Combinatorial Methods in Noncommutative Algebra.
por
Tatiana Gateva-Ivanova, Sofia
Duración: 20 horas, impartido en 10 sesiones, que se detallan seguidamente.
Lugar: Facultad de Ciencias, Fuentenueva s/n, Granada.
Fechas:
Primera sesión: jueves, 18 de mayo de 2006, 12h. - 14h. Aula de Conferencias. Contenidos.
Segunda sesión: jueves, 25 de mayo de 2006, 12h. - 14h. Aula de Conferencias. Contenidos.
Tercera sesión: jueves, 1 de junio de 2006, 12h. - 14h. Aula de Conferencias Contenidos.
Cuarta sesión: lunes, 5 de junio de 2006, 12h. - 14h. Aula M-23, Aulario. Contenidos.
Quinta sesión: jueves, 8 de junio de 2006, 12h. - 14h. Contenidos.
Sexta sesión: martes, 13 de junio de 2006, 12h. - 14h. Contenidos.
Séptima sesión: martes, 20 de junio de 2006, 12h. - 14h. Contenidos.
Octava y novena sesiones: miércoles, 28 de junio de 2006, 11h. - 13:30h. Contenidos.
Program
We shall study finitely presented associative algebras and the algebraic and homological properties implied by their defining relations. Some of the basic tools of our study are the theory of noncommutative Groebner bases, Anick´s resolution, the theory of semigroups of I-type, combinatorial properties of the relations, in particular the cyclic condition and its implications.
We shall illustrate the combinatorial approach with detailed study of various types of finitely presented algebras and semigroups, such as skew polynomial rings with binomial relations, Yang-Baxter algebras and semigroups defined via set-theoretic solutions of the Yang-Baxter equation (YBE), monomial algebras.
Topics
Introduction to finitely presented algebras and noncommutative Groebner basis, the graph of normal words of a standard finitely presente algebra.
Recognisability of algebraic properties, The Hilbert series of a graded algebra, noetherian properties, growth of an algebra, PBW –type algebras.
Skew polynomial rings with binomial relations, combinatorial, algebraic and homological properties.
Anick´s resolution, Artin-Schelter regularity.
Semigroups of I type and their algebraic and homological properties. The lattice structure of a semigroup of I type. Each semigroup of I type defines a set theoretic solution of YBE.
The relation between binomial skew polynomial rings, Artin-Schelter regularity, binomial solutions of YBE, semigroups of I type.
Matched pair approach to set-theoretic solutions of YBE. Extensions of solutions.
Strictly ordered algebras, Noetherian properties and growth. (optional)
Infinite words over a finite alphabet, uniformly recurrent words. Application to monomial algebras. The Jacobson radical of a monomial algebra (optional).
Colaboran : Departamento de Álgebra de la Universidad de Granada, Programa de Doctorado <<Matemáticas>>, Proyectos de Investigación <<Métodos algebraicos en Geometría no Conmutativa>> (MTM2004-01406), <<Aplicaciones del álgebra a la Geometría no Conmutativa>>, (MTM2004-08125).