Seminar of Algebra

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Gröbner bases for modules on skew PBW extensions

Claudia Gallego (Universidad Nacional de Colombia)
Departamento de Álgebra
Fri, 30 jan 2015 10:30
Claudia Gallego is a PhD candidate in mathematical sciences; active mem- ber of the research group “Álgebra constructiva - SAC2”. Her research interests are computational algebra, theory of noncommutative rings and modules, homological algebra, algebraic topology and algebraic geometry (the latter two not active).

In this talk we present the Buchberger’s algorithm for computing Gröbner bases of modules defined on a new class of noncommutative rings: the skew $PBW$ extensions, introduced by us in [4], as a generalization of the $PBW$ extensions established by Bell and Goodearl in [1]. Further, we show some elementary applications of this, such as: membership problem, syzygy module, presentation of a module, kernel and image of a homomorphism.

[1] Bell, A. and Goodearl, K., Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensons, Pacific Journal of Mathematics, 131(1), 1988, 13-37.

[2] Bueso, J., Gómez-Torrecillas, J. and Verschoren, A., Algorithmic Methods in noncommutative Algebra: Applications to Quantum Groups, Kluwer, 2003.

[3] Chyzak, F., Quadrat, A. and Robertz, D., Effective algorithms for parametrizing linear control systems over Ore algebras, Appl. Algebra Engrg. Comm. Comput., 16, 2005, 319-376.

[4] Gallego, C. and Lezama, O., Gröbner bases for ideals of skew $PBW$ extensions, Communications in Algebra, 39, 2011, 50-75.

[5] Gallego, C. and Lezama, O., Matrix approach to noncommutative stably free modules and Hermite rings, Algebra and Discrete Mathematics 18 (1), 2014, 110-139.

[6] Gallego, C., Gröbner basis for bijective skew $PBW$ extensions, Preprint.

[7] Levandovskyy, V., Non-commutatve Computer Algebra for Polynomial Algebras: Gröbner Bases, Applications and Implementation, Doctoral Thesis, Universität Kaiserslautern, 2005.

[8] Lezama, O., Matrix and Gröbner Methods in Homological Algebra, Lambert Academic Publishing, 2011.

[9] Lezama, O. and Reyes, M.A., Some homological properties of skew PBW extensions, Communications in Algebra, 42, 2014, 1200-1230