Algebras, representations e applications

• Grant number: 10/50347-9
• Support Opportunities: Research Projects - Thematic Grants
• Start date: June 01, 2010
• End date: May 31, 2014
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Ivan Chestakov
• Grantee: Ivan Chestakov
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Pesquisadores principais: Alexandre Grichkov ; Plamen Emilov Kochloukov ; Vyacheslav Futorny
• Associated research grant(s): 14/00246-2 - Non-associative algebraic structures and integrable evolution systems, AV.EXT; 14/00210-8 - Invariants of exceptional algebraic groups and algebraic supergroups, AV.EXT; 13/24392-5 - Indecomposable modules over the algebra of polynomial integro-differential operators, AV.EXT; + associated grants 13/22068-6 - Quantizations of Kleinian singularities and Dixmier groups, AV.EXT; 13/13970-8 - Algebras of quantum differential operators, AV.EXT; 13/12050-2 - Lie Algebras over a field of positiv characteristic and their deformations, AV.EXT; 13/01770-4 - Bethe varieties and corresponding representations, AV.EXT; 13/07844-0 - Associative algebras with group action, AV.BR; 13/10596-8 - Algebraic loops and Drinfeld modules, AV.EXT; 12/22965-5 - Representation theory and quantum field theory, AV.EXT; 13/01717-6 - Generalizations of groups and nonassociative structures, AV.EXT; 12/14961-0 - Irreducible modules over the Lie algebra of vector fields on a torus, AV.EXT; 12/21938-4 - Hopf algebras in non-associative Lie theory, AV.EXT; 12/22537-3 - Non-associative Lie theory, AV.EXT; 12/18139-2 - Methods of representation theory in linear algebra, AV.EXT; 12/01359-0 - Tilting theory, matrix problems and representations of linear groups, AV.EXT; 12/04702-7 - Prime alternative and (-1,1)--algebras and superalgebras: subalgebras, ideals, C--operations, and identities, AV.EXT; 12/11068-2 - Moufang loops and related groups, AV.EXT; 12/04704-0 - Finite faithful representations of finite conformal Lie algebras, AV.EXT; 11/52030-5 - Leonid Makar Limanov | Wayne State University - Estados Unidos, AV.EXT; 11/21621-8 - Representations of Lie (super)algebras of vector fields, AV.EXT; 11/51047-1 - Artem Lopatin | Omsk Branch of Institute of Mathematics - Russia, AV.EXT; 11/50984-1 - Vladimir Grebenev | Russian Academy of Sciences - Siberian Division - Russia, AV.EXT; 10/16891-3 - Equivariant deformation quantization, AV.EXT; 10/52121-8 - Frantisek Marko | Pennsylvania State University - Estados Unidos, AV.EXT - associated grants
• Associated scholarship(s): 13/22802-1 - Graded identities in Lie and Jordan algebras, BP.DR; 13/09610-6 - Functional identities and Herstein's problem, BP.PD; 13/02039-1 - Associative, alternative, Jordan algebras and their subalgebras of invariants, BP.PD; + associated scholarships 12/11592-3 - LIE SUPERALGEBRAS SUPPORTED OVER su(3) AND su(2,2), BP.PD; 12/10378-8 - Algebraic structures of the baric algebras, RA loops and linear codes, BP.PD; 11/12079-5 - Representations of Kac-Moody Algebras and quantum groups, BP.PD; 12/02459-8 - Vertex constructions in representation theory of infinite dimensional Lie algebra., BP.PD; 10/51879-4 - Polynomial - identities of matrizes, BP.PD; 10/15781-0 - Stable vector bundles over projective plane and representations of posets in the category of unitary spaces, BP.PD; 10/51880-2 - On finite basis property for varieties of nearly associative algebras., BP.PD; 10/14906-3 - Nonassociative algebras, BE.PQ - associated scholarships

