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E-raamat: Algebraic Operads: An Algorithmic Companion

, (University of Saskatchewan, Saskatoon, Canada)
  • Formaat: 383 pages
  • Ilmumisaeg: 06-Apr-2016
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781482248579
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Algebraic Operads: An Algorithmic CompanionSuurem pilt
  • Formaat: 383 pages
  • Ilmumisaeg: 06-Apr-2016
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781482248579
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Algebraic Operads: An Algorithmic Companion presents a systematic treatment of Gröbner bases in several contexts. The book builds up to the theory of Gröbner bases for operads due to the second author and Khoroshkin as well as various applications of the corresponding diamond lemmas in algebra.

The authors present a variety of topics including: noncommutative Gröbner bases and their applications to the construction of universal enveloping algebras; Gröbner bases for shuffle algebras which can be used to solve questions about combinatorics of permutations; and operadic Gröbner bases, important for applications to algebraic topology, and homological and homotopical algebra.

The last chapters of the book combine classical commutative Gröbner bases with operadic ones to approach some classification problems for operads. Throughout the book, both the mathematical theory and computational methods are emphasized and numerous algorithms, examples, and exercises are provided to clarify and illustrate the concrete meaning of abstract theory.

Arvustused

"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."- Lo¿c Foissy

"By balancing computational methods and abstract reasoning, the authors of the book under review have written an excellent up-to-date introduction to Grobner basis methods applicable to associative structures, especially including operads. The book will be of interest to a wide range of readers, from undergraduates to experts in the field."

