## Koszul duality for operads.(English)Zbl 0855.18006

Operad is a system of data that formalizes properties of a collection of maps $$X^n \to X$$, a certain set for each $$n = 1, 2, \dots$$, which are closed under permutations of arguments of the maps and under all possible superpositions. Operads were introduced by J. P. May in 1972 for the needs of homotopy theory. Since then it has been gradually realized that this concept has in fact fundamental significance for mathematics in general. The present paper, in particular, establishes a deep relationship between operads, moduli spaces of stable curves, graph cohomologies (M. Kontsevich, 1992-93), and Verdier duality on sheaves. The class of quadratic operads and a distinguished subclass of Koszul operads are introduced. A natural duality on quadratic operads, which is analogous to the duality of S. B. Priddy [Trans. Am. Math. Soc. 152, 39-60 (1970; Zbl 0261.18016)] for quadratic associative algebras, is defined.
A brief outline of the paper is as follows. In the first chapter the definition of operad in the terms of the category of trees is recalled and a few examples are given. The significance of the operad $$\mathcal M$$ formed by Grothendieck-Knudsen moduli spaces is explained: any operad can be described as a collection of sheaves on $$\mathcal M$$. The second chapter is devoted to quadratic operads $$\mathcal P$$, the ones generated by binary operations subject to relations involving three arguments only. Most of the structures that one encounters in algebra, e.g., associative, commutative, Lie, Poisson, etc. algebras, correspond to quadratic operads. The quadratic dual operad $${\mathcal P}^!$$ is defined. It is shown that commutative and Lie operads are quadratic dual to each other, and the associative operad is self dual. The duality of Priddy is recovered by the duality of quadratic algebras over quadratic operads. On the category of quadratic operads the internal hom in the spirit of Yu. I. Manin is introduced.
The rôle of the Lie operad as dualizing object is shown. This allows the authors to give a natural interpretation of M. Lazard’s “Lie theory” for formal groups (1955) in terms of Koszul duality. A contravariant duality functor $$D$$ on the category of differential graded operads is introduced in Chapter 3 as opposed to the quadratic duality functor $${\mathcal P} \mapsto {\mathcal P}^!$$. It is shown that from the algebraic point of view, the duality $$D$$ is an analog of the cobar construction and a generalization of the tree part of the graph complex, and that from the geometric point of view, the duality is an analog of the Verdier duality for sheaves. Section 4 is devoted to Koszul operads, the quadratic operads $${\mathcal P}$$ whose quadratic dual is canonically quasi-isomorphic to the $$D$$-dual. Equivalent definitions in terms of the Koszul complex or in terms of vanishing of higher homologies for free $${\mathcal P}$$-algebras is given. It is shown that commutative, associative and Lie operads are Koszul. In a previous paper of the first author the operads formed by Clebsch-Gordan spaces for representations of quantum groups and affine Lie algebras were investigated. The authors plan to study Koszulness of such operads.

### MSC:

 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 17A42 Other $$n$$-ary compositions $$(n \ge 3)$$ 55P47 Infinite loop spaces 16S99 Associative rings and algebras arising under various constructions 18G35 Chain complexes (category-theoretic aspects), dg categories 14H10 Families, moduli of curves (algebraic)
Full Text:

### References:

 [1] J. F. Adams, Infinite loop spaces , Annals of Mathematics Studies, vol. 90, Princeton University Press, Princeton, N.J., 1978. · Zbl 0398.55008 [2] V. I. Arnold, On the representation of continuous functions of three variables by superpositions of continuous functions of two variables , Mat. Sb. (N.S.) 48 (90) (1959), 3-74. · Zbl 0090.27102 [3] V. I. Arnold, Topological invariants of algebraic functions II , Funct. Anal. Appl. 4 (1970), 91-98. · Zbl 0239.14012 [4] H. Bass, E. H. Connell, and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse , Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287-330. · Zbl 0539.13012 [5] A. A. Beilinson, Coherent sheaves on $$P^n$$ and problems in linear algebra , Funct. Anal. Appl. 12 (1978), 214-216. · Zbl 0424.14003 [6] I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Algebraic vector bundles on $$P^n$$ and problems of linear algebra , Funct. Anal. Appl. 12 (1978), 212-214. · Zbl 0424.14002 [7] A. A. Beilinson, V. A. Ginsburg, and V. V. Schechtman, Koszul duality , J. Geom. Phys. 5 (1988), no. 3, 317-350. · Zbl 0695.14009 [8] A. Beilinson, V. Ginzburg, and W. Soergel, Koszul duality patterns in representation theory , to appear in J. Amer. Math. Soc. JSTOR: · Zbl 0864.17006 [9] A. Beilinson and V. Ginzburg, Infinitesimal structure of moduli spaces of $$G$$-bundles , Internat. Math. Res. Notices (1992), no. 4, 63-74. · Zbl 0763.32011 [10] A. Beilinson and V. Ginzburg, Resolution of diagonals homotopy algebra and moduli spaces , · Zbl 0763.32011 [11] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces , Lecture Notes in Mathematics, vol. 347, Springer-Verlag, Berlin, 1973. · Zbl 0285.55012 [12] F. R. Cohen, The homology of $$C_(n+1)$$-spaces , Homology of Iterated Loop Spaces, Lecture Notes in Math., vol. 533, Springer-Verlag, Berlin, 1976, pp. 207-351. · Zbl 0334.55009 [13] P. Deligne, Théorie de Hodge. II , Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5-57. · Zbl 0219.14007 [14] P. Deligne, Resumé des premiérs exposés de A. Grothendieck , Groupes de Monodromie en Géométrie Algébrique, Lecture Notes in Math., vol. 288, Springer-Verlag, Berlin, 1972, pp. 1-24. · Zbl 0267.14003 [15] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields , Ann. of Math. (2) 103 (1976), no. 1, 103-161. JSTOR: · Zbl 0336.20029 [16] P. Deligne and G. Lusztig, Duality for representations of a reductive group over a finite field , J. Algebra 74 (1982), no. 1, 284-291. · Zbl 0482.20027 [17] P. Deligne and G. Lusztig, Duality for representations of a reductive group over a finite field. II , J. Algebra 81 (1983), no. 2, 540-545. · Zbl 0535.20020 [18] V. G. Drinfeld, letter to V. V. Schechtman, unpublished , Septembre, 1988. [19] H. Esnault, V. Schechtman, and E. Viehweg, Cohomology of local systems on the complement of hyperplanes , Invent. Math. 109 (1992), no. 3, 557-561. · Zbl 0788.32005 [20] Z. Fiedorowicz, The symmetric bar-construction , preprint, 1991. [21] W. Fulton and R. MacPherson, A compactification of configuration spaces , Ann. of Math. (2) 139 (1994), no. 1, 183-225. · Zbl 0820.14037 [22] E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories , Comm. Math. Phys. 159 (1994), no. 2, 265-285. · Zbl 0807.17026 [23] E. Getzler and J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces , preprint, 1994. [24] R. Godement, Topologie algébrique et théorie des faisceaux , Hermann, Paris, 1973. · Zbl 0275.55010 [25] I. J. Good, Generalizations to several variables of Lagrange’s expansion, with applications to stochastic processes , Proc. Cambridge Philos. Soc. 56 (1960), 367-380. · Zbl 0135.18802 [26] M. Goresky and R. MacPherson, Stratified Morse theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. [27] V. A. Hinich and V. V. Schectman, On homotopy limit of homotopy algebras , $$K$$-theory, arithmetic and geometry (Moscow, 1984-1986), Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 240-264. · Zbl 0631.55011 [28] V. Hinich and V. Schechtman, Homotopy Lie algebras , I. M. Gelfand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 1-28. · Zbl 0823.18004 [29] S. A. Joni, Lagrange inversion in higher dimensions and umbral operators , Linear and Multilinear Algebra 6 (1978/79), no. 2, 111-122. · Zbl 0395.05005 [30] M. M. Kapranov, On the derived categories and the $$K$$-functor of coherent sheaves on intersections of quadrics , Math. USSR-Izv. 32 (1989), 191-204. · Zbl 0679.14005 [31] M. M. Kapranov, Veronese curves and Grothendieck-Knudsen moduli space $$\overline M_ 0,n$$ , J. Algebraic Geom. 2 (1993), no. 2, 239-262. · Zbl 0790.14020 [32] M. M. Kapranov, Chow quotients of Grassmannians. I , I. M. Gelfand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29-110. · Zbl 0811.14043 [33] M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces , Invent. Math. 92 (1988), no. 3, 479-508. · Zbl 0651.18008 [34] M. Kashiwara and P. Schapira, Sheaves on manifolds , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. · Zbl 0709.18001 [35] A. A. Klyachko, Lie elements in the tensor algebra , Siberian Math. J. 15 (1974), 914-920. · Zbl 0325.15018 [36] F. F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $$M\sbg,n$$ , Math. Scand. 