Abstract

The main part of the project is dedicated to Lie and Jordan algebras and super algebras and their representations. Besides, alternative and Malcev algebras and super algebras will be considered, as well as Moufang loops and various generalizations of algebras mentioned above. The main directions of research will be the following: Lie algebras and their representations; Jordan and alternative algebras and super algebras and their generalizations; identities, graduations, automorphisms; applications e generalizations. Lie algebras and their representations is one of the most important and active areas of modern algebra, with an ample variety of results, applications and relations with other parts of the science. The new developments in non-commutative geometry and mathematical physics suggests a necessity to study infinite-dimensional modules, such as generalized Verma modules, modules with finite-dimensional weight spaces, Gelfand-Tsetlin and Harish¬Chandra modules, etc. The first part of the project treats various aspects of this theory. Besides the Lie algebras, the most important classes of non-associative algebras are alternative and Jordan algebras. The typical questions of the second part treat the study and classification of simple algebras and super algebras and irreducible and indecomposable modules in the varieties of alternative and Jordan algebras and their generalizations. The third part of the project is dedicated to combinatorial questions of the theory of algebras: study of free algebras, identities, gradations and automorphisms. This part of the theory has many common features in associative and non-associative cases, both in the problems and in the methods of their solution. Therefore, in this part of the research we consider both associative and non-associative algebras. We plan to study automorphisms and sub algebras of free algebras, identities and gradations of algebras and super algebras from various classes. Besides, we will try to extend to non-associative algebras some results from the theory of Hopf algebras. In the part of applications and generalizations, we will study Moufang loops, which have strong relations with alternative algebras and some other important classes of loops. We will also study the Albert Problem on the existence of simple commutative power associative finite dimensional nil algebras and will consider algebras that appear in genetics. (AU)

Scientific publications

(44)

(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)

SHESTAKOV, IVAN; TRUSHINA, MARIA. IRREDUCIBLE BIMODULES OVER ALTERNATIVE ALGEBRAS AND SUPERALGEBRAS. Transactions of the American Mathematical Society, v. 368, n. 7, p. 4657-4684, JUL 2016. (10/50347-9)

DMYTRYSHYN, ANDRII; FUTORNY, VYACHESLAV; SERGEICHUK, VLADIMIR V.. Miniversal deformations of matrices under {*}congruence and reducing transformations. Linear Algebra and its Applications, v. 446, p. 388-420, APR 1 2014. (10/50347-9, 10/07278-6, 12/18139-2)

FUTORNY, V.; HARTWIG, J. T.; WILSON, E. A.. QUANTUM AFFINE MODULES FOR NON-TWISTED AFFINE KAC-MOODY ALGEBRAS. Proceedings of the American Mathematical Society, v. 143, n. 12, p. 5159-5171, DEC 2015. (11/12079-5, 10/50347-9)

FERREIRA, B. L. M.; GUZZO, JR., H.; FERREIRA, J. C. M.. The Wedderburn b-decomposition for a class of almost alternative baric algebras. ASIAN-EUROPEAN JOURNAL OF MATHEMATICS, v. 8, n. 1, MAR 2015. (10/50347-9)

FUTORNY, VYACHESLAV; IYER, UMA N.. REPRESENTATIONS OF D-q(k[x]). Israel Journal of Mathematics, v. 212, n. 1, p. 473-506, MAY 2016. (10/50347-9, 13/13970-8)

FUTORNY, VYACHESLAV; GRANTCHAROV, DIMITAR; MAZORCHUK, VOLODYMYR. WEIGHT MODULES OVER INFINITE DIMENSIONAL WEYL ALGEBRAS. Proceedings of the American Mathematical Society, v. 142, n. 9, p. 3049-3057, SEP 2014. (10/50347-9, 11/21621-8)

COX, BEN; FUTORNY, VYACHESLAV; MISRA, KAILASH C.. An imaginary PBW basis for quantum affine algebras of type 1. Journal of Pure and Applied Algebra, v. 219, n. 1, p. 83-100, JAN 2015. (10/50347-9)

FERREIRA, VITOR O.; MURAKAMI, LUCIA S. I.. RATIONALITY OF THE HILBERT SERIES OF HOPF-INVARIANTS OF FREE ALGEBRAS. Proceedings of the American Mathematical Society, v. 142, n. 3, p. 821-826, MAR 2014. (10/50347-9, 09/52665-0)

ZHELYABIN, V. N.; POPOV, A. A.; SHESTAKOV, I. P.. The coordinate ring of an n-dimensional sphere and some examples of differentially simple algebras. Algebra and Logic, v. 52, n. 4, p. 277-289, SEP 2013. (10/50347-9)

COX, BEN; FUTORNY, VYACHESLAV; TIRAO, JUAN A.. DJKM algebras and non-classical orthogonal polynomials. Journal of Differential Equations, v. 255, n. 9, p. 2846-2870, NOV 1 2013. (10/50347-9)

FUTORNY, VYACHESLAV; MARKO, FRANTISEK. DERIVED REPRESENTATION TYPE OF SCHUR SUPERALGEBRAS. COMMUNICATIONS IN ALGEBRA, v. 42, n. 8, p. 3381-3385, AUG 3 2014. (10/50347-9)

FUTORNY, VYACHESLAV; RYBALKINA, TETIANA; SERGEICHUK, VLADIMIR V.. Regularizing decompositions for matrix pencils and a topological classification of pairs of linear mappings. Linear Algebra and its Applications, v. 450, p. 121-137, JUN 1 2014. (10/50347-9, 12/18139-2)