~ Ralf Holtkamp, Mathematical Reviews, March 2018

Preface xiii
Authors xvii
Introduction 1(8)
1 Normal Forms for Vectors and Univariate Polynomials
9(8)
1.1 Standard forms
9(2)
1.1.1 Orders of sets
9(1)
1.1.2 Monomials and polynomials
10(1)
1.2 Normal forms
11(6)
1.2.1 Normal forms of vectors
11(4)
1.2.2 Normal forms of univariate polynomials
15(2)
2 Noncommutative Associative Algebras
17(50)
2.1 Introduction
17(3)
2.1.1 Noncommutative polynomial equations
18(1)
2.1.2 Noncommutative algebras and Koszul duality
19(1)
2.2 Free associative algebras
20(2)
2.2.1 Monomials and polynomials
20(1)
2.2.2 Presentation by generators and relations
21(1)
2.3 Normal forms
22(9)
2.3.1 Monomial orders
22(3)
2.3.2 Long division
25(3)
2.3.3 Grobner bases
28(3)
2.4 Computing Grobner bases
31(8)
2.4.1 Diamond lemma
31(5)
2.4.2 The Buchberger algorithm
36(2)
2.4.3 Triangle lemma
38(1)
2.5 Examples of Grobner bases and their applications
39(19)
2.5.1 Dimensions and Hilbert series of algebras
39(3)
2.5.2 The group algebra of the symmetric group
42(5)
2.5.3 Universal enveloping algebras of Lie algebras
47(4)
2.5.4 PBW bases, Grobner bases, and Koszul duality
51(3)
2.5.5 Viewing commutative algebras as noncommutative ones
54(1)
2.5.6 Computation of noncommutative Grobner bases
55(3)
2.6 Rewriting systems and Grobner bases
58(3)
2.6.1 Abstract rewriting systems
58(2)
2.6.2 Ordered rewriting systems
60(1)
2.7 Exercises
61(6)
3 Nonsymmetric Operads
67(46)
3.1 Introduction
67(2)
3.1.1 Nonsymmetric collections
68(1)
3.1.2 Nonsymmetric endomorphism operad
68(1)
3.2 Nonsymmetric operads
69(3)
3.2.1 Classical definition of a nonsymmetric operad
69(1)
3.2.2 Definition via partial compositions
70(2)
3.3 Free nonsymmetric operads
72(6)
3.3.1 Trees
72(2)
3.3.2 Tree monomials and tree polynomials
74(2)
3.3.3 Grafting trees and the free nonsymmetric operad
76(2)
3.3.4 Presentation by generators and relations
78(1)
3.4 Normal forms
78(13)
3.4.1 Monomial orders
78(3)
3.4.2 Long division
81(8)
3.4.3 Grobner bases
89(2)
3.5 Computing Grobner bases
91(7)
3.5.1 Diamond lemma
91(5)
3.5.2 The Buchberger algorithm
96(1)
3.5.3 Triangle lemma
96(2)
3.6 Examples of Grobner bases for nonsymmetric operads
98(8)
3.6.1 Associative and q-associative operad
99(2)
3.6.2 The dendriform operad
101(5)
3.7 Normal forms for algebras over nonsymmetric operads
106(3)
3.7.1 Extensions and normal forms in algebras
107(1)
3.7.2 Example of normal forms
107(2)
3.8 Exercises
109(4)
4 Twisted Associative Algebras and Shuffle Algebras
113(28)
4.1 Introduction
113(1)
4.2 Twisted associative algebras and shuffle algebras
114(7)
4.2.1 Two definitions of a twisted associative algebra
114(4)
4.2.2 Free twisted associative algebras
118(1)
4.2.3 Shuffle algebras
119(2)
4.3 Free shuffle algebras
121(4)
4.3.1 Monomials and polynomials
121(2)
4.3.2 Presentation by generators and relations
123(1)
4.3.3 Twisted associative algebras as shuffle algebras
124(1)
4.4 Normal forms
125(7)
4.4.1 Monomial orders
125(2)
4.4.2 Long division
127(4)
4.4.3 Grobner bases
131(1)
4.5 Computing Grobner bases
132(4)
4.5.1 Diamond lemma
132(2)
4.5.2 The Buchberger algorithm
134(1)
4.5.3 Triangle lemma
134(2)
4.6 Examples of shuffle algebras and their applications
136(2)
4.6.1 Shuffle algebras and patterns in permutations
136(1)
4.6.2 Antisymmetrizer shuffle algebras
136(1)
4.6.3 Twisted commutative algebras and shuffle algebras
137(1)
4.7 Exercises
138(3)
5 Symmetric Operads and Shuffle Operads
141(46)
5.1 Introduction
141(1)
5.2 Symmetric operads and shuffle operads
142(8)
5.2.1 Two definitions of a symmetric operad
142(2)
5.2.2 Free symmetric operads
144(5)
5.2.3 Shuffle operads
149(1)
5.3 Free shuffle operads
150(9)
5.3.1 Tree monomials and tree polynomials
150(4)
5.3.2 Presentation by generators and relations
154(1)
5.3.3 Symmetric operads as shuffle operads
154(3)
5.3.4 Applying the forgetful functor
157(2)
5.4 Normal forms
159(11)
5.4.1 Monomial orders
159(4)
5.4.2 Long division
163(6)
5.4.3 Grobner bases
169(1)
5.5 Computing Grobner bases
170(5)
5.5.1 Diamond lemma
170(3)
5.5.2 The Buchberger algorithm
173(1)
5.5.3 Triangle lemma
173(2)
5.6 Examples of Grobner bases for shuffle operads
175(7)
5.6.1 Shuffle Lie and associative operads
175(3)
5.6.2 Symmetric and shuffle operad PreLie
178(1)
5.6.3 Symmetric operads as nonsymmetric operads
179(1)
5.6.4 The operad PreLie as a Lie-module
180(2)
5.7 Exercises
182(5)
6 Operadic Homological Algebra and Grobner Bases
187(34)
6.1 Introduction
187(2)
6.1.1 Symmetry isomorphisms and the Koszul sign rule
187(2)
6.2 First instances of Koszul signs for graded operads
189(10)
6.2.1 Determinant operad and operadic suspension
189(2)
6.2.2 Koszul signs in axioms of an operad
191(1)
6.2.3 Totally associative operad
192(1)
6.2.4 Normal forms and higher Koszul duality
193(6)
6.3 Koszul duality for operads
199(11)
6.3.1 Quadratic operads and cooperads
199(2)
6.3.2 Examples of Koszul dual operads
201(2)
6.3.3 The Koszul property of a quadratic operad
203(3)
6.3.4 The Ginzburg--Kapranov criterion
206(2)
6.3.5 Filtered distributive laws between quadratic operads