52 (1983), no. 2, 161-199. · Zbl 0544.14020 [37] A. N. Kolmogorov, On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables , Dokl. Akad. Nauk SSSR (N.S.) 108 (1956), 179-182, (in Russian); reprinted in Selected Works of A. N. Kolmogorov, Vol. I, ed. by V. Tikhomirov, Kluwer, Dordrecht, 1991, 378-382. · Zbl 0128.27803 [38] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function , Comm. Math. Phys. 147 (1992), no. 1, 1-23. · Zbl 0756.35081 [39] M. Kontsevich, Formal (non)commutative symplectic geometry , The Gelfand Mathematical Seminars, 1990-1992 eds. L. Crowin, I. Gelfand, and J. Lepowsky, Birkhäuser Boston, Boston, MA, 1993, pp. 173-187. · Zbl 0821.58018 [40] M. Kontsevich, Graphs, homotopy Lie and low-dimensional topology , preprint, 1992. [41] I. Kriz and J. P. May, Oprads, algebras and modules I , preprint, 1993. [42] I. Kriz and J. P. May, Differential graded algebras up to homotopy and their derived categories , preprint, 1992. [43] M. Lazard, Lois de groupes et analyseurs , Ann. Sci. Ecole Norm. Sup. (3) 72 (1955), 299-400. · Zbl 0068.02702 [44] B. Lian and G. Zuckerman, New perspectives on the BRST-Algebraic structures of string theory , preprint, 1992. · Zbl 0780.17029 [45] J.-L. Loday, Cyclic homology , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1992. · Zbl 0780.18009 [46] C. Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra , Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 291-338. · Zbl 0595.16020 [47] G. Lusztig, The discrete Series of $$GL\sbn$$ over a Finite Field , Annals of Mathematics Studies, vol. 81, Princeton University Press, Princeton, N.J., 1974. · Zbl 0293.20038 [48] J. H. McKay, J. Towber, S. S.-S. Wang, and D. Wright, Reversion of a system of power series with application to combinatorics , preprint, 1992. [49] 1 Saunders Mac Lane, Natural associativity and commutativity , Rice Univ. Studies 49 (1963), no. 4, 28-46. · Zbl 0244.18008 [50] 2 Saunders Mac Lane, Selected papers , Springer-Verlag, New York, 1979. · Zbl 0459.01024 [51] I. G. Macdonald, Symmetric functions and Hall polynomials , The Clarendon Press Oxford University Press, New York, 1979. · Zbl 0487.20007 [52] Y. I. Manin, Some remarks on Koszul algebras and quantum groups , Ann. Inst. Fourier (Grenoble) 37 (1987), no. 4, 191-205. · Zbl 0625.58040 [53] J. P. May, The Geometry of Iterated Loop Spaces , Lectures Notes in Mathematics, vol. 271, Springer-Verlag, Berlin, 1972. · Zbl 0244.55009 [54] J. C. Moore, Differential homological algebra , Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, pp. 335-339. · Zbl 0337.18009 [55] R. C. Penner, The decorated Teichmüller space of punctured surfaces , Comm. Math. Phys. 113 (1987), no. 2, 299-339. · Zbl 0642.32012 [56] S. B. Priddy, Koszul resolutions , Trans. Amer. Math. Soc. 152 (1970), 39-60. JSTOR: · Zbl 0261.18016 [57] D. Quillen, Rational homotopy theory , Ann. of Math. (2) 90 (1969), 205-295. JSTOR: · Zbl 0191.53702 [58] D. Quillen, On the (co-) homology of commutative rings , Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 65-87. · Zbl 0234.18010 [59] V. V. Schechtman and A. N. Varchenko, Arrangements of hyperplanes and Lie algebra homology , Invent. Math. 106 (1991), no. 1, 139-194. · Zbl 0754.17024 [60] M. Schlessinger and J. Stasheff, The Lie algebra structure of tangent cohomology and deformation theory , J. Pure Appl. Algebra 38 (1985), no. 2-3, 313-322. · Zbl 0576.17008 [61] J. D. Stasheff, Homotopy associativity of $$H$$-spaces. I, II , Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293-312. · Zbl 0114.39402 [62] J. J. Sylvester, On the change of systems of independent variables , Quart. J. Math. 1 (1857), 42-56. [63] D. Wright, The tree formulas for reversion of power series , J. Pure Appl. Algebra 57 (1989), no. 2, 191-211. · Zbl 0672.13010
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.