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. (10/50347-9, 05/60337-2)

KOSHLUKOV, PLAMEN; DE MELLO, THIAGO CASTILHO. The centre of generic algebras of small PI algebras. Journal of Algebra, v. 375, p. 109-120, FEB 1 2013. (10/50347-9)

BEKKERT, VIKTOR; DROZD, YURIY; FUTORNY, VYACHESLAV. Tilting, deformations and representations of linear groups over Euclidean algebras. Journal of Pure and Applied Algebra, v. 217, n. 6, p. 1141-1152, JUN 2013. (10/50347-9, 07/05047-4)

RODRIGUES, R. L.; GUZZO, JR., H.; FERREIRA, J. C. M.. When is a Multiplicative Derivation Additive in Alternative Rings?. COMMUNICATIONS IN ALGEBRA, v. 44, n. 6, p. 2561-2566, 2016. (12/10378-8, 10/50347-9)

DMYTRYSHYN, ANDRII; FUTORNY, VYACHESLAV; SERGEICHUK, VLADIMIR V.. Miniversal deformations of matrices under *congruence and reducing transformations. Linear Algebra and its Applications, v. 446, p. 33-pg., 2014-04-01. (10/50347-9, 12/18139-2, 10/07278-6)

SHESTAKOV, IVAN; TRUSHINA, MARIA. IRREDUCIBLE BIMODULES OVER ALTERNATIVE ALGEBRAS AND SUPERALGEBRAS. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v. 368, n. 7, p. 28-pg., 2016-07-01. (10/50347-9)

FUTORNY, VYACHESLAV; OVSIENKO, SERGE. FIBERS OF CHARACTERS IN GELFAND-TSETLIN CATEGORIES. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v. 366, n. 8, p. 36-pg., 2014-08-01. (04/02850-2, 06/60763-4, 10/50347-9)

MACEDO FERREIRA, BRUNO LEONARDO; GUZZO, HENRIQUE, JR.; DA MOTTA FERREIRA, JOAO CARLOS. Additivity of Jordan elementary maps on standard rings. ALGEBRA & DISCRETE MATHEMATICS, v. 18, n. 2, p. 31-pg., 2014-01-01. (10/50347-9)

FUTORNY, VYACHESLAV; SAMOILENKO, YURII; YUSENKO, KOSTYANTYN; ABRAMOV, V; FUCHS, J; PAAL, E; SHESTOPALOV, Y; SILVESTROV, S; STOLIN, A. Representations of posets: linear versus unitary. WAKE CONFERENCE 2021, v. 346, p. 17-pg., 2012-01-01. (10/15781-0, 10/50347-9)

KHARCHENKO, V. K.; SHESTAKOV, I. P.. Generalizations of Lie Algebras. Advances in Applied Clifford Algebras, v. 22, n. 3, p. 23-pg., 2012-09-01. (10/50347-9)

FUTORNY, VYACHESLAV; OVSIENKO, SERGE. FIBERS OF CHARACTERS IN GELFAND-TSETLIN CATEGORIES. Transactions of the American Mathematical Society, v. 366, n. 8, p. 4173-4208, AUG 2014. (10/50347-9, 04/02850-2, 06/60763-4)

BILLIG, YULY; FUTORNY, VYACHESLAV. Classification of simple cuspidal modules for solenoidal Lie algebras. Israel Journal of Mathematics, v. 222, n. 1, p. 109-123, OCT 2017. (10/50347-9, 12/14961-0)

FUTORNY, VYACHESLAV; HARTWIG, JONAS T.. De Concini-Kac filtration and Gelfand-Tsetlin generators for quantum gl(N). Linear Algebra and its Applications, v. 568, p. 173-188, MAY 1 2019. (10/50347-9)

KOLESNIKOV, PAVEL S.; MAKAR-LIMANOV, LEONID G.; SHESTAKOV, IVAN P.. The Freiheitssatz for Generic Poisson Algebras. Symmetry Integrability and Geometry-Methods and Applications, v. 10, 2014. (10/50347-9, 12/04704-0, 11/52030-5)

MACHADO, GUSTAVO GRINGS; KOSHLUKOV, PLAMEN. GK dimension of the relatively free algebra for. MONATSHEFTE FUR MATHEMATIK, v. 175, n. 4, p. 543-553, DEC 2014. (10/50347-9)

MOSTOVOY, J.; PEREZ-IZQUIERDO, J. M.; SHESTAKOV, I. P.. Hopf algebras in non-associative Lie theory. BULLETIN OF MATHEMATICAL SCIENCES, v. 4, n. 1, p. 129-173, APR 2014. (10/50347-9, 12/22537-3, 12/21938-4)

