Algebraic operads. An algorithmic companion (Q5890492)

This book presents a systematic treatment of Gröbner bases, and more generally of the problem of normal forms, departing from linear algebra, going through commutative and noncommutative algebra, to operads. The algorithmic aspects are especially developed, with numerous examples and exercises.NEWLINENEWLINEThe first, short, chapter deals with linear reduction in vector spaces, and long division of polynomials in one variable; its main goal is to develop intuitions for the sequel. In the second chapter, normal forms and Gröbner bases for noncommutative algebras and various applications are discussed. Monomial orders, normal forms, the long division algorithm and the Buchberger algorithm are presented; Gröbner basis are applied to obtain results on Hilbert series, symmetric group algebras and enveloping algebras. The last paragraph makes a link with rewriting systems.NEWLINENEWLINEThese results are then generalized in the next chapter to nonsymmetric operads. The two definitions (the ``classical'' one, and the one through partial compositions) are recalled, and free nonsymmetric operads are built with the help of rooted trees, which here are defined in a especially adapted way for this theory. Monomial orders, normal forms, the long division, the Buchberger algorithm are extended to this context, and it is shown how this theory leads to normal forms for algebras over a nonsymmetric operad.NEWLINENEWLINEChapter 4 turns to twisted associative algebras, that is to say graded algebra which homogeneous components are equipped with symmetric group actions, with a certain compatibility with the product; these objects are of special interests in combinatorics or representation theory. Free objects are here shuffle algebras.NEWLINENEWLINEThe full generality of operads and shuffle operads is developed in chapter 5. The two definitions of operads and shuffle operads are given, and the results of chapter 3 are consolidated to deal with the symmetry. Examples of Gröbner bases are given for shuffle Lie and associative operads, symmetric and shuffle pre-Lie operads.NEWLINENEWLINEThe next chapter gives applications to homological algebra. It is explained how the ``Koszul sign rules'' is set in all the computations of the preceding chapter, and the Koszul duality of quadratic operads is developed.NEWLINENEWLINEChapter 7 turns back to recall on Gröbner bases for commutative algebras. It contains, among other results, a proof of Robbiano's classification of monomial orders, and a historial survey on the algorithmic complexity of Gröbner bases. The results here exposed will be extensively used in the last three chapters. Chapter 8 deals with matrices over a polynomial ring (not necessarily on one variable). The last two chapters are an initial attempt to a classification of nonsymmetric operads with one binary operation satisfying cubic relations, or one ternary operation satisfying quadratic relations. Roughly speaking, the structure coefficients should satisfy certain polynomial relations, which will more easily be handled with Gröbner bases.NEWLINENEWLINEThe appendix is a Maple code for Buchberger's algorithm, written without any call to a procedure of the \textbf{groebner} package.