Deformation Theory of Algebras and Their Diagrams

Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics

Number 116

Martin Markl

American Mathematical Society with support from the National Science Foundation Deformation Theory of Algebras and Their Diagrams

http://dx.doi.org/10.1090/cbms/116

Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics

Number 116

Deformation Theory of Algebras and Their Diagrams

Martin Markl

Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island with support from the National Science Foundation NSF-CBMS Regional Research Conference in the Mathematical Sciences on Deformation Theory of Algebras and Modules held at North Carolina State University, Raleigh, NC, May 16–20, 2011 Partially supported by the National Science Foundation. The author acknowledges support from the Conference Board of the Mathematical Sciences and NSF grant DMS-1040647; the Eduard Cechˇ Institute P201/12/G028; and RVO: 67985840 2010 Mathematics Subject Classiﬁcation. Primary 13D10, 14D15; Secondary 53D55, 55N35.

For additional information and updates on this book, visit www.ams.org/bookpages/cbms-116

Library of Congress Cataloging-in-Publication Data Markl, Martin, 1960-author. [Lectures. Selections] Deformation theory of algebras and their diagrams / Martin Markl. p. cm. — (Regional conference series in mathematics, ISSN 0160-7642 ; number 116) Covers ten lectures given by the author at the NSF-CBMS Regional Conference in the Math- ematical Sciences on Deformation Theory of Algebras and Modules held at North Carolina State University, Raleigh, NC, May 16–20, 2011. Includes bibliographical references and index. ISBN 978-0-8218-8979-4 (alk. paper) 1. Algebra, Homological–Congresses. I. Conference Board of the Mathematical Sciences, spon- soring body. II. NSF-CBMS Regional Conference in the Mathematical Sciences on Deformation Theory of Algebras and Modules (2011 : Raleigh, NC) III. Title. QA169.M355 2012 512.64—dc23 2012025195

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Preface vii Chapter 1. Basic notions 1 Chapter 2. Deformations and cohomology 19 Chapter 3. Finer structures of cohomology 25 Chapter 4. The gauge group 45 Chapter 5. The simplicial Maurer-Cartan space 55 Chapter 6. Strongly homotopy Lie algebras 69 Chapter 7. Homotopy invariance and quantization 75 Chapter 8. Brief introduction to operads 85 Chapter 9. L∞-algebras governing deformations 99 Chapter 10. Examples 113 Index 121 Bibliography

Preface

This monograph covers ten lectures given by the author at the North Carolina State University at Raleigh, NC, during the week May 16–20, 2011. The choice of topics was a result of a compromise, given by the fact that the audience consisted of both graduate students and specialists in the field. The author resisted the temptation to devote the talks to his own results and attempted to present the material scattered in the literature in a compact fashion. Also, the level of generality was determined by the purpose of the book. We decided to focus on deformations over local complete Noetherian rings (which of course includes the Artin case), though more general bases, as formal dg-commutative algebras or formal dg-schemes, can be considered. The complete Noetherian case covers most types of deformations of algebraic structures a working mathematician meets in his/her professional life. The place for the conference was sensibly chosen, because the birth of deformation theory as we understand it today is related to this part of the globe. Mike Schlessinger and Jim Stasheff worked at Chapel Hill, only 28 miles from Raleigh—Mike contributed the Artin ring approach to deformation theory and, together with Jim, introduced the intrinsic bracket. They also wrote a seminal paper that predated modern deformation theory with its emphasis on the moduli space of the Maurer-Cartan equation. Tom Lada, who has spent most of his professional career at Raleigh, together with Jim, introduced L∞-algebras. And, of course, the life of Murray Gerstenhaber, the founding father of algebraic deformation theory, is connected to Philadelphia, a 7 hour drive from Raleigh. Moreover, the author of this monograph worked as a Fulbright fellow for two fruitful semesters in Chapel Hill with Jim and Tom. The book consists of 10 chapters which more or less correspond to the material of the respective talks. Chapters 1–3 review classical Gerstenhaber’s deformation theory of associative algebras over a local complete Noetherian ring. In chapters 4 and 5, which are devoted to Maurer-Cartan elements in differential graded Lie algebras, the moduli space point of view begins to prevail. In chapter 6 we recall L∞-algebras and, in chapter 7, the related simplicial version of the Maurer-Cartan moduli space, and prove its homotopy invariance. As an application, we review the main features of Kontsevich’s approach to deformation quantization of Poisson manifolds. In chapters 8–9 we describe a construction of an L∞-algebra governing deformations of a given class of (diagrams of) algebras. The last chapter contains a couple of explicit examples and indicates possible generalizations. Our intention was to make the presentation self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. Suitable references are [AM69, HS71, ML63a, ML71]. We sometimes omitted