BRESSLER, PAUL; FUTORNY, VYACHESLAV. CHIRAL ANOMALY VIA VERTEX ALGEBROIDS. QUARTERLY JOURNAL OF MATHEMATICS, v. 65, n. 2, p. 581-596, JUN 2014. (10/50347-9, 10/16891-3)

FUTORNY, VYACHESLAV; MOLEV, ALEXANDER. Quantization of the shift of argument subalgebras in type A. ADVANCES IN MATHEMATICS, v. 285, p. 1358-1375, NOV 5 2015. (10/50347-9)

MOSTOVOY, J.; PEREZ-IZQUIERDO, J. M.; SHESTAKOV, I. P.. Nilpotent Sabinin algebras. Journal of Algebra, v. 419, p. 95-123, DEC 1 2014. (10/50347-9, 12/22537-3, 12/21938-4)

DA SILVA, JUACI PICANCO; MURAKAMI, LUCIA S. I.; SHESTAKOV, IVAN. On Right Alternative Superalgebras. COMMUNICATIONS IN ALGEBRA, v. 44, n. 1, p. 240-252, 2016. (10/50347-9)

DMYTRYSHYN, ANDRII; FUTORNY, VYACHESLAV; KAGSTROM, BO; KLIMENKO, LENA; SERGEICHUK, VLADIMIR V.. Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence. Linear Algebra and its Applications, v. 469, p. 305-334, MAR 15 2015. (10/50347-9, 12/18139-2)

KOSHLUKOV, PLAMEN; MATTOS MORTARI, ALDA DAYANA. G-identities for the Lie algebra sl(2)(C). Journal of Pure and Applied Algebra, v. 219, n. 6, p. 2381-2395, JUN 2015. (10/50347-9)

BILLIG, YULY; FUTORNY, VYACHESLAV. Classification of irreducible representations of Lie algebra of vector fields on a torus. JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, v. 720, p. 199-216, NOV 2016. (12/14961-0, 10/50347-9)

PCHELINTSEV, SERGEY V.; SHESTAKOV, IVAN P.. Prime (-1,1) and Jordan monsters and superalgebras of vector type. Journal of Algebra, v. 423, p. 54-86, FEB 1 2015. (10/50347-9, 12/04702-7)

FUTORNY, VYACHESLAV; HARTWIG, JONAS; WILSON, EVAN. Irreducible completely pointed modules of quantum groups of type A. Journal of Algebra, v. 432, p. 252-279, JUN 15 2015. (10/50347-9, 11/12079-5)

RAO, S. ESWARA; FUTORNY, VYACHESLAV. REPRESENTATIONS OF LOOP KAC-MOODY LIE ALGEBRAS. COMMUNICATIONS IN ALGEBRA, v. 41, n. 10, p. 3775-3792, OCT 3 2013. (10/50347-9)

PETROGRADSKY, V. M.; SHESTAKOV, I. P.. Self-similar associative algebras. Journal of Algebra, v. 390, p. 100-125, SEP 15 2013. (10/50347-9, 05/60337-2)

FUTORNY, VYACHESLAV; HARTWIG, JONAS T.. Solution of a -difference Noether problem and the quantum Gelfand-Kirillov conjecture for. MATHEMATISCHE ZEITSCHRIFT, v. 276, n. 1-2, p. 1-37, FEB 2014. (10/50347-9)

KOSHLUKOV, PLAMEN; DE MELLO, THIAGO CASTILHO. On the polynomial identities of the algebra M-11 (E). Linear Algebra and its Applications, v. 438, n. 11, p. 4469-4482, JUN 1 2013. (10/50347-9)

BONDARENKO, VITALIJ M.; FUTORNY, VYACHESLAV; KLIMCHUK, TATIANA; SERGEICHUK, VLADIMIR V.; YUSENKO, KOSTYANTYN. Systems of subspaces of a unitary space. Linear Algebra and its Applications, v. 438, n. 5, p. 2561-2573, MAR 1 2013. (10/50347-9, 10/15781-0)

FERREIRA, B. L. M.; FERREIRA, J. C. M.; GUZZO, H., JR.. JORDAN MAPS ON ALTERNATIVE ALGEBRAS. JP JOURNAL OF ALGEBRA NUMBER THEORY AND APPLICATIONS, v. 31, n. 2, p. 14-pg., 2013-12-01. (10/50347-9)

SHESTAKOV, I.; TSURKOV, A.. Automorphic equivalence of the representations of Lie algebras. ALGEBRA & DISCRETE MATHEMATICS, v. 15, n. 1, p. 31-pg., 2013-01-01. (10/50347-9)