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.
Author(s): Marcelo Aguiar, Swapneel Mahajan
Series: Encyclopedia of Mathematics and its Applications
Publisher: Cambridge University Press
Year: 2020
Language: English
Commentary: Incorrect kerning (e.g. reference 17 in bibliography) and broken intra-links.
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
