Bimonoids for Hyperplane Arrangements [slightly glitchy PDF]

Description

The goal of this monograph is to develop Hopf theory in a new setting which features centrally a real hyperplane arrangement. The new theory is parallel to the classical theory of connected Hopf algebras, and relates to it when specialized to the braid arrangement. Joyal’s theory of combinatorial species, ideas from Tits’ theory of buildings, and Rota’s work on incidence algebras inspire and find common ground in this theory. The authors introduce notions of monoid, comonoid, bimonoid, and Lie monoid relative to a fixed hyperplane arrangement. Faces, flats, and lunes of the arrangement provide the building blocks for these concepts. They also construct universal bimonoids by using generalizations of the classical notions of shuffle and quasishuf- fle, and establish the Borel–Hopf, Poincaré–Birkhoff–Witt, and Cartier–Milnor– Moore theorems in this new setting. A key role is played by noncommutative zeta and Möbius functions which generalize the classical exponential and logarithm, and by the representation theory of the Tits algebra. This monograph opens a vast new area of research. It will be of interest to students and researchers working in the areas of hyperplane arrangements, semigroup theory, Hopf algebras, algebraic Lie theory, operads, and category theory. Preface xi Introduction 1 Part I. Species and operads 17 Chapter 1. Hyperplane arrangements 19 1.1. Faces, bifaces, flats 19 1.2. Nested faces and lunes 25 1.3. Partial-flats 27 1.4. Minimal galleries, distance functions, Varchenko matrices 29 1.5. Incidence algebras, and zeta and M¨ obius functions 34 1.6. Bilune-incidence algebra 40 1.7. Descent, lune, Witt identities 46 1.8. Noncommutative Zaslavsky formula 49 1.9. Birkhoff algebra, Tits algebra, Janus algebra 52 1.10. Takeuchi element 62 1.11. Orientation space and signature space 64 1.12. Lie and Zie elements 66 1.13. Braid arrangement 70 Notes 71 Chapter 2. Species and bimonoids 73 2.1. Species 74 2.2. Monoids, comonoids, bimonoids 77 2.3. (Co)commutative (co)monoids 81 2.4. Deformed bimonoids and signed bimonoids 88 2.5. Signed (co)commutative (co)monoids 91 2.6. Subspecies and quotient species 94 2.7. (Co)abelianizations of (co)monoids 96 2.8. Generating subspecies of monoids 99 2.9. Duality functor on species 100 2.10. Op and cop constructions 102 2.11. Monoids, comonoids, bimonoids as functor categories 105 2.12. Presentation for (co)monoids using covering generators 113 2.13. Partially commutative monoids 118 2.14. Set-species and set-bimonoids 121 2.15. Bimonoids for a rank-one arrangement 123 2.16. Joyal species and Joyal bimonoids 124 Notes 127 Chapter 3. Bimonads on species 134 3.1. Bimonoids as bialgebras over a bimonad 135 3.2. Bicommutative bimonoids as bialgebras over a bimonad 140 3.3. Duality as a bilax functor 145 3.4. Opposite transformation 147 3.5. Lifting of monads to comonoids 150 3.6. Monad for partially commutative monoids 151 3.7. Bimonad for set-species 154 3.8. Symmetries, braidings, lax braidings 158 3.9. LRB species 162 3.10. Mesablishvili–Wisbauer 164 Notes 167 Chapter 4. Operads 168 4.1. Dispecies 169 4.2. Operads 170 4.3. Set-operads 174 4.4. Connected and positive operads 175 4.5. Commutative, associative, Lie operads 176 4.6. May operads 178 4.7. Hadamard product. Hopf operads 179 4.8. Orientation functor and signature functor 181 4.9. Operad presentations 182 4.10. Black and white circle products 187 4.11. Left modules over operads 190 4.12. Bioperads. Mixed distributive laws 196 4.13. Incidence algebra of an operad 199 4.14. Operads for LRB species 202 Notes 203 Part II. Basic theory of bimonoids 205 Chapter 5. Primitive filtrations and decomposable filtrations 207 5.1. Cauchy powers of a species 208 5.2. Graded and filtered bimonoids 211 5.3. Primitive filtrations of comonoids 214 5.4. Decomposable filtrations of monoids 218 5.5. Trivial (co)monoids and (co)derivations 221 5.6. Bimonoid axiom on the primitive part 222 5.7. Primitively generated bimonoids and cocommutativity 226 5.8. Browder–Sweedler and Milnor–Moore 229 Notes 232 Chapter 6. Universal constructions 235 6.1. Free monoids on species 236 6.2. Cofree comonoids on species 240 6.3. (Co)free (co)commutative (co)monoids on species 243 6.4. (Co)free bimonoids associated to species 250 6.5. (Co)free (co)commutative bimonoids associated to species 255 6.6. (Co)abelianizations of (co)free (co)monoids 259 6.7. Primitive filtrations and decomposable filtrations 261 6.8. Alternative descriptions of bimonoids 263 6.9. Norm transformation 265 6.10. (Co)free graded (co)monoids on species 270 6.11. Free partially bicommutative bimonoids 275 Notes 277 Chapter 7. Examples of bimonoids 283 7.1. Species characteristic of chambers 284 