vii viii PREFACE technically complicated proofs when a suitable reference was available. Namely, we did not prove Theorem 6.13 about the Kan property of the induced map between the simplicial Maurer-Cartan spaces. Since operads are not the central topic of this book, we also omitted proofs of statements from operad theory used in chapters 8 and 9. On the other hand, we explained in detail the relation between the uniform continuity of algebraic maps and topologized tensor products and included proofs of the related statements, as this subject does not seem to be commonly known and the literature is scarce. This monograph is not the first text attempting to present algebraic deformation theory. The classical theory is the subject of [GS88], there are lecture notes [DMZ07] and very recent shorter accounts [Fia08, Gia11]. Useful historical re- marks can be found in [Pfl06]; an annotated historical bibliography is contained in [DW], perturbations, deformations, variations and “near misses” are treated in [Maz04]. There is also the influential though still unfinished book [KS]. Acknowledgments. I would like to express my thanks to the organizers of the conference, namely to Tom Lada and Kailash Mishra, for the gigantic work they have done. I am indebted also to the audience, which demonstrated striking tol- erance to my halting English. During my work on the manuscript, I enjoyed the stimulating atmosphere of the Universidad de Talca and of the Max-Planck-Institut für Mathematik in Bonn. In formulating my definition of A∞∞-algebras I profited from conversations with M. Doubek and M. Livernet. Also, comments and suggestions from T. Gi- aquinto, A. Lazarev, M. Manetti, J. Stasheff and D. Yau were very helpful. I wish to thank, in particular, Tom Lada for reading the drafts of the manuscript and correcting typos and the worst of my language insufficiency.

Martin Markl PREFACE ix

Conventions. Most of the algebraic objects will be considered over a ﬁxed ﬁeld of characteristic zero, although some results remain true in arbitrary characteristic,

or even over the ring of integers. The symbol ⊗ will denote the tensor product over

− − and Lin( , )thespaceof -linear maps. By Span(S)wedenotethe -vector space spanned by the set S. The arity of a multilinear map is the number of its

arguments. For instance, a bilinear map has arity two. We will denote by ½X or

simply by ½ when X is understood, the identity endomorphism of an object X (set, vector space, algebra, &c.). The symbol Sn will refer to the symmetric group on n elements. We will observe the usual convention of calling a commutative associative

algebra simply a commutative algebra.

Æ { } We denote by the ring of integers and by the set 1, 2, 3,... of natural

numbers. The adjective “graded” will usually mean -graded, though we will also consider non-negatively or non-positively graded objects; the actual meaning will always be clear from the context. The degree of an homogeneous element a will be denoted by deg(a)orby|a|. The grading will sometimes be indicated by ∗ in sub- or superscript, the simplicial and cosimplicial degrees by •.Ifwewritev ∈ V ∗ for a graded vector space V ∗, we automatically assume that v is homogeneous, i.e. it belongs to a specific component of V . The abbreviation ‘dg’ will mean ‘differential graded.’ Since we decided to re- quire that the Maurer-Cartan elements are placed in degree +1, our preferred degree of differentials is +1. As a consequence, resolutions are non-positively graded. An unpleasant feature of the graded word is the necessity to keep track of complicated signs. We will always use the Koszul sign convention requiring that, whenever we interchange two graded objects of degrees p and q, respectively, we change the overall sign by (−1)pq. This rule however does not determine the signs uniquely. For instance, in Remark 3.51 we explain that the signs in the definition of strongly homotopy algebras depend on the preference for the inversion of the tensor power of the suspension. We will use the sign convention determined by requiring that 1 (i) all terms in the L∞-Maurer-Cartan equation come with the + sign, and that (ii) the intrinsic bracket (9.22) agrees, in the associative algebra case, with the one of [Ger63]. Requirement (i) fixes the signs in L∞-algebras. Requirement (ii) introduces the k+1 correction (−1) to formula (9.4) and affects the sign in (9.13). Since A∞-algebras are Maurer-Cartan elements in the extended Gerstenhaber-Hochschild dg-Lie algebra (3.24), (ii) in turn determines the convention for A∞-algebras. An unfortunate but necessary consequence is the minus sign in the expression (9.15) for the curva- ture and in the related formulas.

