×

The permutoassociahedron, Mac Lane’s coherence theorem and asymptotic zones for the KZ equation. (English) Zbl 0812.18003

All possible bracketings of \(n\) symbols in all possible orders are exhibited as vertices of a combinatorial CW-complex \(KP_ n\). It is clearly relevant to the coherence of symmetric monoidal categories, yet also fits nicely into Drinfel’d’s study of the Knizhnik-Zamolodchikov equations and into the analysis of the Grothendieck-Knudsen moduli space of stable \(n\)-pointed curves of genus 0.

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
14D10 Arithmetic ground fields (finite, local, global) and families or fibrations
35Q99 Partial differential equations of mathematical physics and other areas of application
19D23 Symmetric monoidal categories
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baues, H.-J., Geometry of loop spaces and the cobar-construction, Mem. Amer. Math. Soc., Vol. 230 (1980) · Zbl 0473.55009
[2] Boardman, J. M.; Vogt, R. M., Homotopy invariant algebraic structures on topological spaces, (Lecture Notes in Mathematics, Vol. 347 (1973), Springer: Springer Berlin) · Zbl 0285.55012
[3] Brustein, R.; Ne’eman, Y.; Sternberg, S., Duality, crossing and Mac Lane coherence (1989), Tel Aviv University, Preprint
[4] Cohen, D. I.A., Basic Techniques of Combinatorial Theory (1978), Wiley: Wiley New York · Zbl 0178.01604
[5] Deligne, P., Resumé des premièrs exposés de A. Grothendieck, (SGA7, Exposé I, Vol. 288 (1972), Springer: Springer Berlin), 1-24, Lecture Notes in Mathematics · Zbl 0267.14003
[6] Deligne, P., Letters to V. Drinfel’d and G. Segal (1990), (unpublished).
[7] Drinfel’d, V. G., Quantum groups, (Gleason, A. M., Proceedings of ICM-86 (1987), American Mathematical Society: American Mathematical Society Providence, RI), 798-820 · Zbl 0641.16006
[8] Drinfel’d, V. G., Quasi-Hopf algebras, Leningrad Math. J., 1, 1419-1457 (1990) · Zbl 0718.16033
[9] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Discriminants of polynomials in several variables and triangulations of Newton polytopes, Leningrad Math. J., 2, 449-505 (1991) · Zbl 0741.14033
[10] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Newton polytopes of principal \(A\)-determinants, Soviet Math. Dokl., 40, 278-281 (1990) · Zbl 0742.14042
[11] Gelfand, I. M.; Serganova, V. V., Combinatorial geometries and torus strata on homogeneous spaces, Russian Math. Surveys, 42, 107-134 (1987) · Zbl 0629.14035
[12] Johnson, M., The combinatorics of \(n\)-categorical pasting, J. Pure Appl. Algebra, 62, 211-225 (1989) · Zbl 0694.18007
[13] Kapranov, M. M., Veronese curves and Grothendieck-Knudsen moduli space \(N0, n+1\), J. Algebraic Geom., 2 (1993), to appear. · Zbl 0790.14020
[14] M.M. Kapranov, Chow quotients of Grassmannians I, Adv. Soviet Math., to appear.; M.M. Kapranov, Chow quotients of Grassmannians I, Adv. Soviet Math., to appear. · Zbl 0811.14043
[15] M.M. Kapranov and V.A. Voevodsky, 2-categories and Zamolodchikov’s tetrahedra equations, Proc. Sympos. Pure Math., to appear.; M.M. Kapranov and V.A. Voevodsky, 2-categories and Zamolodchikov’s tetrahedra equations, Proc. Sympos. Pure Math., to appear.
[16] Keel, S., Intersection theory of moduli spaces of stable \(n\)-pointed curves, Trans. Amer. Math. Soc., 330, 545-574 (1992) · Zbl 0768.14002
[17] Knudsen, F. F., The projectivity of the moduli space of stable curves II, The stacks \(M0,n'\), Math. Scand., 52, 163-199 (1983) · Zbl 0544.14020
[18] Lascoux, A.; Schützenberger, M.-P., Symmetry and flag manifolds, (Lecture Notes in Mathematics, Vol. 996 (1983), Springer: Springer Berlin), 118-144 · Zbl 0542.14031
[19] Lawrence, R. J., Algebras and triangle equations (1991), Harvard University, Preprint · Zbl 0762.57012
[20] Lee, C., The associahedron and the triangulations of the \(n\)-gon, European J. Combin., 10, 551-560 (1989) · Zbl 0682.52004
[21] Mac Lane, S., (Natural associativity and commutativity, 49 (1963), Rice University Studies), 28-46 · Zbl 0244.18008
[22] Manin, Y. I.; Schechtman, V. V., Configurations of real hyperplanes and Zamolodchikov equations, (Markov, M. A., Proceedings of the Third Seminar on Group-theoretical methods in Physics, Vol. 1 (1986), Nauka: Nauka Moscow), 316-325, (in Russian)
[23] Milgram, R. J., Iterated loop spaces, Ann. of Math., 84, 386-403 (1966) · Zbl 0145.19901
[24] Moore, G.; Seiberg, N., Classical and quantum conformal field theory, Comm. Math. Phys., 123, 177-254 (1989) · Zbl 0694.53074
[25] Mumford, D., Algebraic Geometry I: Complex Projective Varieties, (Grundlehren der Mathematischen Wissenschaften, 221 (1976), Springer: Springer Berlin), Band · Zbl 0821.14001
[26] Oda, T., Convex Bodies and Algebraic Geometry, (Ergebnisse der Mathematik und ihrer Grenzgebiete, 15 (1988), Springer: Springer Berlin), Band · Zbl 0628.52002
[27] Quillen, D., Higher algebraic \(k\)-theory I, (Lecture Notes in Mathematics, Vol. 341 (1973), Springer: Springer Berlin), 85-147 · Zbl 0292.18004
[28] Serre, J. P., Trees (1980), Springer: Springer Berlin · Zbl 0548.20018
[29] Sleator, D. D.; Tarjan, R. E.; Thurston, W. P., Rotation distance, triangulations and hyperbolic geometry, J. Amer. Math. Soc., 1, 647-681 (1988) · Zbl 0653.51017
[30] Stasheff, J. D., Homotopy associativity of H-spaces I, Trans. Amer. Math. Soc., 108, 275-292 (1963) · Zbl 0114.39402
[31] Stasheff, J. D., Quasi-Hopf algebras and beyond, (Talk at the Mid-Atlantic Conference on Algebraic Structures arising from Quantum Groups (1991), Wake Forest University) · Zbl 0784.17027
[32] Zamolodchikov, A. B., Tetrahedra equations and relativistic \(S\)-matrix for straight strings in 2 + 1 dimensions, Comm. Math. Phys., 79, 489-505 (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.