208(2)
6.4 Models for operads from Grobner bases
210(7)
6.4.1 Models for operads
210(2)
6.4.2 Resolution for monomial relations
212(2)
6.4.3 Resolution for general relations
214(3)
6.5 Exercises
217(4)
7 Commutative Grobner Bases
221(32)
7.1 Introduction
221(1)
7.2 Commutative associative polynomials
222(6)
7.2.1 One variable
222(1)
7.2.2 The general case
222(2)
7.2.3 Monomial orders
224(4)
7.3 Equivalent definitions of commutative Grobner bases
228(5)
7.3.1 Ideals, generators, and zero sets
228(1)
7.3.2 First definition of a Grobner basis: leading monomials
229(1)
7.3.3 Second definition of a Grobner basis: long division
229(1)
7.3.4 Third definition of a Grobner basis: the Church--Rosser property
230(1)
7.3.5 Equivalence of the three definitions
231(1)
7.3.6 S-polynomials and Buchberger's criterion
232(1)
7.4 Classification of commutative monomial orders
233(5)
7.4.1 The classification theorem
233(4)
7.4.2 Examples and non-examples of monomial orders
237(1)
7.5 Zero-dimensional ideals
238(4)
7.5.1 Characterization of zero-dimensional ideals
239(1)
7.5.2 Two examples, and a distribution table
240(2)
7.6 Complexity of Grobner bases: a historical survey
242(7)
7.6.1 A digression on ordinals and computability
243(1)
7.6.2 Exponential space complexity
243(1)
7.6.3 Pioneering work by Hermann and Noether
244(1)
7.6.4 A detour: Seidenberg
245(1)
7.6.5 Mayr and Meyer, Bayer and Stillman
245(2)
7.6.6 An example: the knapsack problem
247(2)
7.7 Exercises
249(4)
8 Linear Algebra over Polynomial Rings
253(28)
8.1 Introduction
253(1)
8.2 Rank of a polynomial matrix; determinantal ideals
254(3)
8.2.1 The rank of the matrix as a function of the variables
254(1)
8.2.2 Definition of the rank of a polynomial matrix
255(1)
8.2.3 Determinantal ideals of a polynomial matrix
255(2)
8.3 Some elementary examples
257(6)
8.3.1 Ranks of pseudorandom matrices
257(2)
8.3.2 Ranks of symmetric matrices
259(2)
8.3.3 Orthonormal bases in Rn
261(2)
8.4 Algorithms for linear algebra over polynomial rings
263(12)
8.4.1 Introduction: elementary row and column operations
263(1)
8.4.2 Partial row/column reduction (partial Smith form)
263(2)
8.4.3 Canonical forms of row submodules
265(10)
8.5 Bibliographical comments
275(1)
8.6 Exercises
276(5)
9 Case Study of Nonsymmetric Binary Cubic Operads
281(26)
9.1 Introduction
281(1)
9.2 Toy model: the quadratic case
282(2)
9.3 The cubic case
284(20)
9.3.1 Preliminary analysis
284(2)
9.3.2 Relation rank 1
286(1)
9.3.3 Relation rank 2
287(5)
9.3.4 Relation rank 3
292(9)
9.3.5 Relation rank 4
301(3)
9.4 Exercises
304(3)
10 Case Study of Nonsymmetric Ternary Quadratic Operads
307(28)
10.1 Introduction
307(2)
10.2 Generalities on nonsymmetric operad with one generator
309(3)
10.2.1 Enumeration and ordering of monomials
309(1)
10.2.2 Quadratic relations and their consequences
310(2)
10.3 Nonsymmetric ternary operads
312(17)
10.3.1 Preliminary analysis
312(1)
10.3.2 Relation rank 1
313(13)
10.3.3 Relation rank 2
326(3)
10.4 Further directions
329(1)
10.4.1 Hilbert series
329(1)
10.4.2 Nilpotency
330(1)
10.5 Exercises
330(5)
A Maple Code for Buchberger's Algorithm
335(8)
A.1 First block: Initialization
335(1)
A.2 Second block: Monomial orders
336(1)
A.3 Third block: Sorting polynomials
337(1)
A.4 Fourth block: Standard forms of polynomials
338(1)
A.5 Fifth block: Reduce and self-reduce
339(2)
A.6 Sixth block: Main loop -- Buchberger's algorithm
341(2)
Bibliography 343(20)
Index 363
Murray R. Bremner, PhD, is a professor at the University of Saskatchewan in Canada. He attended that university as an undergraduate, and received an M. Comp. Sc. degree at Concordia University in Montréal. He obtained a doctorate in mathematics at Yale University with a thesis entitled On Tensor Products of Modules over the Virasoro Algebra. Prior to returning to Saskatchewan, he held shorter positions at MSRI in Berkeley and at the University of Toronto. Dr. Bremner authored the book Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications and is a co-translator with M. V. Kotchetov of Selected Works of A. I. Shirshov in English Translation. His primary research interests are algebraic operads, nonassociative algebra, representation theory, and computer algebra.

Vladimir Dotsenko, PhD, is an assistant professor in pure mathematics at Trinity College Dublin in Ireland. He studied at the Mathematical High School 57 in Moscow, Independent University of Moscow, and Moscow State University. His PhD thesis is titled Analogues of OrlikSolomon Algebras and Related Operads. Dr. Dotsenko also held shorter positions at Dublin Institute for Advanced Studies and the University of Luxembourg. His collaboration with Murray started in February 2013 in CIMAT (Guanajuato, Mexico), where they both lectured in the research school "Associative and Nonassociative Algebras and Dialgebras: Theory and Algorithms." His primary research interests are algebraic operads, homotopical algebra, combinatorics, and representation theory.
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