7.2. Exponential species 285 7.3. Species of chambers 288 7.4. Species of flats 294 7.5. Species of charts and dicharts 298 7.6. Species of faces 301 7.7. Species of top-nested faces and top-lunes 312 7.8. Species of bifaces 320 7.9. Lie and Zie species 326 Notes 330 Chapter 8. Hadamard product 335 8.1. Hadamard functor 336 8.2. Internal hom for the Hadamard product 342 8.3. Biconvolution bimonoids 343 8.4. Internal hom for comonoids. Bimonoid of star families 347 8.5. Species of chamber maps 355 8.6. Universal measuring comonoids 358 8.7. Enrichment of the category of monoids over comonoids 363 8.8. Internal hom for monoids and bimonoids 367 8.9. Hadamard product of set-species 375 8.10. Signature functor 376 Notes 381 Chapter 9. Exponential and logarithm 384 9.1. Exp-log correspondences 386 9.2. Commutative exp-log correspondence 396 9.3. Deformed exp-log correspondences 408 9.4. 0-exp-log correspondence 419 9.5. Primitive and group-like series of bimonoids 422 9.6. Primitive and group-like series of bicomm. bimonoids 427 9.7. Comparisons between exp-log correspondences 431 9.8. Formal power series. Series of Joyal species 433 Notes 439 Chapter 10. Characteristic operations 442 10.1. Characteristic operations 443 10.2. Commutative characteristic operations 452 10.3. Two-sided characteristic operations 457 10.4. Set-theoretic characteristic operations 459 10.5. Idempotent operators on bimonoid components 460 Notes 467 Chapter 11. Modules over monoid algebras and bimonoids in species 469 11.1. Modules over the Tits algebra 470 11.2. Modules over the Birkhoff algebra 472 11.3. Modules over the Janus algebra 472 11.4. Examples 474 11.5. Duality and base change 476 11.6. Signed analogues 479 11.7. A unified viewpoint via partial-biflats 480 11.8. Karoubi envelopes 481 11.9. Monoid-sets and bimonoids in set-species 485 11.10. Bimonoids of h-faces and h-flats 486 Notes 493 Chapter 12. Antipode 495 12.1. Takeuchi formula 495 12.2. Interaction with op and cop constructions 499 12.3. Commutative Takeuchi formula 503 12.4. Logarithm of the antipode 505 12.5. Examples 507 12.6. Antipodes of (co)free bimonoids 512 12.7. Antipodes of (co)free (co)commutative bimonoids 515 12.8. Takeuchi element and characteristic operations 518 12.9. Set-bimonoids 523 Notes 525 Part III. Structure theory for bimonoids 529 Chapter 13. Loday–Ronco, Leray–Samelson, Borel–Hopf 531 13.1. Loday–Ronco for 0-bimonoids 534 13.2. Leray–Samelson for bicommutative bimonoids 538 13.3. Borel–Hopf for cocommutative bimonoids 546 13.4. Borel–Hopf for commutative bimonoids 556 13.5. Unification using partially bicommutative bimonoids 560 13.6. Rigidity of q-bimonoids for q not a root of unity 562 13.7. Monad for Lie monoids 570 Notes 571 Chapter 14. Hoffman–Newman–Radford 575 14.1. Free 0-bimonoids on comonoids 576 14.2. Free bicomm. bimonoids on cocomm. comonoids 580 14.3. Free 0-∼-bicommutative bimonoids 587 14.4. Free bimonoids on cocommutative comonoids 588 14.5. Free q-bimonoids on comonoids 597 14.6. Zeta and Möbius as inverses 605 Notes 607 Chapter 15. Freeness under Hadamard products 609 15.1. Freeness under Hadamard products 609 15.2. Product of free and cofree bimonoids 613 15.3. Product of free comm. and cofree cocomm. bimonoids 623 15.4. Product of bimonoids with one free factor 627 15.5. Species of pairs of chambers 629 Notes 636 Chapter 16. Lie monoids 638 16.1. Lie monoids 639 16.2. Commutator bracket and primitive part of bimonoids 642 16.3. Free Lie monoids on species 645 16.4. Lie species and Zie species as Lie monoids 648 16.5. Universal enveloping monoids 649 16.6. Abelian Lie monoids 659 16.7. Signed Lie monoids 660 16.8. Lie comonoids 664 Notes 671 Chapter 17. Poincar´ e–Birkhoff–Witt and Cartier–Milnor–Moore 678 17.1. Comonoid sections to the abelianization map 679 17.2. Poincar´ e–Birkhoff–Witt 682 17.3. Projecting the free monoid onto the free Lie monoid 687 17.4. Solomon operator 692 17.5. Cartier–Milnor–Moore 698 17.6. Lie monoids for a rank-one arrangement 703 17.7. Joyal Lie monoids 706 17.8. Lie monoids in LRB species 710 Notes 711 Appendices 717 Appendix A. Vector spaces 719 A.1. Kernel, cokernel, image, coimage 719 A.2. Duality functor on vector spaces 719 A.3. Internal hom for the tensor product 720 A.4. Linear maps between direct sums of vector spaces 720 A.5. Idempotent operators 721 Appendix B. Internal hom for monoidal categories 723 B.1. Monoidal and 2-monoidal categories 723 B.2. Internal hom 725 B.3. Powers and copowers 728 B.4. Internal hom for functor categories 730 B.5. Modules over a monoid algebra 733 Notes 735 Appendix C. Higher monads 738 C.1. Higher monads 738 C.2. Higher monad algebras 752 C.3. Adjunctions 758 Notes 760 References 763 List of Notations 802 List of Tables 812 Author Index 814 Subject Index 823