1This convention is used in [Get09a].

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116 Martin Markl, Deformation Theory of Algebras and Their Diagrams, 2012 115 Richard A. Brualdi, The Mutually Beneficial Relationship of Graphs and Matrices, 2011 114 Mark Gross, Tropical Geometry and Mirror Symmetry, 2011 113 Scott A. Wolpert, Families of Riemann Surfaces and Weil-Petersson Geometry, 2010 112 Zhenghan Wang, Topological Quantum Computation, 2010 111 Jonathan Rosenberg, Topology, C∗-Algebras, and String Duality, 2009 110 David Nualart, Malliavin Calculus and Its Applications, 2009 109 Robert J. Zimmer and Dave Witte Morris, Ergodic Theory, Groups, and Geometry, 2008 108 Alexander Koldobsky and Vladyslav Yaskin, The Interface between Convex Geometry and Harmonic Analysis, 2008 107 FanChungandLinyuanLu, Complex Graphs and Networks, 2006 106 Terence Tao, Nonlinear Dispersive Equations, 2006 105 Christoph Thiele, Wave Packet Analysis, 2006 104 Donald G. Saari, Collisions, Rings, and Other Newtonian N-Body Problems, 2005 103 Iain Raeburn, Graph Algebras, 2005 102 Ken Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, 2004 101 Henri Darmon, Rational Points on Modular Elliptic Curves, 2004 100 Alexander Volberg, Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces, 2003 99 Alain Lascoux, Symmetric Functions and Combinatorial Operators on Polynomials, 2003 98 Alexander Varchenko, Special Functions, KZ Type Equations, and Representation Theory, 2003 97 Bernd Sturmfels, Solving Systems of Polynomial Equations, 2002 96 Niky Kamran, Selected Topics in the Geometrical Study of Differential Equations, 2002 95 Benjamin Weiss, Single Orbit Dynamics, 2000 94 David J. Saltman, Lectures on Division Algebras, 1999 93 Goro Shimura, Euler Products and Eisenstein Series, 1997 92 Fan R. K. Chung, Spectral Graph Theory, 1997 91 J.P. May, M. Cole, G. Comezana, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner, Equivariant Homotopy and Cohomology Theory, 1996 90 John Roe, Index Theory, Coarse Geometry, and Topology of Manifolds, 1996 89 Clifford Henry Taubes, Metrics, Connections and Gluing Theorems, 1996 88 Craig L. Huneke, Tight Closure and Its Applications, 1996 87 John Erik Fornæss, Dynamics in Several Complex Variables, 1996 86 Sorin Popa, Classification of Subfactors and Their Endomorphisms, 1995 85 Michio Jimbo and Tetsuji Miwa, Algebraic Analysis of Solvable Lattice Models, 1995 84 Hugh L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, 1994 83 Carlos E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, 1994 82 Susan Montgomery, Hopf Algebras and Their Actions on Rings, 1993 81 Steven G. Krantz, Editor, Geometric Analysis and Function Spaces, 1993 80 Vaughan F. R. Jones, Subfactors and Knots, 1991

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/cbmsseries/. This book brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was available. The relation between the uniform continuity of algebraic maps and topologized tensor products is explained

in detail, however, as this subject does not seem to be Sladká. by Jaroslava Photo taken commonly known and the literature is scarce. The exposition begins by recalling Gerstenhaber’s classical theory for associative algebras. The focus then shifts to a homotopy-invariant setup of Maurer-Cartan moduli spaces. As an application, Kontsevich’s approach to deformation quantization of Poisson manifolds is reviewed. Then, after a brief introduction to operads, a strongly homotopy Lie algebra governing deformations of (diagrams of) algebras of a given type is described, followed by examples and generalizations.

For additional information and updates on this book, visit www.ams.org/bookpages/cbms-116

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