. GRADED CENTRAL POLYNOMIALS FOR T-PRIME ALGEBRAS. COMMUNICATIONS IN ALGEBRA, v. 37, n. 6, p. 2008-2020, 2009. Web of Science Citations: 2.
PETROGRADSKY, V. M.; SHESTAKOV, I. P. Examples of Self-Iterating Lie Algebras, 2. JOURNAL OF LIE THEORY, v. 19, n. 4, p. 697-724, 2009. Web of Science Citations: 8.
ZHUKAVETS, N. M.; SHESTAKOV, I. P. A Base of the Free Alternative Superalgebra on One Odd Generator. DOKLADY MATHEMATICS, v. 78, n. 2, p. 693-695, OCT 2008. Web of Science Citations: 0.
KOSHLUKOV, PLAMEN. Graded polynomial identities for the Lie algebra sl(2)(K). INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, v. 18, n. 5, p. 825-836, AUG 2008. Web of Science Citations: 14.
ZHELYABIN‚ V.N.; SHESTAKOV‚ I.P. The Chevalley and Costant theorems for Mal’tsev algebras. Algebra and Logic, v. 46, n. 5, p. 303-317, 2007.

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